Fibonacci sequence

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href="#Relation_to_the_golden_ratio"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Relation to the golden ratio</span> </div> </a> <button aria-controls="toc-Relation_to_the_golden_ratio-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Relation to the golden ratio subsection</span> </button> <ul id="toc-Relation_to_the_golden_ratio-sublist" class="vector-toc-list"> <li id="toc-Closed-form_expression" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Closed-form_expression"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Closed-form expression</span> </div> </a> <ul id="toc-Closed-form_expression-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computation_by_rounding" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computation_by_rounding"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Computation by rounding</span> </div> </a> <ul id="toc-Computation_by_rounding-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Magnitude" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Magnitude"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Magnitude</span> </div> </a> <ul id="toc-Magnitude-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Limit_of_consecutive_quotients" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Limit_of_consecutive_quotients"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Limit of consecutive quotients</span> </div> </a> <ul id="toc-Limit_of_consecutive_quotients-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Decomposition_of_powers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Decomposition_of_powers"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Decomposition of powers</span> </div> </a> <ul id="toc-Decomposition_of_powers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Identification" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Identification"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Identification</span> </div> </a> <ul id="toc-Identification-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Matrix_form" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Matrix_form"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Matrix form</span> </div> </a> <ul id="toc-Matrix_form-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Combinatorial_identities" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Combinatorial_identities"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Combinatorial identities</span> </div> </a> <button aria-controls="toc-Combinatorial_identities-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Combinatorial identities subsection</span> </button> <ul id="toc-Combinatorial_identities-sublist" class="vector-toc-list"> <li id="toc-Combinatorial_proofs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Combinatorial_proofs"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Combinatorial proofs</span> </div> </a> <ul id="toc-Combinatorial_proofs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Symbolic_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Symbolic_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Symbolic method</span> </div> </a> <ul id="toc-Symbolic_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Induction_proofs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Induction_proofs"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Induction proofs</span> </div> </a> <ul id="toc-Induction_proofs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Binet_formula_proofs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Binet_formula_proofs"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Binet formula proofs</span> </div> </a> <ul id="toc-Binet_formula_proofs-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_identities" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Other_identities"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Other identities</span> </div> </a> <button aria-controls="toc-Other_identities-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Other identities subsection</span> </button> <ul id="toc-Other_identities-sublist" class="vector-toc-list"> <li id="toc-Cassini's_and_Catalan's_identities" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cassini's_and_Catalan's_identities"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Cassini's and Catalan's identities</span> </div> </a> <ul id="toc-Cassini's_and_Catalan's_identities-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-d'Ocagne's_identity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#d'Ocagne's_identity"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>d'Ocagne's identity</span> </div> </a> <ul id="toc-d'Ocagne's_identity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generating_function" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generating_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Generating function</span> </div> </a> <ul id="toc-Generating_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reciprocal_sums" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Reciprocal_sums"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Reciprocal sums</span> </div> </a> <ul id="toc-Reciprocal_sums-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Primes_and_divisibility" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Primes_and_divisibility"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Primes and divisibility</span> </div> </a> <button aria-controls="toc-Primes_and_divisibility-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Primes and divisibility subsection</span> </button> <ul id="toc-Primes_and_divisibility-sublist" class="vector-toc-list"> <li id="toc-Divisibility_properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Divisibility_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Divisibility properties</span> </div> </a> <ul id="toc-Divisibility_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Primality_testing" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Primality_testing"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2</span> <span>Primality testing</span> </div> </a> <ul id="toc-Primality_testing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fibonacci_primes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fibonacci_primes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.3</span> <span>Fibonacci primes</span> </div> </a> <ul id="toc-Fibonacci_primes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Prime_divisors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Prime_divisors"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.4</span> <span>Prime divisors</span> </div> </a> <ul id="toc-Prime_divisors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Periodicity_modulo_n" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Periodicity_modulo_n"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.5</span> <span>Periodicity modulo <i>n</i></span> </div> </a> <ul id="toc-Periodicity_modulo_n-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Generalizations</span> </div> </a> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Applications</span> </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Applications subsection</span> </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Mathematics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mathematics"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.1</span> <span>Mathematics</span> </div> </a> <ul id="toc-Mathematics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computer_science" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computer_science"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.2</span> <span>Computer science</span> </div> </a> <ul id="toc-Computer_science-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nature" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nature"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.3</span> <span>Nature</span> </div> </a> <ul id="toc-Nature-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.4</span> <span>Other</span> </div> </a> <ul id="toc-Other-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>References</span> </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Explanatory_footnotes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Explanatory_footnotes"> <div class="vector-toc-text"> <span class="vector-toc-numb">13.1</span> <span>Explanatory footnotes</span> </div> </a> <ul id="toc-Explanatory_footnotes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">13.2</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Works_cited" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Works_cited"> <div class="vector-toc-text"> <span class="vector-toc-numb">13.3</span> <span>Works cited</span> </div> </a> <ul id="toc-Works_cited-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Fibonacci sequence</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 84 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-84" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">84 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Fibonacci-reeks" title="Fibonacci-reeks – Afrikaans" lang="af" hreflang="af" data-title="Fibonacci-reeks" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AA%D8%AA%D8%A7%D9%84%D9%8A%D8%A9_%D9%81%D9%8A%D8%A8%D9%88%D9%86%D8%A7%D8%AA%D8%B4%D9%8A" title="متتالية فيبوناتشي – Arabic" lang="ar" hreflang="ar" data-title="متتالية فيبوناتشي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-hyw mw-list-item"><a href="https://hyw.wikipedia.org/wiki/%D5%96%D5%AB%D5%BA%D5%B8%D5%B6%D5%A1%D5%B9%D5%B9%D5%AB%D5%AB_%D5%A9%D5%AB%D6%82" title="Ֆիպոնաչչիի թիւ – Western Armenian" lang="hyw" hreflang="hyw" data-title="Ֆիպոնաչչիի թիւ" data-language-autonym="Արեւմտահայերէն" data-language-local-name="Western Armenian" class="interlanguage-link-target"><span>Արեւմտահայերէն</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Fibona%C3%A7%C3%A7i_%C9%99d%C9%99dl%C9%99ri" title="Fibonaççi ədədləri – Azerbaijani" lang="az" hreflang="az" data-title="Fibonaççi ədədləri" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AB%E0%A6%BF%E0%A6%AC%E0%A7%8B%E0%A6%A8%E0%A6%BE%E0%A6%9A%E0%A7%8D%E0%A6%9A%E0%A6%BF_%E0%A6%B0%E0%A6%BE%E0%A6%B6%E0%A6%BF%E0%A6%AE%E0%A6%BE%E0%A6%B2%E0%A6%BE" title="ফিবোনাচ্চি রাশিমালা – Bangla" lang="bn" hreflang="bn" data-title="ফিবোনাচ্চি রাশিমালা" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A4%D0%B8%D0%B1%D0%BE%D0%BD%D0%B0%D1%87%D1%87%D0%B8_%D2%BB%D0%B0%D0%BD%D0%B4%D0%B0%D1%80%D1%8B" title="Фибоначчи һандары – Bashkir" lang="ba" hreflang="ba" data-title="Фибоначчи һандары" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9B%D1%96%D0%BA%D1%96_%D0%A4%D1%96%D0%B1%D0%B0%D0%BD%D0%B0%D1%87%D1%8B" title="Лікі Фібаначы – Belarusian" lang="be" hreflang="be" data-title="Лікі Фібаначы" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%B0_%D0%BD%D0%B0_%D0%A4%D0%B8%D0%B1%D0%BE%D0%BD%D0%B0%D1%87%D0%B8" title="Числа на Фибоначи – Bulgarian" lang="bg" hreflang="bg" data-title="Числа на Фибоначи" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Fibonaccijev_broj" title="Fibonaccijev broj – Bosnian" lang="bs" hreflang="bs" data-title="Fibonaccijev broj" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Successi%C3%B3_de_Fibonacci" title="Successió de Fibonacci – Catalan" lang="ca" hreflang="ca" data-title="Successió de Fibonacci" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A4%D0%B8%D0%B1%D0%BE%D0%BD%D0%B0%D1%87%D1%87%D0%B8_%D1%85%D0%B8%D1%81%D0%B5%D0%BF%C4%95" title="Фибоначчи хисепĕ – Chuvash" lang="cv" hreflang="cv" data-title="Фибоначчи хисепĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Fibonacciho_posloupnost" title="Fibonacciho posloupnost – Czech" lang="cs" hreflang="cs" data-title="Fibonacciho posloupnost" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Rhif_Fibonacci" title="Rhif Fibonacci – Welsh" lang="cy" hreflang="cy" data-title="Rhif Fibonacci" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Fibonacci-tal" title="Fibonacci-tal – Danish" lang="da" hreflang="da" data-title="Fibonacci-tal" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Fibonacci-Folge" title="Fibonacci-Folge – German" lang="de" hreflang="de" data-title="Fibonacci-Folge" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Fibonacci_jada" title="Fibonacci jada – Estonian" lang="et" hreflang="et" data-title="Fibonacci jada" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%BA%CE%BF%CE%BB%CE%BF%CF%85%CE%B8%CE%AF%CE%B1_%CE%A6%CE%B9%CE%BC%CF%80%CE%BF%CE%BD%CE%AC%CF%84%CF%83%CE%B9" title="Ακολουθία Φιμπονάτσι – Greek" lang="el" hreflang="el" data-title="Ακολουθία Φιμπονάτσι" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Sucesi%C3%B3n_de_Fibonacci" title="Sucesión de Fibonacci – Spanish" lang="es" hreflang="es" data-title="Sucesión de Fibonacci" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Fibona%C4%89i-nombro" title="Fibonaĉi-nombro – Esperanto" lang="eo" hreflang="eo" data-title="Fibonaĉi-nombro" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Fibonacciren_segida" title="Fibonacciren segida – Basque" lang="eu" hreflang="eu" data-title="Fibonacciren segida" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%B9%D8%AF%D8%A7%D8%AF_%D9%81%DB%8C%D8%A8%D9%88%D9%86%D8%A7%DA%86%DB%8C" title="اعداد فیبوناچی – Persian" lang="fa" hreflang="fa" data-title="اعداد فیبوناچی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Suite_de_Fibonacci" title="Suite de Fibonacci – French" lang="fr" hreflang="fr" data-title="Suite de Fibonacci" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Seicheamh_Fibonacci" title="Seicheamh Fibonacci – Irish" lang="ga" hreflang="ga" data-title="Seicheamh Fibonacci" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/Straih_Fibonacci" title="Straih Fibonacci – Manx" lang="gv" hreflang="gv" data-title="Straih Fibonacci" data-language-autonym="Gaelg" data-language-local-name="Manx" class="interlanguage-link-target"><span>Gaelg</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Sucesi%C3%B3n_de_Fibonacci" title="Sucesión de Fibonacci – Galician" lang="gl" hreflang="gl" data-title="Sucesión de Fibonacci" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%AB%E0%AA%BF%E0%AA%AC%E0%AB%8B%E0%AA%A8%E0%AA%BE%E0%AA%95%E0%AA%BF" title="ફિબોનાકિ – Gujarati" lang="gu" hreflang="gu" data-title="ફિબોનાકિ" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%94%BC%EB%B3%B4%EB%82%98%EC%B9%98_%EC%88%98" title="피보나치 수 – Korean" lang="ko" hreflang="ko" data-title="피보나치 수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%96%D5%AB%D5%A2%D5%B8%D5%B6%D5%A1%D5%B9%D5%AB%D5%AB_%D5%B0%D5%A1%D5%BB%D5%B8%D6%80%D5%A4%D5%A1%D5%AF%D5%A1%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Ֆիբոնաչիի հաջորդականություն – Armenian" lang="hy" hreflang="hy" data-title="Ֆիբոնաչիի հաջորդականություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AB%E0%A4%BF%E0%A4%AC%E0%A5%8B%E0%A4%A8%E0%A4%BE%E0%A4%9A%E0%A5%80_%E0%A4%85%E0%A4%A8%E0%A5%81%E0%A4%95%E0%A5%8D%E0%A4%B0%E0%A4%AE" title="फिबोनाची अनुक्रम – Hindi" lang="hi" hreflang="hi" data-title="फिबोनाची अनुक्रम" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Fibonaccijev_broj" title="Fibonaccijev broj – Croatian" lang="hr" hreflang="hr" data-title="Fibonaccijev broj" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Barisan_Fibonacci" title="Barisan Fibonacci – Indonesian" lang="id" hreflang="id" data-title="Barisan Fibonacci" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Fibonacci-runan" title="Fibonacci-runan – Icelandic" lang="is" hreflang="is" data-title="Fibonacci-runan" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Successione_di_Fibonacci" title="Successione di Fibonacci – Italian" lang="it" hreflang="it" data-title="Successione di Fibonacci" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A1%D7%93%D7%A8%D7%AA_%D7%A4%D7%99%D7%91%D7%95%D7%A0%D7%90%D7%A6%27%D7%99" title="סדרת פיבונאצ'י – Hebrew" lang="he" hreflang="he" data-title="סדרת פיבונאצ'י" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AB%E0%B2%BF%E0%B2%AC%E0%B3%8A%E0%B2%A8%E0%B2%BE%E0%B2%B6%E0%B2%BF_%E0%B2%B8%E0%B2%82%E0%B2%96%E0%B3%8D%E0%B2%AF%E0%B3%86%E0%B2%97%E0%B2%B3%E0%B3%81" title="ಫಿಬೊನಾಶಿ ಸಂಖ್ಯೆಗಳು – Kannada" lang="kn" hreflang="kn" data-title="ಫಿಬೊನಾಶಿ ಸಂಖ್ಯೆಗಳು" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A4%D0%B8%D0%B1%D0%BE%D0%BD%D0%B0%D1%87%D1%87%D0%B8_%D1%81%D0%B0%D0%BD%D0%B4%D0%B0%D1%80%D1%8B" title="Фибоначчи сандары – Kazakh" lang="kk" hreflang="kk" data-title="Фибоначчи сандары" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Numeri_Fibonacciani" title="Numeri Fibonacciani – Latin" lang="la" hreflang="la" data-title="Numeri Fibonacciani" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Fibona%C4%8Di_skait%C4%BCi" title="Fibonači skaitļi – Latvian" lang="lv" hreflang="lv" data-title="Fibonači skaitļi" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Fibona%C4%8Dio_skai%C4%8Di%C5%B3_seka" title="Fibonačio skaičių seka – Lithuanian" lang="lt" hreflang="lt" data-title="Fibonačio skaičių seka" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Fibonacci-sz%C3%A1mok" title="Fibonacci-számok – Hungarian" lang="hu" hreflang="hu" data-title="Fibonacci-számok" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A4%D0%B8%D0%B1%D0%BE%D0%BD%D0%B0%D1%87%D0%B8%D0%B5%D0%B2%D0%B0_%D0%BD%D0%B8%D0%B7%D0%B0" title="Фибоначиева низа – Macedonian" lang="mk" hreflang="mk" data-title="Фибоначиева низа" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AB%E0%B4%BF%E0%B4%AC%E0%B4%A8%E0%B4%BE%E0%B4%9A%E0%B5%8D%E0%B4%9A%E0%B4%BF_%E0%B4%B6%E0%B5%8D%E0%B4%B0%E0%B5%87%E0%B4%A3%E0%B4%BF" title="ഫിബനാച്ചി ശ്രേണി – Malayalam" lang="ml" hreflang="ml" data-title="ഫിബനാച്ചി ശ്രേണി" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%AB%E0%A4%BF%E0%A4%AC%E0%A5%8B%E0%A4%A8%E0%A4%BE%E0%A4%9A%E0%A5%80_%E0%A4%B6%E0%A5%8D%E0%A4%B0%E0%A5%87%E0%A4%A3%E0%A5%80" title="फिबोनाची श्रेणी – Marathi" lang="mr" hreflang="mr" data-title="फिबोनाची श्रेणी" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Jujukan_Fibonacci" title="Jujukan Fibonacci – Malay" lang="ms" hreflang="ms" data-title="Jujukan Fibonacci" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A4%D0%B8%D0%B1%D0%BE%D0%BD%D0%B0%D1%87%D1%87%D0%B8%D0%B9%D0%BD_%D1%82%D0%BE%D0%BE" title="Фибоначчийн тоо – Mongolian" lang="mn" hreflang="mn" data-title="Фибоначчийн тоо" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Rij_van_Fibonacci" title="Rij van Fibonacci – Dutch" lang="nl" hreflang="nl" data-title="Rij van Fibonacci" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%95%E3%82%A3%E3%83%9C%E3%83%8A%E3%83%83%E3%83%81%E6%95%B0" title="フィボナッチ数 – Japanese" lang="ja" hreflang="ja" data-title="フィボナッチ数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Fibonaccitaalen" title="Fibonaccitaalen – Northern Frisian" lang="frr" hreflang="frr" data-title="Fibonaccitaalen" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no badge-Q70894304 mw-list-item" title=""><a href="https://no.wikipedia.org/wiki/Fibonaccirekken" title="Fibonaccirekken – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Fibonaccirekken" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Fibonaccif%C3%B8lgja" title="Fibonaccifølgja – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Fibonaccifølgja" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Fibonacci_sonlari" title="Fibonacci sonlari – Uzbek" lang="uz" hreflang="uz" data-title="Fibonacci sonlari" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AB%E0%A8%BC%E0%A9%80%E0%A8%AC%E0%A9%8B%E0%A8%A8%E0%A8%BE%E0%A8%9A%E0%A9%80_%E0%A8%A4%E0%A8%B0%E0%A8%A4%E0%A9%80%E0%A8%AC" title="ਫ਼ੀਬੋਨਾਚੀ ਤਰਤੀਬ – Punjabi" lang="pa" hreflang="pa" data-title="ਫ਼ੀਬੋਨਾਚੀ ਤਰਤੀਬ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D9%81%DB%8C%D8%A8%D9%88%D9%86%D8%A7%DA%86%DB%90_%D8%A7%D8%B9%D8%AF%D8%A7%D8%AF" title="فیبوناچې اعداد – Pashto" lang="ps" hreflang="ps" data-title="فیبوناچې اعداد" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Sequensa_%C3%ABd_Fibonacci" title="Sequensa ëd Fibonacci – Piedmontese" lang="pms" hreflang="pms" data-title="Sequensa ëd Fibonacci" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Ci%C4%85g_Fibonacciego" title="Ciąg Fibonacciego – Polish" lang="pl" hreflang="pl" data-title="Ciąg Fibonacciego" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Sequ%C3%AAncia_de_Fibonacci" title="Sequência de Fibonacci – Portuguese" lang="pt" hreflang="pt" data-title="Sequência de Fibonacci" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/Fibonachchi_sanlar%C4%B1" title="Fibonachchi sanları – Kara-Kalpak" lang="kaa" hreflang="kaa" data-title="Fibonachchi sanları" data-language-autonym="Qaraqalpaqsha" data-language-local-name="Kara-Kalpak" class="interlanguage-link-target"><span>Qaraqalpaqsha</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Num%C4%83r_Fibonacci" title="Număr Fibonacci – Romanian" lang="ro" hreflang="ro" data-title="Număr Fibonacci" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru badge-Q70894304 mw-list-item" title=""><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D1%81%D0%BB%D0%B5%D0%B4%D0%BE%D0%B2%D0%B0%D1%82%D0%B5%D0%BB%D1%8C%D0%BD%D0%BE%D1%81%D1%82%D1%8C_%D0%A4%D0%B8%D0%B1%D0%BE%D0%BD%D0%B0%D1%87%D1%87%D0%B8" title="Последовательность Фибоначчи – Russian" lang="ru" hreflang="ru" data-title="Последовательность Фибоначчи" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Numrat_e_Fibonaccit" title="Numrat e Fibonaccit – Albanian" lang="sq" hreflang="sq" data-title="Numrat e Fibonaccit" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Succissioni_di_Fibonacci" title="Succissioni di Fibonacci – Sicilian" lang="scn" hreflang="scn" data-title="Succissioni di Fibonacci" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B7%86%E0%B7%92%E0%B6%B6%E0%B7%9C%E0%B6%B1%E0%B7%8F%E0%B6%A0%E0%B7%8A%E0%B6%A0%E0%B7%92_%E0%B7%83%E0%B6%82%E0%B6%9B%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F" title="ෆිබොනාච්චි සංඛ්‍යා – Sinhala" lang="si" hreflang="si" data-title="ෆිබොනාච්චි සංඛ්‍යා" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Fibonacci_number" title="Fibonacci number – Simple English" lang="en-simple" hreflang="en-simple" data-title="Fibonacci number" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Fibonacciho_postupnos%C5%A5" title="Fibonacciho postupnosť – Slovak" lang="sk" hreflang="sk" data-title="Fibonacciho postupnosť" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Fibonaccijevo_%C5%A1tevilo" title="Fibonaccijevo število – Slovenian" lang="sl" hreflang="sl" data-title="Fibonaccijevo število" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-cu mw-list-item"><a href="https://cu.wikipedia.org/wiki/%D0%A4%D1%97%D0%B2%D0%BE%D0%BD%D0%B0%D0%BA%D1%97%D0%B8%D0%BD%D0%BE%D0%B2%D0%B8_%D1%87%D0%B8%D1%81%D0%BC%D1%94%D0%BD%D0%B0" title="Фївонакїинови чисмєна – Church Slavic" lang="cu" hreflang="cu" data-title="Фївонакїинови чисмєна" data-language-autonym="Словѣньскъ / ⰔⰎⰑⰂⰡⰐⰠⰔⰍⰟ" data-language-local-name="Church Slavic" class="interlanguage-link-target"><span>Словѣньскъ / ⰔⰎⰑⰂⰡⰐⰠⰔⰍⰟ</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%98%D9%85%D8%A7%D8%B1%DB%95%DB%8C_%D9%81%DB%8C%D8%A8%DB%86%D9%86%D8%A7%DA%86%DB%8C" title="ژمارەی فیبۆناچی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ژمارەی فیبۆناچی" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A4%D0%B8%D0%B1%D0%BE%D0%BD%D0%B0%D1%87%D0%B8%D1%98%D0%B5%D0%B2_%D0%BD%D0%B8%D0%B7" title="Фибоначијев низ – Serbian" lang="sr" hreflang="sr" data-title="Фибоначијев низ" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Fibonaccijev_broj" title="Fibonaccijev broj – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Fibonaccijev broj" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Fibonaccin_lukujono" title="Fibonaccin lukujono – Finnish" lang="fi" hreflang="fi" data-title="Fibonaccin lukujono" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Fibonaccital" title="Fibonaccital – Swedish" lang="sv" hreflang="sv" data-title="Fibonaccital" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Bilang_na_Fibonacci" title="Bilang na Fibonacci – Tagalog" lang="tl" hreflang="tl" data-title="Bilang na Fibonacci" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AE%BF%E0%AE%AA%E0%AE%A9%E0%AE%BE%E0%AE%9A%E0%AF%8D%E0%AE%9A%E0%AE%BF_%E0%AE%8E%E0%AE%A3%E0%AF%8D%E0%AE%95%E0%AE%B3%E0%AF%8D" title="பிபனாச்சி எண்கள் – Tamil" lang="ta" hreflang="ta" data-title="பிபனாச்சி எண்கள்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%AB%E0%B0%BF%E0%B0%AC%E0%B1%8B%E0%B0%A8%E0%B0%BE%E0%B0%9A%E0%B1%80_%E0%B0%B8%E0%B0%82%E0%B0%96%E0%B1%8D%E0%B0%AF%E0%B0%B2%E0%B1%81" title="ఫిబోనాచీ సంఖ్యలు – Telugu" lang="te" hreflang="te" data-title="ఫిబోనాచీ సంఖ్యలు" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5" title="จำนวนฟีโบนัชชี – Thai" lang="th" hreflang="th" data-title="จำนวนฟีโบนัชชี" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Fibonacci_dizisi" title="Fibonacci dizisi – Turkish" lang="tr" hreflang="tr" data-title="Fibonacci dizisi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%BE%D1%81%D0%BB%D1%96%D0%B4%D0%BE%D0%B2%D0%BD%D1%96%D1%81%D1%82%D1%8C_%D0%A4%D1%96%D0%B1%D0%BE%D0%BD%D0%B0%D1%87%D1%87%D1%96" title="Послідовність Фібоначчі – Ukrainian" lang="uk" hreflang="uk" data-title="Послідовність Фібоначчі" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/D%C3%A3y_Fibonacci" title="Dãy Fibonacci – Vietnamese" lang="vi" hreflang="vi" data-title="Dãy Fibonacci" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Fibonacci_arv" title="Fibonacci arv – Võro" lang="vro" hreflang="vro" data-title="Fibonacci arv" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Reke_van_Fibonacci" title="Reke van Fibonacci – West Flemish" lang="vls" hreflang="vls" data-title="Reke van Fibonacci" data-language-autonym="West-Vlams" data-language-local-name="West Flemish" class="interlanguage-link-target"><span>West-Vlams</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Ihap_Fibonacci" title="Ihap Fibonacci – Waray" lang="war" hreflang="war" data-title="Ihap Fibonacci" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%96%90%E6%B3%A2%E9%82%A3%E5%A5%91%E6%95%B0%E5%88%97" title="斐波那契数列 – Wu" lang="wuu" hreflang="wuu" data-title="斐波那契数列" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%B2%BB%E6%B0%8F%E6%95%B8%E5%88%97" title="費氏數列 – Cantonese" lang="yue" hreflang="yue" data-title="費氏數列" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link 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.hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For the chamber ensemble, see <a href="/wiki/Fibonacci_Sequence_(ensemble)" title="Fibonacci Sequence (ensemble)">Fibonacci Sequence (ensemble)</a>.</div> <p>In mathematics, the <b>Fibonacci sequence</b> is a <a href="/wiki/Integer_sequence" title="Integer sequence">sequence</a> in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as <b>Fibonacci numbers</b>, commonly denoted <span class="nowrap"><span class="texhtml"><i>F<sub>n</sub></i></span><span class="nowrap"> </span></span>. Many writers begin the sequence with 0 and 1, although some authors start it from 1 and 1<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> and some (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the sequence begins </p> <dl><dd>0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... (sequence <span class="nowrap external"><a href="//oeis.org/A000045" class="extiw" title="oeis:A000045">A000045</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Fibonacci_Squares.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Fibonacci_Squares.svg/220px-Fibonacci_Squares.svg.png" decoding="async" width="220" height="136" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Fibonacci_Squares.svg/330px-Fibonacci_Squares.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/15/Fibonacci_Squares.svg/440px-Fibonacci_Squares.svg.png 2x" data-file-width="512" data-file-height="317" /></a><figcaption>A tiling with <a href="/wiki/Square" title="Square">squares</a> whose side lengths are successive Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13 and 21</figcaption></figure> <p>The Fibonacci numbers were first described in <a href="/wiki/Indian_mathematics" title="Indian mathematics">Indian mathematics</a> as early as 200 BC in work by <a href="/wiki/Pingala" title="Pingala">Pingala</a> on enumerating possible patterns of <a href="/wiki/Sanskrit" title="Sanskrit">Sanskrit</a> poetry formed from syllables of two lengths.<sup id="cite_ref-GlobalScience_3-0" class="reference"><a href="#cite_note-GlobalScience-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-HistoriaMathematica_4-0" class="reference"><a href="#cite_note-HistoriaMathematica-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Donald_Knuth_2006_50_5-0" class="reference"><a href="#cite_note-Donald_Knuth_2006_50-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> They are named after the Italian mathematician Leonardo of Pisa, also known as <a href="/wiki/Fibonacci" title="Fibonacci">Fibonacci</a>, who introduced the sequence to Western European mathematics in his 1202 book <span title="Latin-language text"><i lang="la"><a href="/wiki/Liber_Abaci" title="Liber Abaci">Liber Abaci</a></i></span>.<sup id="cite_ref-FOOTNOTESigler2002404–05_6-0" class="reference"><a href="#cite_note-FOOTNOTESigler2002404–05-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the <i><a href="/wiki/Fibonacci_Quarterly" title="Fibonacci Quarterly">Fibonacci Quarterly</a></i>. Applications of Fibonacci numbers include computer algorithms such as the <a href="/wiki/Fibonacci_search_technique" title="Fibonacci search technique">Fibonacci search technique</a> and the <a href="/wiki/Fibonacci_heap" title="Fibonacci heap">Fibonacci heap</a> <a href="/wiki/Data_structure" title="Data structure">data structure</a>, and <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graphs</a> called <a href="/wiki/Fibonacci_cube" title="Fibonacci cube">Fibonacci cubes</a> used for interconnecting parallel and distributed systems. They also appear <a href="/wiki/Patterns_in_nature#Spirals" title="Patterns in nature">in biological settings</a>, such as branching in trees, <a href="/wiki/Phyllotaxis" title="Phyllotaxis">the arrangement of leaves on a stem</a>, the fruit sprouts of a <a href="/wiki/Pineapple" title="Pineapple">pineapple</a>, the flowering of an <a href="/wiki/Artichoke" title="Artichoke">artichoke</a>, and the arrangement of a <a href="/wiki/Pine_cone" class="mw-redirect" title="Pine cone">pine cone</a>'s bracts, though they do not occur in all species. </p><p>Fibonacci numbers are also strongly related to the <a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</a>: <a href="#Binet's_formula">Binet's formula</a> expresses the <span class="texhtml mvar" style="font-style:italic;">n</span>-th Fibonacci number in terms of <span class="texhtml mvar" style="font-style:italic;">n</span> and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as <span class="texhtml mvar" style="font-style:italic;">n</span> increases. Fibonacci numbers are also closely related to <a href="/wiki/Lucas_number" title="Lucas number">Lucas numbers</a>, which obey the same <a href="/wiki/Recurrence_relation" title="Recurrence relation">recurrence relation</a> and with the Fibonacci numbers form a complementary pair of <a href="/wiki/Lucas_sequence" title="Lucas sequence">Lucas sequences</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Fibonacci_Spiral.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Fibonacci_Spiral.svg/220px-Fibonacci_Spiral.svg.png" decoding="async" width="220" height="136" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Fibonacci_Spiral.svg/330px-Fibonacci_Spiral.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Fibonacci_Spiral.svg/440px-Fibonacci_Spiral.svg.png 2x" data-file-width="512" data-file-height="317" /></a><figcaption>The Fibonacci spiral: an approximation of the <a href="/wiki/Golden_spiral" title="Golden spiral">golden spiral</a> created by drawing <a href="/wiki/Circular_arc" title="Circular arc">circular arcs</a> connecting the opposite corners of squares in the Fibonacci tiling (see preceding image)</figcaption></figure> <p>The Fibonacci numbers may be defined by the <a href="/wiki/Recurrence_relation" title="Recurrence relation">recurrence relation</a><sup id="cite_ref-FOOTNOTELucas18913_7-0" class="reference"><a href="#cite_note-FOOTNOTELucas18913-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{0}=0,\quad F_{1}=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{0}=0,\quad F_{1}=1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c667d91153450b3a161371582ee8227af85951f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.623ex; height:2.509ex;" alt="{\displaystyle F_{0}=0,\quad F_{1}=1,}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=F_{n-1}+F_{n-2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=F_{n-1}+F_{n-2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fa6d281e7a54e08aeffeef7458ddc0884333686" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.279ex; height:2.509ex;" alt="{\displaystyle F_{n}=F_{n-1}+F_{n-2}}"></span> for <span class="texhtml"><i>n</i> > 1</span>. </p><p>Under some older definitions, the value <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{0}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{0}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58ebe8b2d5551fb272cd4258940fe1e492592d02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{0}=0}"></span> is omitted, so that the sequence starts with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}=F_{2}=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}=F_{2}=1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39ce4b302203aa4afd0eccf11b8ccbb207fadd06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.104ex; height:2.509ex;" alt="{\displaystyle F_{1}=F_{2}=1,}"></span> and the recurrence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=F_{n-1}+F_{n-2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=F_{n-1}+F_{n-2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fa6d281e7a54e08aeffeef7458ddc0884333686" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.279ex; height:2.509ex;" alt="{\displaystyle F_{n}=F_{n-1}+F_{n-2}}"></span> is valid for <span class="texhtml"><i>n</i> > 2</span>.<sup id="cite_ref-FOOTNOTEBeckGeoghegan2010_8-0" class="reference"><a href="#cite_note-FOOTNOTEBeckGeoghegan2010-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEBóna2011180_9-0" class="reference"><a href="#cite_note-FOOTNOTEBóna2011180-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>The first 20 Fibonacci numbers <span class="texhtml"><i>F<sub>n</sub></i></span> are: </p> <dl><dd><table class="wikitable" style="text-align:right"> <tbody><tr> <th><span class="texhtml"><i>F</i><sub>0</sub></span> </th> <th><span class="texhtml"><i>F</i><sub>1</sub></span> </th> <th><span class="texhtml"><i>F</i><sub>2</sub></span> </th> <th><span class="texhtml"><i>F</i><sub>3</sub></span> </th> <th><span class="texhtml"><i>F</i><sub>4</sub></span> </th> <th><span class="texhtml"><i>F</i><sub>5</sub></span> </th> <th><span class="texhtml"><i>F</i><sub>6</sub></span> </th> <th><span class="texhtml"><i>F</i><sub>7</sub></span> </th> <th><span class="texhtml"><i>F</i><sub>8</sub></span> </th> <th><span class="texhtml"><i>F</i><sub>9</sub></span> </th> <th><span class="texhtml"><i>F</i><sub>10</sub></span> </th> <th><span class="texhtml"><i>F</i><sub>11</sub></span> </th> <th><span class="texhtml"><i>F</i><sub>12</sub></span> </th> <th><span class="texhtml"><i>F</i><sub>13</sub></span> </th> <th><span class="texhtml"><i>F</i><sub>14</sub></span> </th> <th><span class="texhtml"><i>F</i><sub>15</sub></span> </th> <th><span class="texhtml"><i>F</i><sub>16</sub></span> </th> <th><span class="texhtml"><i>F</i><sub>17</sub></span> </th> <th><span class="texhtml"><i>F</i><sub>18</sub></span> </th> <th><span class="texhtml"><i>F</i><sub>19</sub></span> </th></tr> <tr> <td>0 </td> <td>1 </td> <td>1 </td> <td>2 </td> <td>3 </td> <td>5 </td> <td>8 </td> <td>13 </td> <td>21 </td> <td>34 </td> <td>55 </td> <td>89 </td> <td>144 </td> <td>233 </td> <td>377 </td> <td>610 </td> <td>987 </td> <td>1597 </td> <td>2584 </td> <td>4181 </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=2" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="India">India</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=3" title="Edit section: India"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Golden_ratio#History" title="Golden ratio">Golden ratio § History</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Fibonacci_Sanskrit_prosody.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Fibonacci_Sanskrit_prosody.svg/220px-Fibonacci_Sanskrit_prosody.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Fibonacci_Sanskrit_prosody.svg/330px-Fibonacci_Sanskrit_prosody.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/Fibonacci_Sanskrit_prosody.svg/440px-Fibonacci_Sanskrit_prosody.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>Thirteen (<span class="texhtml"><i>F</i><sub>7</sub></span>) ways of arranging long and short syllables in a cadence of length six. Eight (<span class="texhtml"><i>F</i><sub>6</sub></span>) end with a short syllable and five (<span class="texhtml"><i>F</i><sub>5</sub></span>) end with a long syllable.</figcaption></figure> <p>The Fibonacci sequence appears in <a href="/wiki/Indian_mathematics" title="Indian mathematics">Indian mathematics</a>, in connection with <a href="/wiki/Sanskrit_prosody" title="Sanskrit prosody">Sanskrit prosody</a>.<sup id="cite_ref-HistoriaMathematica_4-1" class="reference"><a href="#cite_note-HistoriaMathematica-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-knuth-v1_10-0" class="reference"><a href="#cite_note-knuth-v1-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTELivio2003197_11-0" class="reference"><a href="#cite_note-FOOTNOTELivio2003197-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> In the Sanskrit poetic tradition, there was interest in enumerating all patterns of long (L) syllables of 2 units duration, juxtaposed with short (S) syllables of 1 unit duration. Counting the different patterns of successive L and S with a given total duration results in the Fibonacci numbers: the number of patterns of duration <span class="texhtml mvar" style="font-style:italic;">m</span> units is <span class="texhtml"><i>F</i><sub><i>m</i>+1</sub></span>.<sup id="cite_ref-Donald_Knuth_2006_50_5-1" class="reference"><a href="#cite_note-Donald_Knuth_2006_50-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Knowledge of the Fibonacci sequence was expressed as early as <a href="/wiki/Pingala" title="Pingala">Pingala</a> (<abbr title="circa">c.</abbr> 450 BC–200 BC). Singh cites Pingala's cryptic formula <i>misrau cha</i> ("the two are mixed") and scholars who interpret it in context as saying that the number of patterns for <span class="texhtml mvar" style="font-style:italic;">m</span> beats (<span class="texhtml"><i>F</i><sub><i>m</i>+1</sub></span>) is obtained by adding one [S] to the <span class="texhtml"><i>F</i><sub><i>m</i></sub></span> cases and one [L] to the <span class="texhtml"><i>F</i><sub><i>m</i>−1</sub></span> cases.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Bharata_Muni" class="mw-redirect" title="Bharata Muni">Bharata Muni</a> also expresses knowledge of the sequence in the <i><a href="/wiki/Natya_Shastra" title="Natya Shastra">Natya Shastra</a></i> (c. 100 BC–c. 350 AD).<sup id="cite_ref-HistoriaMathematica_4-2" class="reference"><a href="#cite_note-HistoriaMathematica-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-GlobalScience_3-1" class="reference"><a href="#cite_note-GlobalScience-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> However, the clearest exposition of the sequence arises in the work of <a href="/wiki/Virahanka" title="Virahanka">Virahanka</a> (c. 700 AD), whose own work is lost, but is available in a quotation by Gopala (c. 1135):<sup id="cite_ref-FOOTNOTELivio2003197_11-1" class="reference"><a href="#cite_note-FOOTNOTELivio2003197-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <blockquote><p>Variations of two earlier meters [is the variation] ... For example, for [a meter of length] four, variations of meters of two [and] three being mixed, five happens. [works out examples 8, 13, 21] ... In this way, the process should be followed in all <i>mātrā-vṛttas</i> [prosodic combinations].<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup></p></blockquote> <p><a href="/wiki/Hemachandra" title="Hemachandra">Hemachandra</a> (c. 1150) is credited with knowledge of the sequence as well,<sup id="cite_ref-GlobalScience_3-2" class="reference"><a href="#cite_note-GlobalScience-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> writing that "the sum of the last and the one before the last is the number ... of the next mātrā-vṛtta."<sup id="cite_ref-FOOTNOTELivio2003197–198_15-0" class="reference"><a href="#cite_note-FOOTNOTELivio2003197–198-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Europe">Europe</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=4" title="Edit section: Europe"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Liber_abbaci_magliab_f124r.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/Liber_abbaci_magliab_f124r.jpg/280px-Liber_abbaci_magliab_f124r.jpg" decoding="async" width="280" height="428" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/Liber_abbaci_magliab_f124r.jpg/420px-Liber_abbaci_magliab_f124r.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/04/Liber_abbaci_magliab_f124r.jpg/560px-Liber_abbaci_magliab_f124r.jpg 2x" data-file-width="744" data-file-height="1137" /></a><figcaption>A page of <a href="/wiki/Fibonacci" title="Fibonacci">Fibonacci</a>'s <span title="Latin-language text"><i lang="la"><a href="/wiki/Liber_Abaci" title="Liber Abaci">Liber Abaci</a></i></span> from the <a href="/wiki/National_Central_Library_(Florence)" title="National Central Library (Florence)">Biblioteca Nazionale di Firenze</a> showing (in box on right) 13 entries of the Fibonacci sequence:<br /> the indices from present to XII (months) as Latin ordinals and Roman numerals and the numbers (of rabbit pairs) as Hindu-Arabic numerals starting with 1, 2, 3, 5 and ending with 377.</figcaption></figure> <p>The Fibonacci sequence first appears in the book <span title="Latin-language text"><i lang="la"><a href="/wiki/Liber_Abaci" title="Liber Abaci">Liber Abaci</a></i></span> (<i>The Book of Calculation</i>, 1202) by <a href="/wiki/Fibonacci" title="Fibonacci">Fibonacci</a><sup id="cite_ref-FOOTNOTESigler2002404–405_17-0" class="reference"><a href="#cite_note-FOOTNOTESigler2002404–405-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> where it is used to calculate the growth of rabbit populations.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> Fibonacci considers the growth of an idealized (<a href="/wiki/Biology" title="Biology">biologically</a> unrealistic) <a href="/wiki/Rabbit" title="Rabbit">rabbit</a> population, assuming that: a newly born breeding pair of rabbits are put in a field; each breeding pair mates at the age of one month, and at the end of their second month they always produce another pair of rabbits; and rabbits never die, but continue breeding forever. Fibonacci posed the rabbit <a href="/wiki/Mathematical_problem" title="Mathematical problem">math problem</a>: how many pairs will there be in one year? </p> <ul><li>At the end of the first month, they mate, but there is still only 1 pair.</li> <li>At the end of the second month they produce a new pair, so there are 2 pairs in the field.</li> <li>At the end of the third month, the original pair produce a second pair, but the second pair only mate to gestate for a month, so there are 3 pairs in all.</li> <li>At the end of the fourth month, the original pair has produced yet another new pair, and the pair born two months ago also produces their first pair, making 5 pairs.</li></ul> <p>At the end of the <span class="texhtml mvar" style="font-style:italic;">n</span>-th month, the number of pairs of rabbits is equal to the number of mature pairs (that is, the number of pairs in month <span class="texhtml"><i>n</i> – 2</span>) plus the number of pairs alive last month (month <span class="texhtml"><i>n</i> – 1</span>). The number in the <span class="texhtml mvar" style="font-style:italic;">n</span>-th month is the <span class="texhtml mvar" style="font-style:italic;">n</span>-th Fibonacci number.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>The name "Fibonacci sequence" was first used by the 19th-century number theorist <a href="/wiki/%C3%89douard_Lucas" title="Édouard Lucas">Édouard Lucas</a>.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Fibonacci_Rabbits.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Fibonacci_Rabbits.svg/330px-Fibonacci_Rabbits.svg.png" decoding="async" width="330" height="256" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Fibonacci_Rabbits.svg/495px-Fibonacci_Rabbits.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/Fibonacci_Rabbits.svg/660px-Fibonacci_Rabbits.svg.png 2x" data-file-width="512" data-file-height="397" /></a><figcaption>Solution to Fibonacci rabbit <a href="/wiki/Mathematical_problem" title="Mathematical problem">problem</a>: In a growing idealized population, the number of rabbit pairs form the Fibonacci sequence. At <i>the end of the n</i>th month, the number of pairs is equal to <i>F<sub>n.</sub></i></figcaption></figure> <div style="clear:left;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Relation_to_the_golden_ratio">Relation to the golden ratio</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=5" title="Edit section: Relation to the golden ratio"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Golden_ratio" title="Golden ratio">Golden ratio</a></div> <div class="mw-heading mw-heading3"><h3 id="Closed-form_expression">Closed-form expression <span class="anchor" id="Binet's_formula"></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=6" title="Edit section: Closed-form expression"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Like every <a href="/wiki/Sequence" title="Sequence">sequence</a> defined by a homogeneous <a href="/wiki/Linear_recurrence_with_constant_coefficients" title="Linear recurrence with constant coefficients">linear recurrence with constant coefficients</a>, the Fibonacci numbers have a <a href="/wiki/Closed-form_expression" title="Closed-form expression">closed-form expression</a>.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> It has become known as <b>Binet's formula</b>, named after French mathematician <a href="/wiki/Jacques_Philippe_Marie_Binet" title="Jacques Philippe Marie Binet">Jacques Philippe Marie Binet</a>, though it was already known by <a href="/wiki/Abraham_de_Moivre" title="Abraham de Moivre">Abraham de Moivre</a> and <a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a>:<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}={\frac {\varphi ^{n}-\psi ^{n}}{\varphi -\psi }}={\frac {\varphi ^{n}-\psi ^{n}}{\sqrt {5}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mi>φ</mi> <mo>−</mo> <mi>ψ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <msqrt> <mn>5</mn> </msqrt> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}={\frac {\varphi ^{n}-\psi ^{n}}{\varphi -\psi }}={\frac {\varphi ^{n}-\psi ^{n}}{\sqrt {5}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80f3adaabc7795348c3647d253891b5790ef6dd7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:27.85ex; height:6.343ex;" alt="{\displaystyle F_{n}={\frac {\varphi ^{n}-\psi ^{n}}{\varphi -\psi }}={\frac {\varphi ^{n}-\psi ^{n}}{\sqrt {5}}},}"></span> </p><p>where </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}\approx 1.61803\,39887\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>≈</mo> <mn>1.61803</mn> <mspace width="thinmathspace" /> <mn>39887</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}\approx 1.61803\,39887\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de8aaf6a6b60f0f58cc274515efd7f177bd65802" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.586ex; height:5.843ex;" alt="{\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}\approx 1.61803\,39887\ldots }"></span> </p><p>is the <a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</a>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> is its <a href="/wiki/Conjugate_(square_roots)" title="Conjugate (square roots)">conjugate</a>:<sup id="cite_ref-FOOTNOTEBall2003156_25-0" class="reference"><a href="#cite_note-FOOTNOTEBall2003156-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ={\frac {1-{\sqrt {5}}}{2}}=1-\varphi =-{1 \over \varphi }\approx -0.61803\,39887\ldots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>φ</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>φ</mi> </mfrac> </mrow> <mo>≈</mo> <mo>−</mo> <mn>0.61803</mn> <mspace width="thinmathspace" /> <mn>39887</mn> <mo>…</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ={\frac {1-{\sqrt {5}}}{2}}=1-\varphi =-{1 \over \varphi }\approx -0.61803\,39887\ldots .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a531a6c08c01aacc84fb8ab1311ab471c6b22820" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:51.305ex; height:6.343ex;" alt="{\displaystyle \psi ={\frac {1-{\sqrt {5}}}{2}}=1-\varphi =-{1 \over \varphi }\approx -0.61803\,39887\ldots .}"></span> </p><p>Since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi =-\varphi ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <mo>−</mo> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi =-\varphi ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d966e5006d42590d61ad2416b47a178c783bfde6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.273ex; height:3.176ex;" alt="{\displaystyle \psi =-\varphi ^{-1}}"></span>, this formula can also be written as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}={\frac {\varphi ^{n}-(-\varphi )^{-n}}{\sqrt {5}}}={\frac {\varphi ^{n}-(-\varphi )^{-n}}{2\varphi -1}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>φ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> </mrow> <msqrt> <mn>5</mn> </msqrt> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>φ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>φ</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}={\frac {\varphi ^{n}-(-\varphi )^{-n}}{\sqrt {5}}}={\frac {\varphi ^{n}-(-\varphi )^{-n}}{2\varphi -1}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/451078e3f7c9ae5d67abd6a1e770602d3c9ebc63" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:37.656ex; height:6.676ex;" alt="{\displaystyle F_{n}={\frac {\varphi ^{n}-(-\varphi )^{-n}}{\sqrt {5}}}={\frac {\varphi ^{n}-(-\varphi )^{-n}}{2\varphi -1}}.}"></span> </p><p>To see the relation between the sequence and these constants,<sup id="cite_ref-FOOTNOTEBall2003155–156_26-0" class="reference"><a href="#cite_note-FOOTNOTEBall2003155–156-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> note that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> are both solutions of the equation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x^{2}=x+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x^{2}=x+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afcaaabe7eb8ba6c20cca6efe1f46c8acdf13a65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.815ex; height:2.676ex;" alt="{\textstyle x^{2}=x+1}"></span> and thus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{n}=x^{n-1}+x^{n-2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{n}=x^{n-1}+x^{n-2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f548d9fc5c4668b9a3fa2dcbc012f33e008cfc08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.431ex; height:3.009ex;" alt="{\displaystyle x^{n}=x^{n-1}+x^{n-2},}"></span> so the powers of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> satisfy the Fibonacci recursion. In other words, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\varphi ^{n}&=\varphi ^{n-1}+\varphi ^{n-2},\\[3mu]\psi ^{n}&=\psi ^{n-1}+\psi ^{n-2}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.467em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varphi ^{n}&=\varphi ^{n-1}+\varphi ^{n-2},\\[3mu]\psi ^{n}&=\psi ^{n-1}+\psi ^{n-2}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/282a0b620cb2eca72787ef444e6d07d39678d383" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.754ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}\varphi ^{n}&=\varphi ^{n-1}+\varphi ^{n-2},\\[3mu]\psi ^{n}&=\psi ^{n-1}+\psi ^{n-2}.\end{aligned}}}"></span> </p><p>It follows that for any values <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span>, the sequence defined by </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{n}=a\varphi ^{n}+b\psi ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>a</mi> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{n}=a\varphi ^{n}+b\psi ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4eb55069f47b9ddaa49f1c88f557ce103e557b36" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.442ex; height:2.843ex;" alt="{\displaystyle U_{n}=a\varphi ^{n}+b\psi ^{n}}"></span> </p><p>satisfies the same recurrence, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}U_{n}&=a\varphi ^{n}+b\psi ^{n}\\[3mu]&=a(\varphi ^{n-1}+\varphi ^{n-2})+b(\psi ^{n-1}+\psi ^{n-2})\\[3mu]&=a\varphi ^{n-1}+b\psi ^{n-1}+a\varphi ^{n-2}+b\psi ^{n-2}\\[3mu]&=U_{n-1}+U_{n-2}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.467em 0.467em 0.467em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>a</mi> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> <mo stretchy="false">(</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>a</mi> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}U_{n}&=a\varphi ^{n}+b\psi ^{n}\\[3mu]&=a(\varphi ^{n-1}+\varphi ^{n-2})+b(\psi ^{n-1}+\psi ^{n-2})\\[3mu]&=a\varphi ^{n-1}+b\psi ^{n-1}+a\varphi ^{n-2}+b\psi ^{n-2}\\[3mu]&=U_{n-1}+U_{n-2}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49e155e3c74b9d82976473240c1fc8f5db5d0f1f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:40.366ex; height:13.676ex;" alt="{\displaystyle {\begin{aligned}U_{n}&=a\varphi ^{n}+b\psi ^{n}\\[3mu]&=a(\varphi ^{n-1}+\varphi ^{n-2})+b(\psi ^{n-1}+\psi ^{n-2})\\[3mu]&=a\varphi ^{n-1}+b\psi ^{n-1}+a\varphi ^{n-2}+b\psi ^{n-2}\\[3mu]&=U_{n-1}+U_{n-2}.\end{aligned}}}"></span> </p><p>If <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> are chosen so that <span class="texhtml"><i>U</i><sub>0</sub> = 0</span> and <span class="texhtml"><i>U</i><sub>1</sub> = 1</span> then the resulting sequence <span class="texhtml"><i>U</i><sub><i>n</i></sub></span> must be the Fibonacci sequence. This is the same as requiring <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> satisfy the system of equations: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{{\begin{aligned}a+b&=0\\\varphi a+\psi b&=1\end{aligned}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>φ</mi> <mi>a</mi> <mo>+</mo> <mi>ψ</mi> <mi>b</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{aligned}a+b&=0\\\varphi a+\psi b&=1\end{aligned}}\right.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d0900f5323cf3537ffc6ed9765c4976ea192469" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.857ex; height:6.176ex;" alt="{\displaystyle \left\{{\begin{aligned}a+b&=0\\\varphi a+\psi b&=1\end{aligned}}\right.}"></span> </p><p>which has solution </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a={\frac {1}{\varphi -\psi }}={\frac {1}{\sqrt {5}}},\quad b=-a,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>φ</mi> <mo>−</mo> <mi>ψ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>5</mn> </msqrt> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>b</mi> <mo>=</mo> <mo>−</mo> <mi>a</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\frac {1}{\varphi -\psi }}={\frac {1}{\sqrt {5}}},\quad b=-a,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/214c97528ee0994fd5d62b3589e88a712b13f878" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:29.208ex; height:6.176ex;" alt="{\displaystyle a={\frac {1}{\varphi -\psi }}={\frac {1}{\sqrt {5}}},\quad b=-a,}"></span> </p><p>producing the required formula. </p><p>Taking the starting values <span class="texhtml"><i>U</i><sub>0</sub></span> and <span class="texhtml"><i>U</i><sub>1</sub></span> to be arbitrary constants, a more general solution is: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{n}=a\varphi ^{n}+b\psi ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>a</mi> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{n}=a\varphi ^{n}+b\psi ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4eb55069f47b9ddaa49f1c88f557ce103e557b36" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.442ex; height:2.843ex;" alt="{\displaystyle U_{n}=a\varphi ^{n}+b\psi ^{n}}"></span> </p><p>where </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}a&={\frac {U_{1}-U_{0}\psi }{\sqrt {5}}},\\[3mu]b&={\frac {U_{0}\varphi -U_{1}}{\sqrt {5}}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.467em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>ψ</mi> </mrow> <msqrt> <mn>5</mn> </msqrt> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>φ</mi> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <msqrt> <mn>5</mn> </msqrt> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a&={\frac {U_{1}-U_{0}\psi }{\sqrt {5}}},\\[3mu]b&={\frac {U_{0}\varphi -U_{1}}{\sqrt {5}}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b7a1f5df3f3d8a6b6e48e76a19c75a2b5c65645" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:16.207ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}a&={\frac {U_{1}-U_{0}\psi }{\sqrt {5}}},\\[3mu]b&={\frac {U_{0}\varphi -U_{1}}{\sqrt {5}}}.\end{aligned}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Computation_by_rounding">Computation by rounding</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=7" title="Edit section: Computation by rounding"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left|{\frac {\psi ^{n}}{\sqrt {5}}}\right|<{\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msqrt> <mn>5</mn> </msqrt> </mfrac> </mrow> <mo>|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \left|{\frac {\psi ^{n}}{\sqrt {5}}}\right|<{\frac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f36056259148225cdc34612e19c42887fb89055" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.077ex; height:4.843ex;" alt="{\textstyle \left|{\frac {\psi ^{n}}{\sqrt {5}}}\right|<{\frac {1}{2}}}"></span> for all <span class="texhtml"><i>n</i> ≥ 0</span>, the number <span class="texhtml"><i>F</i><sub><i>n</i></sub></span> is the closest <a href="/wiki/Integer" title="Integer">integer</a> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\varphi ^{n}}{\sqrt {5}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msqrt> <mn>5</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\varphi ^{n}}{\sqrt {5}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67180162747d0e482d1bbb421cee2742f2e303c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:3.934ex; height:6.343ex;" alt="{\displaystyle {\frac {\varphi ^{n}}{\sqrt {5}}}}"></span>. Therefore, it can be found by <a href="/wiki/Rounding" title="Rounding">rounding</a>, using the nearest integer function: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=\left\lfloor {\frac {\varphi ^{n}}{\sqrt {5}}}\right\rceil ,\ n\geq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msqrt> <mn>5</mn> </msqrt> </mfrac> </mrow> <mo>⌉</mo> </mrow> <mo>,</mo> <mtext> </mtext> <mi>n</mi> <mo>≥</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=\left\lfloor {\frac {\varphi ^{n}}{\sqrt {5}}}\right\rceil ,\ n\geq 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1472b2568affd51a4ff35dbc6b0e0b27094d093" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:20.761ex; height:6.509ex;" alt="{\displaystyle F_{n}=\left\lfloor {\frac {\varphi ^{n}}{\sqrt {5}}}\right\rceil ,\ n\geq 0.}"></span> </p><p>In fact, the rounding error quickly becomes very small as <span class="texhtml mvar" style="font-style:italic;">n</span> grows, being less than 0.1 for <span class="texhtml"><i>n</i> ≥ 4</span>, and less than 0.01 for <span class="texhtml"><i>n</i> ≥ 8</span>. This formula is easily inverted to find an index of a Fibonacci number <span class="texhtml mvar" style="font-style:italic;">F</span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(F)=\left\lfloor \log _{\varphi }{\sqrt {5}}F\right\rceil ,\ F\geq 1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>⌊</mo> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mi>F</mi> </mrow> <mo>⌉</mo> </mrow> <mo>,</mo> <mtext> </mtext> <mi>F</mi> <mo>≥</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n(F)=\left\lfloor \log _{\varphi }{\sqrt {5}}F\right\rceil ,\ F\geq 1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b3f644578160a16c71f8fa46c1e5fc4f841e0ec" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:28.393ex; height:3.343ex;" alt="{\displaystyle n(F)=\left\lfloor \log _{\varphi }{\sqrt {5}}F\right\rceil ,\ F\geq 1.}"></span> </p><p>Instead using the <a href="/wiki/Floor_function" class="mw-redirect" title="Floor function">floor function</a> gives the largest index of a Fibonacci number that is not greater than <span class="texhtml mvar" style="font-style:italic;">F</span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{\mathrm {largest} }(F)=\left\lfloor \log _{\varphi }{\sqrt {5}}(F+1/2)\right\rfloor ,\ F\geq 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>⌊</mo> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">(</mo> <mi>F</mi> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mo>⌋</mo> </mrow> <mo>,</mo> <mtext> </mtext> <mi>F</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{\mathrm {largest} }(F)=\left\lfloor \log _{\varphi }{\sqrt {5}}(F+1/2)\right\rfloor ,\ F\geq 0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa53c285938a5e1d062edaf1b611aaedbcd69e90" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:41.526ex; height:3.343ex;" alt="{\displaystyle n_{\mathrm {largest} }(F)=\left\lfloor \log _{\varphi }{\sqrt {5}}(F+1/2)\right\rfloor ,\ F\geq 0,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{\varphi }(x)=\ln(x)/\ln(\varphi )=\log _{10}(x)/\log _{10}(\varphi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{\varphi }(x)=\ln(x)/\ln(\varphi )=\log _{10}(x)/\log _{10}(\varphi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/010eea776ca1d745f74b0e4a1aefdc563315a3c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:43.225ex; height:3.176ex;" alt="{\displaystyle \log _{\varphi }(x)=\ln(x)/\ln(\varphi )=\log _{10}(x)/\log _{10}(\varphi )}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(\varphi )=0.481211\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.481211</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(\varphi )=0.481211\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbe83479430c5a199b4657e1476f11e01f9bc99b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.262ex; height:2.843ex;" alt="{\displaystyle \ln(\varphi )=0.481211\ldots }"></span>,<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{10}(\varphi )=0.208987\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.208987</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{10}(\varphi )=0.208987\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/411e51d091b7b5f93fa086bb2bdfd2f72d2f27d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.17ex; height:2.843ex;" alt="{\displaystyle \log _{10}(\varphi )=0.208987\ldots }"></span>.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Magnitude">Magnitude</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=8" title="Edit section: Magnitude"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since <i>F<sub>n</sub></i> is <a href="/wiki/Asymptotic_analysis" title="Asymptotic analysis">asymptotic</a> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{n}/{\sqrt {5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ^{n}/{\sqrt {5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e44feeb18cba5ae7d9cef89d5bc2c4b016b71161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.999ex; height:3.009ex;" alt="{\displaystyle \varphi ^{n}/{\sqrt {5}}}"></span>, the number of digits in <span class="texhtml"><i>F</i><sub><i>n</i></sub></span> is asymptotic to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\log _{10}\varphi \approx 0.2090\,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mi>φ</mi> <mo>≈</mo> <mn>0.2090</mn> <mspace width="thinmathspace" /> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\log _{10}\varphi \approx 0.2090\,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c17e5414f184956bc08aa6f0447847e1eddbfbb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.876ex; height:2.676ex;" alt="{\displaystyle n\log _{10}\varphi \approx 0.2090\,n}"></span>. As a consequence, for every integer <span class="texhtml"><i>d</i> > 1</span> there are either 4 or 5 Fibonacci numbers with <span class="texhtml mvar" style="font-style:italic;">d</span> decimal digits. </p><p>More generally, in the <a href="/wiki/Radix" title="Radix">base</a> <span class="texhtml mvar" style="font-style:italic;">b</span> representation, the number of digits in <span class="texhtml"><i>F</i><sub><i>n</i></sub></span> is asymptotic to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\log _{b}\varphi ={\frac {n\log \varphi }{\log b}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>log</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> <mrow> <mi>log</mi> <mo>⁡</mo> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\log _{b}\varphi ={\frac {n\log \varphi }{\log b}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7f35205ae578d5571a57374c0f225b169a5554" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.841ex; height:5.843ex;" alt="{\displaystyle n\log _{b}\varphi ={\frac {n\log \varphi }{\log b}}.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Limit_of_consecutive_quotients">Limit of consecutive quotients</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=9" title="Edit section: Limit of consecutive quotients"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Johannes Kepler</a> observed that the ratio of consecutive Fibonacci numbers <a href="/wiki/Convergent_sequence" class="mw-redirect" title="Convergent sequence">converges</a>. He wrote that "as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost", and concluded that these ratios approach the golden ratio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \colon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \colon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d67e4a3ca72a183b43017db264aea368b4f49b90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.167ex; height:2.176ex;" alt="{\displaystyle \varphi \colon }"></span> <sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }{\frac {F_{n+1}}{F_{n}}}=\varphi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }{\frac {F_{n+1}}{F_{n}}}=\varphi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c97c57b45024325087cc20cbfd9af27fc6c5a5bf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.575ex; height:5.676ex;" alt="{\displaystyle \lim _{n\to \infty }{\frac {F_{n+1}}{F_{n}}}=\varphi .}"></span> </p><p>This convergence holds regardless of the starting values <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52116e300c6199496d7b4a60d417ba34a4e569dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle U_{0}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc9e7f892894bc50c32ce1b9f9a68a15562146ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle U_{1}}"></span>, unless <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{1}=-U_{0}/\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{1}=-U_{0}/\varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a6f0e3daace50ce6d41cd734bd07fe0e9a38572" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.873ex; height:2.843ex;" alt="{\displaystyle U_{1}=-U_{0}/\varphi }"></span>. This can be verified using <a href="#Binet's_formula">Binet's formula</a>. For example, the initial values 3 and 2 generate the sequence 3, 2, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, ... . The ratio of consecutive elements in this sequence shows the same convergence towards the golden ratio. </p><p>In general, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }{\frac {F_{n+m}}{F_{n}}}=\varphi ^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>m</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }{\frac {F_{n+m}}{F_{n}}}=\varphi ^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57c7440ca49a11b48c02fdd2366e5f8b87036bda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.224ex; height:5.676ex;" alt="{\displaystyle \lim _{n\to \infty }{\frac {F_{n+m}}{F_{n}}}=\varphi ^{m}}"></span>, because the ratios between consecutive Fibonacci numbers approaches <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span>. </p> <dl><dd><figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Fibonacci_tiling_of_the_plane_and_approximation_to_Golden_Ratio.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Fibonacci_tiling_of_the_plane_and_approximation_to_Golden_Ratio.gif/480px-Fibonacci_tiling_of_the_plane_and_approximation_to_Golden_Ratio.gif" decoding="async" width="480" height="297" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Fibonacci_tiling_of_the_plane_and_approximation_to_Golden_Ratio.gif/720px-Fibonacci_tiling_of_the_plane_and_approximation_to_Golden_Ratio.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Fibonacci_tiling_of_the_plane_and_approximation_to_Golden_Ratio.gif/960px-Fibonacci_tiling_of_the_plane_and_approximation_to_Golden_Ratio.gif 2x" data-file-width="1166" data-file-height="721" /></a><figcaption>Successive tilings of the plane and a graph of approximations to the golden ratio calculated by dividing each Fibonacci number by the previous</figcaption></figure></dd></dl> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Decomposition_of_powers">Decomposition of powers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=10" title="Edit section: Decomposition of powers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since the golden ratio satisfies the equation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{2}=\varphi +1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>φ</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ^{2}=\varphi +1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12c53552ccd728ee54e6c0b7bbd8c5b8000cbaf7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.843ex; height:3.176ex;" alt="{\displaystyle \varphi ^{2}=\varphi +1,}"></span> </p><p>this expression can be used to decompose higher powers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bfefb926a7e3f934344edc2d382566ad9c37c30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.739ex; height:2.843ex;" alt="{\displaystyle \varphi ^{n}}"></span> as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> and 1. The resulting <a href="/wiki/Recurrence_relation" title="Recurrence relation">recurrence relationships</a> yield Fibonacci numbers as the linear <a href="/wiki/Coefficient" title="Coefficient">coefficients</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{n}=F_{n}\varphi +F_{n-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>φ</mi> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ^{n}=F_{n}\varphi +F_{n-1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fcfa1f460d7b8f99a370d02facecbaa259fa173" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.371ex; height:2.843ex;" alt="{\displaystyle \varphi ^{n}=F_{n}\varphi +F_{n-1}.}"></span> This equation can be <a href="/wiki/Mathematical_proof" title="Mathematical proof">proved</a> by <a href="/wiki/Mathematical_induction" title="Mathematical induction">induction</a> on <span class="texhtml"><i>n</i> ≥ 1</span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{n+1}=(F_{n}\varphi +F_{n-1})\varphi =F_{n}\varphi ^{2}+F_{n-1}\varphi =F_{n}(\varphi +1)+F_{n-1}\varphi =(F_{n}+F_{n-1})\varphi +F_{n}=F_{n+1}\varphi +F_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>φ</mi> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>φ</mi> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mi>φ</mi> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mi>φ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>φ</mi> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mi>φ</mi> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ^{n+1}=(F_{n}\varphi +F_{n-1})\varphi =F_{n}\varphi ^{2}+F_{n-1}\varphi =F_{n}(\varphi +1)+F_{n-1}\varphi =(F_{n}+F_{n-1})\varphi +F_{n}=F_{n+1}\varphi +F_{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568cd7e416d7126af8b8aeafc331a9c4d67326d7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:101.013ex; height:3.176ex;" alt="{\displaystyle \varphi ^{n+1}=(F_{n}\varphi +F_{n-1})\varphi =F_{n}\varphi ^{2}+F_{n-1}\varphi =F_{n}(\varphi +1)+F_{n-1}\varphi =(F_{n}+F_{n-1})\varphi +F_{n}=F_{n+1}\varphi +F_{n}.}"></span> For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi =-1/\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi =-1/\varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91b27868e4f91891f5d1fa260c9acce9018dd638" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.265ex; height:2.843ex;" alt="{\displaystyle \psi =-1/\varphi }"></span>, it is also the case that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{2}=\psi +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>ψ</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ^{2}=\psi +1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fff2f022bcd58d953fb30897d6ba19a94f6c8b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.182ex; height:3.009ex;" alt="{\displaystyle \psi ^{2}=\psi +1}"></span> and it is also the case that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{n}=F_{n}\psi +F_{n-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>ψ</mi> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ^{n}=F_{n}\psi +F_{n-1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58016be5b6eadf0faad3650e1e55bfd1f5254a02" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.357ex; height:2.676ex;" alt="{\displaystyle \psi ^{n}=F_{n}\psi +F_{n-1}.}"></span> </p><p>These expressions are also true for <span class="texhtml"><i>n</i> < 1</span> if the Fibonacci sequence <i>F<sub>n</sub></i> is <a href="/wiki/Generalizations_of_Fibonacci_numbers#Extension_to_negative_integers" title="Generalizations of Fibonacci numbers">extended to negative integers</a> using the Fibonacci rule <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=F_{n+2}-F_{n+1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=F_{n+2}-F_{n+1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aa79f32a1e7651f9ad22a00a09646a0898d30ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.926ex; height:2.509ex;" alt="{\displaystyle F_{n}=F_{n+2}-F_{n+1}.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Identification">Identification</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=11" title="Edit section: Identification"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Binet's formula provides a proof that a positive integer <span class="texhtml mvar" style="font-style:italic;">x</span> is a Fibonacci number <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> at least one of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5x^{2}+4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5x^{2}+4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63265a5787c1994492f335bf4957ff17524ca4e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.549ex; height:2.843ex;" alt="{\displaystyle 5x^{2}+4}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5x^{2}-4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5x^{2}-4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcba7f88ca4dd3784151c47c6107dd16124bc7c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.549ex; height:2.843ex;" alt="{\displaystyle 5x^{2}-4}"></span> is a <a href="/wiki/Square_number" title="Square number">perfect square</a>.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> This is because Binet's formula, which can be written as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=(\varphi ^{n}-(-1)^{n}\varphi ^{-n})/{\sqrt {5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=(\varphi ^{n}-(-1)^{n}\varphi ^{-n})/{\sqrt {5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f55805af793c7b658d259a3d00a98533e49ff72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.476ex; height:3.009ex;" alt="{\displaystyle F_{n}=(\varphi ^{n}-(-1)^{n}\varphi ^{-n})/{\sqrt {5}}}"></span>, can be multiplied by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {5}}\varphi ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {5}}\varphi ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f439472ea2a40ccc0863e2c1c6a0280002493b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.837ex; height:3.009ex;" alt="{\displaystyle {\sqrt {5}}\varphi ^{n}}"></span> and solved as a <a href="/wiki/Quadratic_equation" title="Quadratic equation">quadratic equation</a> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bfefb926a7e3f934344edc2d382566ad9c37c30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.739ex; height:2.843ex;" alt="{\displaystyle \varphi ^{n}}"></span> via the <a href="/wiki/Quadratic_formula" title="Quadratic formula">quadratic formula</a>: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{n}={\frac {F_{n}{\sqrt {5}}\pm {\sqrt {5{F_{n}}^{2}+4(-1)^{n}}}}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ^{n}={\frac {F_{n}{\sqrt {5}}\pm {\sqrt {5{F_{n}}^{2}+4(-1)^{n}}}}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2245a1396e2a69f6ff79e3782a6da469ff41e5e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.227ex; height:7.676ex;" alt="{\displaystyle \varphi ^{n}={\frac {F_{n}{\sqrt {5}}\pm {\sqrt {5{F_{n}}^{2}+4(-1)^{n}}}}{2}}.}"></span> </p><p>Comparing this to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{n}=F_{n}\varphi +F_{n-1}=(F_{n}{\sqrt {5}}+F_{n}+2F_{n-1})/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>φ</mi> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ^{n}=F_{n}\varphi +F_{n-1}=(F_{n}{\sqrt {5}}+F_{n}+2F_{n-1})/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66791cc13a236899e54dc6135a7b787e74696bee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.138ex; height:3.009ex;" alt="{\displaystyle \varphi ^{n}=F_{n}\varphi +F_{n-1}=(F_{n}{\sqrt {5}}+F_{n}+2F_{n-1})/2}"></span>, it follows that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5{F_{n}}^{2}+4(-1)^{n}=(F_{n}+2F_{n-1})^{2}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5{F_{n}}^{2}+4(-1)^{n}=(F_{n}+2F_{n-1})^{2}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a27ce32af06f3f737b2ffc5f9756048a451beda" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.456ex; height:3.176ex;" alt="{\displaystyle 5{F_{n}}^{2}+4(-1)^{n}=(F_{n}+2F_{n-1})^{2}\,.}"></span></dd></dl> <p>In particular, the left-hand side is a perfect square. </p> <div class="mw-heading mw-heading2"><h2 id="Matrix_form">Matrix form</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=12" title="Edit section: Matrix form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A 2-dimensional system of linear <a href="/wiki/Difference_equation" class="mw-redirect" title="Difference equation">difference equations</a> that describes the Fibonacci sequence is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {F_{k+2} \choose F_{k+1}}={\begin{pmatrix}1&1\\1&0\end{pmatrix}}{F_{k+1} \choose F_{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {F_{k+2} \choose F_{k+1}}={\begin{pmatrix}1&1\\1&0\end{pmatrix}}{F_{k+1} \choose F_{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c795926ab8f80e6207800de0451de69a69b8f641" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.129ex; height:6.176ex;" alt="{\displaystyle {F_{k+2} \choose F_{k+1}}={\begin{pmatrix}1&1\\1&0\end{pmatrix}}{F_{k+1} \choose F_{k}}}"></span> alternatively denoted <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{k+1}=\mathbf {A} {\vec {F}}_{k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{k+1}=\mathbf {A} {\vec {F}}_{k},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db94da4b40dbbefa66db42e9b42ecf9b3bda51e7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.585ex; height:3.176ex;" alt="{\displaystyle {\vec {F}}_{k+1}=\mathbf {A} {\vec {F}}_{k},}"></span> </p><p>which yields <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{n}=\mathbf {A} ^{n}{\vec {F}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{n}=\mathbf {A} ^{n}{\vec {F}}_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/876b987b76e01bf64355805883dd71b3fbec01e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.151ex; height:3.176ex;" alt="{\displaystyle {\vec {F}}_{n}=\mathbf {A} ^{n}{\vec {F}}_{0}}"></span>. The <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> of the <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> <span class="texhtml"><b>A</b></span> are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}~\!{\bigr )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mtext> </mtext> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}~\!{\bigr )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf5ab0819fbf64ef2942ee3e805000ef40f24849" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.701ex; height:3.509ex;" alt="{\displaystyle \varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}~\!{\bigr )}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi =-\varphi ^{-1}={\tfrac {1}{2}}{\bigl (}1-{\sqrt {5}}~\!{\bigr )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <mo>−</mo> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mtext> </mtext> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi =-\varphi ^{-1}={\tfrac {1}{2}}{\bigl (}1-{\sqrt {5}}~\!{\bigr )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a24efb7dab0274af1074f2a7a09eabfaa80bec0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:24.454ex; height:3.509ex;" alt="{\displaystyle \psi =-\varphi ^{-1}={\tfrac {1}{2}}{\bigl (}1-{\sqrt {5}}~\!{\bigr )}}"></span> corresponding to the respective <a href="/wiki/Eigenvector" class="mw-redirect" title="Eigenvector">eigenvectors</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\mu }}={\varphi \choose 1},\quad {\vec {\nu }}={-\varphi ^{-1} \choose 1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>φ</mi> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ν</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mo>−</mo> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\mu }}={\varphi \choose 1},\quad {\vec {\nu }}={-\varphi ^{-1} \choose 1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35d454f8a347071314eb99c66d96a574d0fb48a3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.908ex; height:6.343ex;" alt="{\displaystyle {\vec {\mu }}={\varphi \choose 1},\quad {\vec {\nu }}={-\varphi ^{-1} \choose 1}.}"></span> </p><p>As the initial value is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{0}={1 \choose 0}={\frac {1}{\sqrt {5}}}{\vec {\mu }}-{\frac {1}{\sqrt {5}}}{\vec {\nu }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mn>1</mn> <mn>0</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>5</mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>5</mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ν</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{0}={1 \choose 0}={\frac {1}{\sqrt {5}}}{\vec {\mu }}-{\frac {1}{\sqrt {5}}}{\vec {\nu }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b34669f0f64cb5db98489c5e16a71e7ac003e73" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:27.646ex; height:6.509ex;" alt="{\displaystyle {\vec {F}}_{0}={1 \choose 0}={\frac {1}{\sqrt {5}}}{\vec {\mu }}-{\frac {1}{\sqrt {5}}}{\vec {\nu }},}"></span> </p><p>it follows that the <span class="texhtml mvar" style="font-style:italic;">n</span>th element is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\vec {F}}_{n}&={\frac {1}{\sqrt {5}}}A^{n}{\vec {\mu }}-{\frac {1}{\sqrt {5}}}A^{n}{\vec {\nu }}\\&={\frac {1}{\sqrt {5}}}\varphi ^{n}{\vec {\mu }}-{\frac {1}{\sqrt {5}}}(-\varphi )^{-n}{\vec {\nu }}\\&={\cfrac {1}{\sqrt {5}}}\left({\cfrac {1+{\sqrt {5}}}{2}}\right)^{\!n}{\varphi \choose 1}\,-\,{\cfrac {1}{\sqrt {5}}}\left({\cfrac {1-{\sqrt {5}}}{2}}\right)^{\!n}{-\varphi ^{-1} \choose 1}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>5</mn> </msqrt> </mfrac> </mrow> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>5</mn> </msqrt> </mfrac> </mrow> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ν</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>5</mn> </msqrt> </mfrac> </mrow> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>μ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>5</mn> </msqrt> </mfrac> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>φ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ν</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mstyle> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="negativethinmathspace" /> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>φ</mi> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>−</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mstyle> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="negativethinmathspace" /> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mo>−</mo> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\vec {F}}_{n}&={\frac {1}{\sqrt {5}}}A^{n}{\vec {\mu }}-{\frac {1}{\sqrt {5}}}A^{n}{\vec {\nu }}\\&={\frac {1}{\sqrt {5}}}\varphi ^{n}{\vec {\mu }}-{\frac {1}{\sqrt {5}}}(-\varphi )^{-n}{\vec {\nu }}\\&={\cfrac {1}{\sqrt {5}}}\left({\cfrac {1+{\sqrt {5}}}{2}}\right)^{\!n}{\varphi \choose 1}\,-\,{\cfrac {1}{\sqrt {5}}}\left({\cfrac {1-{\sqrt {5}}}{2}}\right)^{\!n}{-\varphi ^{-1} \choose 1}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4806b3dcaf325ee487ea84cc68a5785689741ebe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.505ex; width:57.893ex; height:20.176ex;" alt="{\displaystyle {\begin{aligned}{\vec {F}}_{n}&={\frac {1}{\sqrt {5}}}A^{n}{\vec {\mu }}-{\frac {1}{\sqrt {5}}}A^{n}{\vec {\nu }}\\&={\frac {1}{\sqrt {5}}}\varphi ^{n}{\vec {\mu }}-{\frac {1}{\sqrt {5}}}(-\varphi )^{-n}{\vec {\nu }}\\&={\cfrac {1}{\sqrt {5}}}\left({\cfrac {1+{\sqrt {5}}}{2}}\right)^{\!n}{\varphi \choose 1}\,-\,{\cfrac {1}{\sqrt {5}}}\left({\cfrac {1-{\sqrt {5}}}{2}}\right)^{\!n}{-\varphi ^{-1} \choose 1}.\end{aligned}}}"></span> </p><p>From this, the <span class="texhtml mvar" style="font-style:italic;">n</span>th element in the Fibonacci series may be read off directly as a <a href="/wiki/Closed-form_expression" title="Closed-form expression">closed-form expression</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}={\cfrac {1}{\sqrt {5}}}\left({\cfrac {1+{\sqrt {5}}}{2}}\right)^{\!n}-\,{\cfrac {1}{\sqrt {5}}}\left({\cfrac {1-{\sqrt {5}}}{2}}\right)^{\!n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mstyle> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="negativethinmathspace" /> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mstyle> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="negativethinmathspace" /> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}={\cfrac {1}{\sqrt {5}}}\left({\cfrac {1+{\sqrt {5}}}{2}}\right)^{\!n}-\,{\cfrac {1}{\sqrt {5}}}\left({\cfrac {1-{\sqrt {5}}}{2}}\right)^{\!n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d045a58e82c3c606d5dd60516b9bfd0cbaccf30" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.455ex; height:7.509ex;" alt="{\displaystyle F_{n}={\cfrac {1}{\sqrt {5}}}\left({\cfrac {1+{\sqrt {5}}}{2}}\right)^{\!n}-\,{\cfrac {1}{\sqrt {5}}}\left({\cfrac {1-{\sqrt {5}}}{2}}\right)^{\!n}.}"></span> </p><p>Equivalently, the same computation may be performed by <a href="/wiki/Matrix_diagonalization" class="mw-redirect" title="Matrix diagonalization">diagonalization</a> of <span class="texhtml"><b>A</b></span> through use of its <a href="/wiki/Eigendecomposition" class="mw-redirect" title="Eigendecomposition">eigendecomposition</a>: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}A&=S\Lambda S^{-1},\\[3mu]A^{n}&=S\Lambda ^{n}S^{-1},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.467em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>S</mi> <mi mathvariant="normal">Λ</mi> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>S</mi> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}A&=S\Lambda S^{-1},\\[3mu]A^{n}&=S\Lambda ^{n}S^{-1},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fba6879ab00e30ad2d346663f77b8945d261b71" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.644ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}A&=S\Lambda S^{-1},\\[3mu]A^{n}&=S\Lambda ^{n}S^{-1},\end{aligned}}}"></span> </p><p>where </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda ={\begin{pmatrix}\varphi &0\\0&-\varphi ^{-1}\!\end{pmatrix}},\quad S={\begin{pmatrix}\varphi &-\varphi ^{-1}\\1&1\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>φ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="negativethinmathspace" /> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>φ</mi> </mtd> <mtd> <mo>−</mo> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda ={\begin{pmatrix}\varphi &0\\0&-\varphi ^{-1}\!\end{pmatrix}},\quad S={\begin{pmatrix}\varphi &-\varphi ^{-1}\\1&1\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed53a0d32c7434137113d478f333a7991cd9022" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.278ex; height:6.176ex;" alt="{\displaystyle \Lambda ={\begin{pmatrix}\varphi &0\\0&-\varphi ^{-1}\!\end{pmatrix}},\quad S={\begin{pmatrix}\varphi &-\varphi ^{-1}\\1&1\end{pmatrix}}.}"></span> </p><p>The closed-form expression for the <span class="texhtml mvar" style="font-style:italic;">n</span>th element in the Fibonacci series is therefore given by </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{F_{n+1} \choose F_{n}}&=A^{n}{F_{1} \choose F_{0}}\\&=S\Lambda ^{n}S^{-1}{F_{1} \choose F_{0}}\\&=S{\begin{pmatrix}\varphi ^{n}&0\\0&(-\varphi )^{-n}\end{pmatrix}}S^{-1}{F_{1} \choose F_{0}}\\&={\begin{pmatrix}\varphi &-\varphi ^{-1}\\1&1\end{pmatrix}}{\begin{pmatrix}\varphi ^{n}&0\\0&(-\varphi )^{-n}\end{pmatrix}}{\frac {1}{\sqrt {5}}}{\begin{pmatrix}1&\varphi ^{-1}\\-1&\varphi \end{pmatrix}}{1 \choose 0},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>S</mi> <msup> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo stretchy="false">(</mo> <mo>−</mo> <mi>φ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>φ</mi> </mtd> <mtd> <mo>−</mo> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo stretchy="false">(</mo> <mo>−</mo> <mi>φ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>5</mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mi>φ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mn>1</mn> <mn>0</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{F_{n+1} \choose F_{n}}&=A^{n}{F_{1} \choose F_{0}}\\&=S\Lambda ^{n}S^{-1}{F_{1} \choose F_{0}}\\&=S{\begin{pmatrix}\varphi ^{n}&0\\0&(-\varphi )^{-n}\end{pmatrix}}S^{-1}{F_{1} \choose F_{0}}\\&={\begin{pmatrix}\varphi &-\varphi ^{-1}\\1&1\end{pmatrix}}{\begin{pmatrix}\varphi ^{n}&0\\0&(-\varphi )^{-n}\end{pmatrix}}{\frac {1}{\sqrt {5}}}{\begin{pmatrix}1&\varphi ^{-1}\\-1&\varphi \end{pmatrix}}{1 \choose 0},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb33bcdb756109d4b6514cfd918084866c48b33e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.171ex; width:65.113ex; height:25.509ex;" alt="{\displaystyle {\begin{aligned}{F_{n+1} \choose F_{n}}&=A^{n}{F_{1} \choose F_{0}}\\&=S\Lambda ^{n}S^{-1}{F_{1} \choose F_{0}}\\&=S{\begin{pmatrix}\varphi ^{n}&0\\0&(-\varphi )^{-n}\end{pmatrix}}S^{-1}{F_{1} \choose F_{0}}\\&={\begin{pmatrix}\varphi &-\varphi ^{-1}\\1&1\end{pmatrix}}{\begin{pmatrix}\varphi ^{n}&0\\0&(-\varphi )^{-n}\end{pmatrix}}{\frac {1}{\sqrt {5}}}{\begin{pmatrix}1&\varphi ^{-1}\\-1&\varphi \end{pmatrix}}{1 \choose 0},\end{aligned}}}"></span> </p><p>which again yields <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}={\cfrac {\varphi ^{n}-(-\varphi )^{-n}}{\sqrt {5}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>φ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}={\cfrac {\varphi ^{n}-(-\varphi )^{-n}}{\sqrt {5}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba298d5c723813c09d537fbd5e41bd933e703bbc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.508ex; height:7.176ex;" alt="{\displaystyle F_{n}={\cfrac {\varphi ^{n}-(-\varphi )^{-n}}{\sqrt {5}}}.}"></span> </p><p>The matrix <span class="texhtml"><b>A</b></span> has a <a href="/wiki/Determinant" title="Determinant">determinant</a> of −1, and thus it is a 2 × 2 <a href="/wiki/Unimodular_matrix" title="Unimodular matrix">unimodular matrix</a>. </p><p>This property can be understood in terms of the <a href="/wiki/Continued_fraction" title="Continued fraction">continued fraction</a> representation for the golden ratio <span class="texhtml mvar" style="font-style:italic;">φ</span>: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mo>⋱</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab0b33385e6f677e6bcfe29ca876ffa6041f69d5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.671ex; width:26.764ex; height:15.843ex;" alt="{\displaystyle \varphi =1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}.}"></span> </p><p>The <a href="/wiki/Convergent_(continued_fraction)" class="mw-redirect" title="Convergent (continued fraction)">convergents</a> of the continued fraction for <span class="texhtml mvar" style="font-style:italic;">φ</span> are ratios of successive Fibonacci numbers: <span class="texhtml"><i>φ</i><sub><i>n</i></sub> = <i>F</i><sub><i>n</i>+1</sub> / <i>F</i><sub><i>n</i></sub></span> is the <span class="texhtml mvar" style="font-style:italic;">n</span>-th convergent, and the <span class="texhtml">(<i>n</i> + 1)</span>-st convergent can be found from the recurrence relation <span class="texhtml"><i>φ</i><sub><i>n</i>+1</sub> = 1 + 1 / <i>φ</i><sub><i>n</i></sub></span>.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> The matrix formed from successive convergents of any continued fraction has a determinant of +1 or −1. The matrix representation gives the following closed-form expression for the Fibonacci numbers: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}1&1\\1&0\end{pmatrix}}^{n}={\begin{pmatrix}F_{n+1}&F_{n}\\F_{n}&F_{n-1}\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1&1\\1&0\end{pmatrix}}^{n}={\begin{pmatrix}F_{n+1}&F_{n}\\F_{n}&F_{n-1}\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f90aae99d109a6d152d80d03d0353a5e849c560e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.227ex; margin-bottom: -0.278ex; width:29.906ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}1&1\\1&0\end{pmatrix}}^{n}={\begin{pmatrix}F_{n+1}&F_{n}\\F_{n}&F_{n-1}\end{pmatrix}}.}"></span> </p><p>For a given <span class="texhtml mvar" style="font-style:italic;">n</span>, this matrix can be computed in <span class="texhtml"><i>O</i>(log <i>n</i>)</span> arithmetic operations, using the <a href="/wiki/Exponentiation_by_squaring" title="Exponentiation by squaring">exponentiation by squaring</a> method. </p><p>Taking the determinant of both sides of this equation yields <a href="/wiki/Cassini%27s_identity" class="mw-redirect" title="Cassini's identity">Cassini's identity</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)^{n}=F_{n+1}F_{n-1}-{F_{n}}^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1)^{n}=F_{n+1}F_{n-1}-{F_{n}}^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4d056d9cfdd53e2a48c4c675caae2a974f98ff6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.979ex; height:3.176ex;" alt="{\displaystyle (-1)^{n}=F_{n+1}F_{n-1}-{F_{n}}^{2}.}"></span> </p><p>Moreover, since <span class="texhtml"><b>A</b><sup><i>n</i></sup><b>A</b><sup><i>m</i></sup> = <b>A</b><sup><i>n</i>+<i>m</i></sup></span> for any <a href="/wiki/Square_matrix" title="Square matrix">square matrix</a> <span class="texhtml"><b>A</b></span>, the following <a href="/wiki/Identity_(mathematics)" title="Identity (mathematics)">identities</a> can be derived (they are obtained from two different coefficients of the <a href="/wiki/Matrix_product" class="mw-redirect" title="Matrix product">matrix product</a>, and one may easily deduce the second one from the first one by changing <span class="texhtml mvar" style="font-style:italic;">n</span> into <span class="texhtml"><i>n</i> + 1</span>), <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{F_{m}}{F_{n}}+{F_{m-1}}{F_{n-1}}&=F_{m+n-1},\\[3mu]F_{m}F_{n+1}+F_{m-1}F_{n}&=F_{m+n}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.467em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{F_{m}}{F_{n}}+{F_{m-1}}{F_{n-1}}&=F_{m+n-1},\\[3mu]F_{m}F_{n+1}+F_{m-1}F_{n}&=F_{m+n}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9759138bd138ff200cb7ce83c6585d4b0a05b5a1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.304ex; margin-bottom: -0.2ex; width:30.838ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}{F_{m}}{F_{n}}+{F_{m-1}}{F_{n-1}}&=F_{m+n-1},\\[3mu]F_{m}F_{n+1}+F_{m-1}F_{n}&=F_{m+n}.\end{aligned}}}"></span> </p><p>In particular, with <span class="texhtml"><i>m</i> = <i>n</i></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}F_{2n-1}&={F_{n}}^{2}+{F_{n-1}}^{2}\\[6mu]F_{2n{\phantom {{}-1}}}&=(F_{n-1}+F_{n+1})F_{n}\\[3mu]&=(2F_{n-1}+F_{n})F_{n}\\[3mu]&=(2F_{n+1}-F_{n})F_{n}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.633em 0.467em 0.467em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>−</mo> <mn>1</mn> </mphantom> </mrow> </mrow> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}F_{2n-1}&={F_{n}}^{2}+{F_{n-1}}^{2}\\[6mu]F_{2n{\phantom {{}-1}}}&=(F_{n-1}+F_{n+1})F_{n}\\[3mu]&=(2F_{n-1}+F_{n})F_{n}\\[3mu]&=(2F_{n+1}-F_{n})F_{n}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c52b4967a19817553ce67e6d0ff1d411d4141731" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:26.475ex; height:14.009ex;" alt="{\displaystyle {\begin{aligned}F_{2n-1}&={F_{n}}^{2}+{F_{n-1}}^{2}\\[6mu]F_{2n{\phantom {{}-1}}}&=(F_{n-1}+F_{n+1})F_{n}\\[3mu]&=(2F_{n-1}+F_{n})F_{n}\\[3mu]&=(2F_{n+1}-F_{n})F_{n}.\end{aligned}}}"></span> </p><p>These last two identities provide a way to compute Fibonacci numbers <a href="/wiki/Recursion_(computer_science)" title="Recursion (computer science)">recursively</a> in <span class="texhtml"><i>O</i>(log <i>n</i>)</span> arithmetic operations. This matches the time for computing the <span class="texhtml mvar" style="font-style:italic;">n</span>-th Fibonacci number from the closed-form matrix formula, but with fewer redundant steps if one avoids recomputing an already computed Fibonacci number (recursion with <a href="/wiki/Memoization" title="Memoization">memoization</a>).<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Combinatorial_identities">Combinatorial identities</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=13" title="Edit section: Combinatorial identities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Combinatorial_proofs">Combinatorial proofs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=14" title="Edit section: Combinatorial proofs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Most identities involving Fibonacci numbers can be proved using <a href="/wiki/Combinatorial_proof" title="Combinatorial proof">combinatorial arguments</a> using the fact that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"></span> can be interpreted as the number of (possibly empty) sequences of 1s and 2s whose sum is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}"></span>. This can be taken as the definition of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"></span> with the conventions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{0}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{0}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58ebe8b2d5551fb272cd4258940fe1e492592d02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{0}=0}"></span>, meaning no such sequence exists whose sum is −1, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c374ba08c140de90c6cbb4c9b9fcd26e3f99ef56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{1}=1}"></span>, meaning the empty sequence "adds up" to 0. In the following, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{...}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{...}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75f05f18aeb1ba679b1139aa940ce5b761017c13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.008ex; height:2.843ex;" alt="{\displaystyle |{...}|}"></span> is the <a href="/wiki/Cardinality" title="Cardinality">cardinality</a> of a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{0}=0=|\{\}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{0}=0=|\{\}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11d5f5532a775ac0eff7962124e5498d6a3cc83a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.527ex; height:2.843ex;" alt="{\displaystyle F_{0}=0=|\{\}|}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}=1=|\{()\}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}=1=|\{()\}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50caafd76b616284b322433b7cfb86999e1e65ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.336ex; height:2.843ex;" alt="{\displaystyle F_{1}=1=|\{()\}|}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{2}=1=|\{(1)\}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{2}=1=|\{(1)\}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae508092bb6c760fd77a5054b1dae8682d289904" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.499ex; height:2.843ex;" alt="{\displaystyle F_{2}=1=|\{(1)\}|}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{3}=2=|\{(1,1),(2)\}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{3}=2=|\{(1,1),(2)\}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a13cd11cdf10fec9961904486bc69f9d491403b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.701ex; height:2.843ex;" alt="{\displaystyle F_{3}=2=|\{(1,1),(2)\}|}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{4}=3=|\{(1,1,1),(1,2),(2,1)\}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mn>3</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{4}=3=|\{(1,1,1),(1,2),(2,1)\}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ac5e0303af18302f99f1c9a2325fb830a007524" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.296ex; height:2.843ex;" alt="{\displaystyle F_{4}=3=|\{(1,1,1),(1,2),(2,1)\}|}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{5}=5=|\{(1,1,1,1),(1,1,2),(1,2,1),(2,1,1),(2,2)\}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>=</mo> <mn>5</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{5}=5=|\{(1,1,1,1),(1,1,2),(1,2,1),(2,1,1),(2,2)\}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a16446b92531757c7914fa9b5e9bdb3234ec82fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.485ex; height:2.843ex;" alt="{\displaystyle F_{5}=5=|\{(1,1,1,1),(1,1,2),(1,2,1),(2,1,1),(2,2)\}|}"></span></dd></dl> <p>In this manner the recurrence relation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=F_{n-1}+F_{n-2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=F_{n-1}+F_{n-2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fa6d281e7a54e08aeffeef7458ddc0884333686" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.279ex; height:2.509ex;" alt="{\displaystyle F_{n}=F_{n-1}+F_{n-2}}"></span> may be understood by dividing the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"></span> sequences into two non-overlapping sets where all sequences either begin with 1 or 2: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=|\{(1,...),(1,...),...\}|+|\{(2,...),(2,...),...\}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">)</mo> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">)</mo> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=|\{(1,...),(1,...),...\}|+|\{(2,...),(2,...),...\}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/467f5698fc61bf1e34d132dc0fa6cc4bd9f23db1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.659ex; height:2.843ex;" alt="{\displaystyle F_{n}=|\{(1,...),(1,...),...\}|+|\{(2,...),(2,...),...\}|}"></span> Excluding the first element, the remaining terms in each sequence sum to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff40d66ad535411eedb9c686a9008a5089c35ac0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-2}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ee3741ee7dd3d098f3f16980e15c0435471dda0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-3}"></span> and the cardinality of each set is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61373b860d2d2e4842b10ac0b1c3f90362c2c7d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.814ex; height:2.509ex;" alt="{\displaystyle F_{n-1}}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n-2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n-2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff0954dd46f66d84e032b21cfae83fef10c5fcbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.814ex; height:2.509ex;" alt="{\displaystyle F_{n-2}}"></span> giving a total of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n-1}+F_{n-2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n-1}+F_{n-2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92185922908ebe1ca9d3fa6a5edd6a53fbfd0a15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.468ex; height:2.509ex;" alt="{\displaystyle F_{n-1}+F_{n-2}}"></span> sequences, showing this is equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"></span>. </p><p>In a similar manner it may be shown that the sum of the first Fibonacci numbers up to the <span class="texhtml mvar" style="font-style:italic;">n</span>-th is equal to the <span class="texhtml">(<i>n</i> + 2)</span>-th Fibonacci number minus 1.<sup id="cite_ref-FOOTNOTELucas18914_34-0" class="reference"><a href="#cite_note-FOOTNOTELucas18914-34"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> In symbols: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f7cc529b34dec365fba8c962b8948d4d02b9cbc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.951ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1}"></span> </p><p>This may be seen by dividing all sequences summing to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a135e65a42f2d73cccbfc4569523996ca0036f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}"></span> based on the location of the first 2. Specifically, each set consists of those sequences that start <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2,...),(1,2,...),...,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">)</mo> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2,...),(1,2,...),...,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebf29c32df12b6c2c70391fa0c5e29ecca0415a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.228ex; height:2.843ex;" alt="{\displaystyle (2,...),(1,2,...),...,}"></span> until the last two sets <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(1,1,...,1,2)\},\{(1,1,...,1)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(1,1,...,1,2)\},\{(1,1,...,1)\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fd7aff3a5285dfb5b0aea238d0ed8b1c192c0b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.881ex; height:2.843ex;" alt="{\displaystyle \{(1,1,...,1,2)\},\{(1,1,...,1)\}}"></span> each with cardinality 1. </p><p>Following the same logic as before, by summing the cardinality of each set we see that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n+2}=F_{n}+F_{n-1}+...+|\{(1,1,...,1,2)\}|+|\{(1,1,...,1)\}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n+2}=F_{n}+F_{n-1}+...+|\{(1,1,...,1,2)\}|+|\{(1,1,...,1)\}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3d7499d2f1cb8b0d34907c42161e9788bf1eda8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:60.272ex; height:2.843ex;" alt="{\displaystyle F_{n+2}=F_{n}+F_{n-1}+...+|\{(1,1,...,1,2)\}|+|\{(1,1,...,1)\}|}"></span></dd></dl> <p>... where the last two terms have the value <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c374ba08c140de90c6cbb4c9b9fcd26e3f99ef56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{1}=1}"></span>. From this it follows that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f7cc529b34dec365fba8c962b8948d4d02b9cbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.951ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1}"></span>. </p><p>A similar argument, grouping the sums by the position of the first 1 rather than the first 2 gives two more identities: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=0}^{n-1}F_{2i+1}=F_{2n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=0}^{n-1}F_{2i+1}=F_{2n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a04201a2d4d96048635bf1e3e6db2e0d504f119" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.592ex; height:7.343ex;" alt="{\displaystyle \sum _{i=0}^{n-1}F_{2i+1}=F_{2n}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}F_{2i}=F_{2n+1}-1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}F_{2i}=F_{2n+1}-1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4ce0af1cd650cf21bdab6238a948588495eb09f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.242ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}F_{2i}=F_{2n+1}-1.}"></span> In words, the sum of the first Fibonacci numbers with <a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">odd</a> index up to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{2n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{2n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da1f7e7003160cda89bfedb51e7971f700d1457c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.636ex; height:2.509ex;" alt="{\displaystyle F_{2n-1}}"></span> is the <span class="texhtml">(2<i>n</i>)</span>-th Fibonacci number, and the sum of the first Fibonacci numbers with <a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">even</a> index up to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{2n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{2n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/191c2d8c9a1430c1d33704b6240aa12881382496" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.535ex; height:2.509ex;" alt="{\displaystyle F_{2n}}"></span> is the <span class="texhtml">(2<i>n</i> + 1)</span>-th Fibonacci number minus 1.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p><p>A different trick may be used to prove <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}F_{i}^{2}=F_{n}F_{n+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}F_{i}^{2}=F_{n}F_{n+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b41a8a56e6c68130672f6477f44b89f900aafffe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.236ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}F_{i}^{2}=F_{n}F_{n+1}}"></span> or in words, the sum of the squares of the first Fibonacci numbers up to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"></span> is the product of the <span class="texhtml mvar" style="font-style:italic;">n</span>-th and <span class="texhtml">(<i>n</i> + 1)</span>-th Fibonacci numbers. To see this, begin with a Fibonacci rectangle of size <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}\times F_{n+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>×</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}\times F_{n+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cb44dffb7e5fa0286f5dd41eb1e9a69b7a00afd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.367ex; height:2.509ex;" alt="{\displaystyle F_{n}\times F_{n+1}}"></span> and decompose it into squares of size <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n},F_{n-1},...,F_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n},F_{n-1},...,F_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96ffdb7c33371dc43890e95708d85ff6ddcfaba5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.279ex; height:2.509ex;" alt="{\displaystyle F_{n},F_{n-1},...,F_{1}}"></span>; from this the identity follows by comparing <a href="/wiki/Area" title="Area">areas</a>: </p><p><span typeof="mw:File/Frameless"><a href="/wiki/File:Fibonacci_Squares.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Fibonacci_Squares.svg/260px-Fibonacci_Squares.svg.png" decoding="async" width="260" height="161" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Fibonacci_Squares.svg/390px-Fibonacci_Squares.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/15/Fibonacci_Squares.svg/520px-Fibonacci_Squares.svg.png 2x" data-file-width="512" data-file-height="317" /></a></span> </p> <div class="mw-heading mw-heading3"><h3 id="Symbolic_method">Symbolic method</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=15" title="Edit section: Symbolic method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (F_{n})_{n\in \mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (F_{n})_{n\in \mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d217766d14654c8751bbcedfd13baefcd874ef1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.024ex; height:2.843ex;" alt="{\displaystyle (F_{n})_{n\in \mathbb {N} }}"></span> is also considered using the <a href="/wiki/Symbolic_method_(combinatorics)" title="Symbolic method (combinatorics)">symbolic method</a>.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> More precisely, this sequence corresponds to a <a href="/wiki/Specifiable_combinatorial_class" class="mw-redirect" title="Specifiable combinatorial class">specifiable combinatorial class</a>. The specification of this sequence is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Seq} ({\mathcal {Z+Z^{2}}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Seq</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">Z</mi> <mo class="MJX-tex-caligraphic" mathvariant="script">+</mo> <msup> <mi class="MJX-tex-caligraphic" mathvariant="script">Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">2</mn> </mrow> </msup> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Seq} ({\mathcal {Z+Z^{2}}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/006473f23bb1f07aac16e461aa7624934d9e847e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.851ex; height:2.843ex;" alt="{\displaystyle \operatorname {Seq} ({\mathcal {Z+Z^{2}}})}"></span>. Indeed, as stated above, the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-th Fibonacci number equals the number of <a href="/wiki/Composition_(combinatorics)" title="Composition (combinatorics)">combinatorial compositions</a> (ordered <a href="/wiki/Integer_partition" title="Integer partition">partitions</a>) of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}"></span> using terms 1 and 2. </p><p>It follows that the <a href="/wiki/Ordinary_generating_function" class="mw-redirect" title="Ordinary generating function">ordinary generating function</a> of the Fibonacci sequence, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=0}^{\infty }F_{i}z^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=0}^{\infty }F_{i}z^{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4ad50a8a3dd65b86a3385caa4ecfd30e2b3c96b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:7.926ex; height:6.843ex;" alt="{\displaystyle \sum _{i=0}^{\infty }F_{i}z^{i}}"></span>, is the <a href="/wiki/Rational_function" title="Rational function">rational function</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {z}{1-z-z^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>z</mi> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {z}{1-z-z^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b69f403752340c2419d446976709e0806304b074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.559ex; height:5.176ex;" alt="{\displaystyle {\frac {z}{1-z-z^{2}}}.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Induction_proofs">Induction proofs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=16" title="Edit section: Induction proofs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Fibonacci identities often can be easily proved using <a href="/wiki/Mathematical_induction" title="Mathematical induction">mathematical induction</a>. </p><p>For example, reconsider <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d35c36bbee08b20fe2cbd2708ccf16dcc5b04f6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.598ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1.}"></span> Adding <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bfbe34f204a6b7b01dd49571e6b287c2bdf7735" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.814ex; height:2.509ex;" alt="{\displaystyle F_{n+1}}"></span> to both sides gives </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}F_{i}+F_{n+1}=F_{n+1}+F_{n+2}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}F_{i}+F_{n+1}=F_{n+1}+F_{n+2}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f2641e2f9bbfb8cbb364883ae5a5f1751d69ff1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.259ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}F_{i}+F_{n+1}=F_{n+1}+F_{n+2}-1}"></span></dd></dl> <p>and so we have the formula for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a135e65a42f2d73cccbfc4569523996ca0036f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n+1}F_{i}=F_{n+3}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n+1}F_{i}=F_{n+3}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ae59ddee4467eb31b82821a8b8a059c4ac8cbcb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.951ex; height:7.343ex;" alt="{\displaystyle \sum _{i=1}^{n+1}F_{i}=F_{n+3}-1}"></span> </p><p>Similarly, add <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {F_{n+1}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {F_{n+1}}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7197da79ba73a67b86f387fe8ffa56edab7a87cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.868ex; height:3.009ex;" alt="{\displaystyle {F_{n+1}}^{2}}"></span> to both sides of <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}F_{i}^{2}=F_{n}F_{n+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}F_{i}^{2}=F_{n}F_{n+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b41a8a56e6c68130672f6477f44b89f900aafffe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.236ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}F_{i}^{2}=F_{n}F_{n+1}}"></span> to give <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}F_{i}^{2}+{F_{n+1}}^{2}=F_{n+1}\left(F_{n}+F_{n+1}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}F_{i}^{2}+{F_{n+1}}^{2}=F_{n+1}\left(F_{n}+F_{n+1}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc5c1103913db195272f22d4621f4fcbae10d188" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.795ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}F_{i}^{2}+{F_{n+1}}^{2}=F_{n+1}\left(F_{n}+F_{n+1}\right)}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n+1}F_{i}^{2}=F_{n+1}F_{n+2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n+1}F_{i}^{2}=F_{n+1}F_{n+2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5a96aeafb1c1fd0251b4bbeb14ca6da444c7a5c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.337ex; height:7.343ex;" alt="{\displaystyle \sum _{i=1}^{n+1}F_{i}^{2}=F_{n+1}F_{n+2}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Binet_formula_proofs">Binet formula proofs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=17" title="Edit section: Binet formula proofs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Binet formula is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {5}}F_{n}=\varphi ^{n}-\psi ^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {5}}F_{n}=\varphi ^{n}-\psi ^{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b436885595c4c0ffcf815a0ddad385b86bdcef85" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.867ex; height:3.009ex;" alt="{\displaystyle {\sqrt {5}}F_{n}=\varphi ^{n}-\psi ^{n}.}"></span> This can be used to prove Fibonacci identities. </p><p>For example, to prove that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6c2dfcb3f4762e5c4f8b5323583c0862d8b4a5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.95ex; height:3.176ex;" alt="{\textstyle \sum _{i=1}^{n}F_{i}=F_{n+2}-1}"></span> note that the left hand side multiplied by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b78ccdb7e18e02d4fc567c66aac99bf524acb5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:2.843ex;" alt="{\displaystyle {\sqrt {5}}}"></span> becomes <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}1+&\varphi +\varphi ^{2}+\dots +\varphi ^{n}-\left(1+\psi +\psi ^{2}+\dots +\psi ^{n}\right)\\&={\frac {\varphi ^{n+1}-1}{\varphi -1}}-{\frac {\psi ^{n+1}-1}{\psi -1}}\\&={\frac {\varphi ^{n+1}-1}{-\psi }}-{\frac {\psi ^{n+1}-1}{-\varphi }}\\&={\frac {-\varphi ^{n+2}+\varphi +\psi ^{n+2}-\psi }{\varphi \psi }}\\&=\varphi ^{n+2}-\psi ^{n+2}-(\varphi -\psi )\\&={\sqrt {5}}(F_{n+2}-1)\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mn>1</mn> <mo>+</mo> </mtd> <mtd> <mi>φ</mi> <mo>+</mo> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>ψ</mi> <mo>+</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>φ</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>ψ</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mo>−</mo> <mi>ψ</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mo>−</mo> <mi>φ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>φ</mi> <mo>+</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>ψ</mi> </mrow> <mrow> <mi>φ</mi> <mi>ψ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>−</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}1+&\varphi +\varphi ^{2}+\dots +\varphi ^{n}-\left(1+\psi +\psi ^{2}+\dots +\psi ^{n}\right)\\&={\frac {\varphi ^{n+1}-1}{\varphi -1}}-{\frac {\psi ^{n+1}-1}{\psi -1}}\\&={\frac {\varphi ^{n+1}-1}{-\psi }}-{\frac {\psi ^{n+1}-1}{-\varphi }}\\&={\frac {-\varphi ^{n+2}+\varphi +\psi ^{n+2}-\psi }{\varphi \psi }}\\&=\varphi ^{n+2}-\psi ^{n+2}-(\varphi -\psi )\\&={\sqrt {5}}(F_{n+2}-1)\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a3345c0b04b975a90f8e8af57b4e304a116ca2e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.905ex; margin-bottom: -0.266ex; width:48.829ex; height:29.509ex;" alt="{\displaystyle {\begin{aligned}1+&\varphi +\varphi ^{2}+\dots +\varphi ^{n}-\left(1+\psi +\psi ^{2}+\dots +\psi ^{n}\right)\\&={\frac {\varphi ^{n+1}-1}{\varphi -1}}-{\frac {\psi ^{n+1}-1}{\psi -1}}\\&={\frac {\varphi ^{n+1}-1}{-\psi }}-{\frac {\psi ^{n+1}-1}{-\varphi }}\\&={\frac {-\varphi ^{n+2}+\varphi +\psi ^{n+2}-\psi }{\varphi \psi }}\\&=\varphi ^{n+2}-\psi ^{n+2}-(\varphi -\psi )\\&={\sqrt {5}}(F_{n+2}-1)\\\end{aligned}}}"></span> as required, using the facts <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \varphi \psi =-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>φ</mi> <mi>ψ</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \varphi \psi =-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9ed2751085d1c14679f552b2bf4b4e434a38fb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.102ex; height:2.676ex;" alt="{\textstyle \varphi \psi =-1}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \varphi -\psi ={\sqrt {5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>φ</mi> <mo>−</mo> <mi>ψ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \varphi -\psi ={\sqrt {5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ef489f2b58745f57aadfd0a1bc704fdc4eeaa7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.07ex; height:3.176ex;" alt="{\textstyle \varphi -\psi ={\sqrt {5}}}"></span> to simplify the equations. </p> <div class="mw-heading mw-heading2"><h2 id="Other_identities">Other identities</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=18" title="Edit section: Other identities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Numerous other identities can be derived using various methods. Here are some of them:<sup id="cite_ref-MathWorld_37-0" class="reference"><a href="#cite_note-MathWorld-37"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Cassini's_and_Catalan's_identities"><span id="Cassini.27s_and_Catalan.27s_identities"></span>Cassini's and Catalan's identities</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=19" title="Edit section: Cassini's and Catalan's identities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Cassini_and_Catalan_identities" title="Cassini and Catalan identities">Cassini and Catalan identities</a></div> <p>Cassini's identity states that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {F_{n}}^{2}-F_{n+1}F_{n-1}=(-1)^{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {F_{n}}^{2}-F_{n+1}F_{n-1}=(-1)^{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d3b6b2f84510b00027e024e15dfa9b1e2d783c9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.432ex; height:3.176ex;" alt="{\displaystyle {F_{n}}^{2}-F_{n+1}F_{n-1}=(-1)^{n-1}}"></span> Catalan's identity is a generalization: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {F_{n}}^{2}-F_{n+r}F_{n-r}=(-1)^{n-r}{F_{r}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>r</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>r</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {F_{n}}^{2}-F_{n+r}F_{n-r}=(-1)^{n-r}{F_{r}}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc2cdb4060916b0bc252f441a430f36a9ff8456d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.713ex; height:3.176ex;" alt="{\displaystyle {F_{n}}^{2}-F_{n+r}F_{n-r}=(-1)^{n-r}{F_{r}}^{2}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="d'Ocagne's_identity"><span id="d.27Ocagne.27s_identity"></span>d'Ocagne's identity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=20" title="Edit section: d'Ocagne's identity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{m}F_{n+1}-F_{m+1}F_{n}=(-1)^{n}F_{m-n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{m}F_{n+1}-F_{m+1}F_{n}=(-1)^{n}F_{m-n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8aec9670ab0862f7ce7064cb63251cbe5d1d9b7a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.338ex; height:2.843ex;" alt="{\displaystyle F_{m}F_{n+1}-F_{m+1}F_{n}=(-1)^{n}F_{m-n}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{2n}={F_{n+1}}^{2}-{F_{n-1}}^{2}=F_{n}\left(F_{n+1}+F_{n-1}\right)=F_{n}L_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{2n}={F_{n+1}}^{2}-{F_{n-1}}^{2}=F_{n}\left(F_{n+1}+F_{n-1}\right)=F_{n}L_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6e3b5d6d8737d80ad7c28017e6dc83e72a2f180" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.298ex; height:3.176ex;" alt="{\displaystyle F_{2n}={F_{n+1}}^{2}-{F_{n-1}}^{2}=F_{n}\left(F_{n+1}+F_{n-1}\right)=F_{n}L_{n}}"></span> where <span class="texhtml"><i>L</i><sub><i>n</i></sub></span> is the <span class="texhtml mvar" style="font-style:italic;">n</span>-th <a href="/wiki/Lucas_number" title="Lucas number">Lucas number</a>. The last is an identity for doubling <span class="texhtml mvar" style="font-style:italic;">n</span>; other identities of this type are <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{3n}=2{F_{n}}^{3}+3F_{n}F_{n+1}F_{n-1}=5{F_{n}}^{3}+3(-1)^{n}F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>5</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{3n}=2{F_{n}}^{3}+3F_{n}F_{n+1}F_{n-1}=5{F_{n}}^{3}+3(-1)^{n}F_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6033f2fe48454cad2405702a0feedd046271b0b6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.649ex; height:3.176ex;" alt="{\displaystyle F_{3n}=2{F_{n}}^{3}+3F_{n}F_{n+1}F_{n-1}=5{F_{n}}^{3}+3(-1)^{n}F_{n}}"></span> by Cassini's identity. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{3n+1}={F_{n+1}}^{3}+3F_{n+1}{F_{n}}^{2}-{F_{n}}^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{3n+1}={F_{n+1}}^{3}+3F_{n+1}{F_{n}}^{2}-{F_{n}}^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b9f3b8cdf3f43bb248a2e5c8e65542274b65a69" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.793ex; height:3.009ex;" alt="{\displaystyle F_{3n+1}={F_{n+1}}^{3}+3F_{n+1}{F_{n}}^{2}-{F_{n}}^{3}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{3n+2}={F_{n+1}}^{3}+3{F_{n+1}}^{2}F_{n}+{F_{n}}^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{3n+2}={F_{n+1}}^{3}+3{F_{n+1}}^{2}F_{n}+{F_{n}}^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31b6a472518de34c8dd38f662352f2e3889ffb00" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.793ex; height:3.009ex;" alt="{\displaystyle F_{3n+2}={F_{n+1}}^{3}+3{F_{n+1}}^{2}F_{n}+{F_{n}}^{3}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{4n}=4F_{n}F_{n+1}\left({F_{n+1}}^{2}+2{F_{n}}^{2}\right)-3{F_{n}}^{2}\left({F_{n}}^{2}+2{F_{n+1}}^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>4</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>3</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{4n}=4F_{n}F_{n+1}\left({F_{n+1}}^{2}+2{F_{n}}^{2}\right)-3{F_{n}}^{2}\left({F_{n}}^{2}+2{F_{n+1}}^{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88718bc2c80218dd63e07f8673ba6bc1d410624d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:55.403ex; height:3.343ex;" alt="{\displaystyle F_{4n}=4F_{n}F_{n+1}\left({F_{n+1}}^{2}+2{F_{n}}^{2}\right)-3{F_{n}}^{2}\left({F_{n}}^{2}+2{F_{n+1}}^{2}\right)}"></span> These can be found experimentally using <a href="/wiki/Lattice_reduction" title="Lattice reduction">lattice reduction</a>, and are useful in setting up the <a href="/wiki/Special_number_field_sieve" title="Special number field sieve">special number field sieve</a> to <a href="/wiki/Factorization" title="Factorization">factorize</a> a Fibonacci number. </p><p>More generally,<sup id="cite_ref-MathWorld_37-1" class="reference"><a href="#cite_note-MathWorld-37"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{kn+c}=\sum _{i=0}^{k}{k \choose i}F_{c-i}{F_{n}}^{i}{F_{n+1}}^{k-i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>n</mi> <mo>+</mo> <mi>c</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>k</mi> <mi>i</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo>−</mo> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mi>i</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{kn+c}=\sum _{i=0}^{k}{k \choose i}F_{c-i}{F_{n}}^{i}{F_{n+1}}^{k-i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d8c08c41b06d1e2f7b9f03561a5d5536ca7417e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.226ex; height:7.343ex;" alt="{\displaystyle F_{kn+c}=\sum _{i=0}^{k}{k \choose i}F_{c-i}{F_{n}}^{i}{F_{n+1}}^{k-i}.}"></span> </p><p>or alternatively </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{kn+c}=\sum _{i=0}^{k}{k \choose i}F_{c+i}{F_{n}}^{i}{F_{n-1}}^{k-i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>n</mi> <mo>+</mo> <mi>c</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>k</mi> <mi>i</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo>+</mo> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mi>i</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{kn+c}=\sum _{i=0}^{k}{k \choose i}F_{c+i}{F_{n}}^{i}{F_{n-1}}^{k-i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f887b0401601b91cadc9e622ca53adc402f5a165" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.226ex; height:7.343ex;" alt="{\displaystyle F_{kn+c}=\sum _{i=0}^{k}{k \choose i}F_{c+i}{F_{n}}^{i}{F_{n-1}}^{k-i}.}"></span> </p><p>Putting <span class="texhtml"><i>k</i> = 2</span> in this formula, one gets again the formulas of the end of above section <a href="#Matrix_form">Matrix form</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Generating_function">Generating function</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=21" title="Edit section: Generating function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Generating_function" title="Generating function">generating function</a> of the Fibonacci sequence is the <a href="/wiki/Power_series" title="Power series">power series</a> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(z)=\sum _{k=0}^{\infty }F_{k}z^{k}=0+z+z^{2}+2z^{3}+3z^{4}+5z^{5}+\dots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> <mo>+</mo> <mi>z</mi> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>5</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mo>…</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(z)=\sum _{k=0}^{\infty }F_{k}z^{k}=0+z+z^{2}+2z^{3}+3z^{4}+5z^{5}+\dots .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/582b282895fccd248a49e6d9c3a5ec3110b14afa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:53.805ex; height:7.009ex;" alt="{\displaystyle s(z)=\sum _{k=0}^{\infty }F_{k}z^{k}=0+z+z^{2}+2z^{3}+3z^{4}+5z^{5}+\dots .}"></span> </p><p>This series is convergent for any <a href="/wiki/Complex_number" title="Complex number">complex number</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> satisfying <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|<1/\varphi \approx 0.618,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>φ</mi> <mo>≈</mo> <mn>0.618</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|<1/\varphi \approx 0.618,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ce62aaa0b60075574b8a9506e1dd2f0fdf4392e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.367ex; height:2.843ex;" alt="{\displaystyle |z|<1/\varphi \approx 0.618,}"></span> and its sum has a simple closed form:<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(z)={\frac {z}{1-z-z^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>z</mi> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(z)={\frac {z}{1-z-z^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29025ca1c45e39c40263294973a31a46255bf4ee" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.645ex; height:5.176ex;" alt="{\displaystyle s(z)={\frac {z}{1-z-z^{2}}}.}"></span> </p><p>This can be proved by multiplying by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (1-z-z^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (1-z-z^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca577c4a9a39370f2542f0db98c7cdba2b471414" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.885ex; height:3.009ex;" alt="{\textstyle (1-z-z^{2})}"></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(1-z-z^{2})s(z)&=\sum _{k=0}^{\infty }F_{k}z^{k}-\sum _{k=0}^{\infty }F_{k}z^{k+1}-\sum _{k=0}^{\infty }F_{k}z^{k+2}\\&=\sum _{k=0}^{\infty }F_{k}z^{k}-\sum _{k=1}^{\infty }F_{k-1}z^{k}-\sum _{k=2}^{\infty }F_{k-2}z^{k}\\&=0z^{0}+1z^{1}-0z^{1}+\sum _{k=2}^{\infty }(F_{k}-F_{k-1}-F_{k-2})z^{k}\\&=z,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>0</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>z</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(1-z-z^{2})s(z)&=\sum _{k=0}^{\infty }F_{k}z^{k}-\sum _{k=0}^{\infty }F_{k}z^{k+1}-\sum _{k=0}^{\infty }F_{k}z^{k+2}\\&=\sum _{k=0}^{\infty }F_{k}z^{k}-\sum _{k=1}^{\infty }F_{k-1}z^{k}-\sum _{k=2}^{\infty }F_{k-2}z^{k}\\&=0z^{0}+1z^{1}-0z^{1}+\sum _{k=2}^{\infty }(F_{k}-F_{k-1}-F_{k-2})z^{k}\\&=z,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/687b0d1fc32c8ed8cfd0392a920fe1eff59a622f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.671ex; width:63.141ex; height:24.343ex;" alt="{\displaystyle {\begin{aligned}(1-z-z^{2})s(z)&=\sum _{k=0}^{\infty }F_{k}z^{k}-\sum _{k=0}^{\infty }F_{k}z^{k+1}-\sum _{k=0}^{\infty }F_{k}z^{k+2}\\&=\sum _{k=0}^{\infty }F_{k}z^{k}-\sum _{k=1}^{\infty }F_{k-1}z^{k}-\sum _{k=2}^{\infty }F_{k-2}z^{k}\\&=0z^{0}+1z^{1}-0z^{1}+\sum _{k=2}^{\infty }(F_{k}-F_{k-1}-F_{k-2})z^{k}\\&=z,\end{aligned}}}"></span> </p><p>where all terms involving <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/152bec16756f5bbf6fa6038d8f8d022923cb7b06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.179ex; height:2.676ex;" alt="{\displaystyle z^{k}}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\geq 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c797a67c0a51167d373c013a9a020f4568a11754" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.472ex; height:2.343ex;" alt="{\displaystyle k\geq 2}"></span> cancel out because of the defining Fibonacci recurrence relation. </p><p>The <a href="/wiki/Partial_fraction_decomposition" title="Partial fraction decomposition">partial fraction decomposition</a> is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(z)={\frac {1}{\sqrt {5}}}\left({\frac {1}{1-\varphi z}}-{\frac {1}{1-\psi z}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>5</mn> </msqrt> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>φ</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>ψ</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(z)={\frac {1}{\sqrt {5}}}\left({\frac {1}{1-\varphi z}}-{\frac {1}{1-\psi z}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/520e18661f1c3c47f389d2fd1b179d5bdbdb5268" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:32.557ex; height:6.509ex;" alt="{\displaystyle s(z)={\frac {1}{\sqrt {5}}}\left({\frac {1}{1-\varphi z}}-{\frac {1}{1-\psi z}}\right)}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \varphi ={\tfrac {1}{2}}\left(1+{\sqrt {5}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \varphi ={\tfrac {1}{2}}\left(1+{\sqrt {5}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b287380393d68ac21a9b14367d36530b4685c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.895ex; height:3.509ex;" alt="{\textstyle \varphi ={\tfrac {1}{2}}\left(1+{\sqrt {5}}\right)}"></span> is the golden ratio and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ={\tfrac {1}{2}}\left(1-{\sqrt {5}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ={\tfrac {1}{2}}\left(1-{\sqrt {5}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57a0e2b9cc155c9737fea8786161f3422f432849" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.888ex; height:3.509ex;" alt="{\displaystyle \psi ={\tfrac {1}{2}}\left(1-{\sqrt {5}}\right)}"></span> is its <a href="/wiki/Conjugate_(square_roots)" title="Conjugate (square roots)">conjugate</a>. </p><p>The related function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle z\mapsto -s\left(-1/z\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>z</mi> <mo stretchy="false">↦</mo> <mo>−</mo> <mi>s</mi> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>z</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle z\mapsto -s\left(-1/z\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d91eb017da4a8e875ba62889165897bd8ee40ade" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.018ex; height:2.843ex;" alt="{\textstyle z\mapsto -s\left(-1/z\right)}"></span> is the generating function for the <a href="/wiki/Negafibonacci" class="mw-redirect" title="Negafibonacci">negafibonacci</a> numbers, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bf6ea9e3135132b01ef408198af8a1470edfe15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.988ex; height:2.843ex;" alt="{\displaystyle s(z)}"></span> satisfies the <a href="/wiki/Functional_equation" title="Functional equation">functional equation</a> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(z)=s\!\left(-{\frac {1}{z}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>s</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(z)=s\!\left(-{\frac {1}{z}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eff5b62bfe172d956c574ba6ccac986cfaeb1aee" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.439ex; height:6.176ex;" alt="{\displaystyle s(z)=s\!\left(-{\frac {1}{z}}\right).}"></span> </p><p>Using <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> equal to any of 0.01, 0.001, 0.0001, etc. lays out the first Fibonacci numbers in the decimal expansion of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bf6ea9e3135132b01ef408198af8a1470edfe15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.988ex; height:2.843ex;" alt="{\displaystyle s(z)}"></span>. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(0.001)={\frac {0.001}{0.998999}}={\frac {1000}{998999}}=0.001001002003005008013021\ldots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mn>0.001</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>0.001</mn> <mn>0.998999</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1000</mn> <mn>998999</mn> </mfrac> </mrow> <mo>=</mo> <mn>0.001001002003005008013021</mn> <mo>…</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(0.001)={\frac {0.001}{0.998999}}={\frac {1000}{998999}}=0.001001002003005008013021\ldots .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71b1cfea70bdd62a2e447aef380cdc233095ade8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:68.775ex; height:5.176ex;" alt="{\displaystyle s(0.001)={\frac {0.001}{0.998999}}={\frac {1000}{998999}}=0.001001002003005008013021\ldots .}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Reciprocal_sums">Reciprocal sums</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=22" title="Edit section: Reciprocal sums"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Infinite sums over <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">reciprocal</a> Fibonacci numbers can sometimes be evaluated in terms of <a href="/wiki/Theta_function" title="Theta function">theta functions</a>. For example, the sum of every odd-indexed reciprocal Fibonacci number can be written as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{F_{2k-1}}}={\frac {\sqrt {5}}{4}}\;\vartheta _{2}\!\left(0,{\frac {3-{\sqrt {5}}}{2}}\right)^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>5</mn> </msqrt> <mn>4</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="negativethinmathspace" /> <msup> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{F_{2k-1}}}={\frac {\sqrt {5}}{4}}\;\vartheta _{2}\!\left(0,{\frac {3-{\sqrt {5}}}{2}}\right)^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09907bb47e3739f7d0f54e6929861a2c192ab59e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.059ex; height:7.343ex;" alt="{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{F_{2k-1}}}={\frac {\sqrt {5}}{4}}\;\vartheta _{2}\!\left(0,{\frac {3-{\sqrt {5}}}{2}}\right)^{2},}"></span> </p><p>and the sum of squared reciprocal Fibonacci numbers as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{{F_{k}}^{2}}}={\frac {5}{24}}\!\left(\vartheta _{2}\!\left(0,{\frac {3-{\sqrt {5}}}{2}}\right)^{4}-\vartheta _{4}\!\left(0,{\frac {3-{\sqrt {5}}}{2}}\right)^{4}+1\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>24</mn> </mfrac> </mrow> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="negativethinmathspace" /> <msup> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <msub> <mi>ϑ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mspace width="negativethinmathspace" /> <msup> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{{F_{k}}^{2}}}={\frac {5}{24}}\!\left(\vartheta _{2}\!\left(0,{\frac {3-{\sqrt {5}}}{2}}\right)^{4}-\vartheta _{4}\!\left(0,{\frac {3-{\sqrt {5}}}{2}}\right)^{4}+1\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e6d516f46bb35c751c35917bd2413985f4d603b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:59.334ex; height:7.509ex;" alt="{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{{F_{k}}^{2}}}={\frac {5}{24}}\!\left(\vartheta _{2}\!\left(0,{\frac {3-{\sqrt {5}}}{2}}\right)^{4}-\vartheta _{4}\!\left(0,{\frac {3-{\sqrt {5}}}{2}}\right)^{4}+1\right).}"></span> </p><p>If we add 1 to each Fibonacci number in the first sum, there is also the closed form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{1+F_{2k-1}}}={\frac {\sqrt {5}}{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>5</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{1+F_{2k-1}}}={\frac {\sqrt {5}}{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/776f057aaf77aad7a7524d524a6f1f80cab8a775" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.767ex; height:7.009ex;" alt="{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{1+F_{2k-1}}}={\frac {\sqrt {5}}{2}},}"></span> </p><p>and there is a <i>nested</i> sum of squared Fibonacci numbers giving the reciprocal of the <a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{\sum _{j=1}^{k}{F_{j}}^{2}}}={\frac {{\sqrt {5}}-1}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{\sum _{j=1}^{k}{F_{j}}^{2}}}={\frac {{\sqrt {5}}-1}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/730936940fc9ae723e0d8342dd840c66517aa796" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:25.571ex; height:7.509ex;" alt="{\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{\sum _{j=1}^{k}{F_{j}}^{2}}}={\frac {{\sqrt {5}}-1}{2}}.}"></span> </p><p>The sum of all even-indexed reciprocal Fibonacci numbers is<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{F_{2k}}}={\sqrt {5}}\left(L(\psi ^{2})-L(\psi ^{4})\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>−</mo> <mi>L</mi> <mo stretchy="false">(</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{F_{2k}}}={\sqrt {5}}\left(L(\psi ^{2})-L(\psi ^{4})\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c64d80a6a7361ac14696af0ac8e1c2425b664beb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.457ex; height:6.843ex;" alt="{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{F_{2k}}}={\sqrt {5}}\left(L(\psi ^{2})-L(\psi ^{4})\right)}"></span> with the <a href="/wiki/Lambert_series" title="Lambert series">Lambert series</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle L(q):=\sum _{k=1}^{\infty }{\frac {q^{k}}{1-q^{k}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle L(q):=\sum _{k=1}^{\infty }{\frac {q^{k}}{1-q^{k}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/136916fcbd27ca5379d3b54f4e05b7609fa5c6e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.444ex; height:5.009ex;" alt="{\displaystyle \textstyle L(q):=\sum _{k=1}^{\infty }{\frac {q^{k}}{1-q^{k}}},}"></span> since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {1}{F_{2k}}}={\sqrt {5}}\left({\frac {\psi ^{2k}}{1-\psi ^{2k}}}-{\frac {\psi ^{4k}}{1-\psi ^{4k}}}\right)\!.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>k</mi> </mrow> </msup> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="negativethinmathspace" /> <mo>.</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {1}{F_{2k}}}={\sqrt {5}}\left({\frac {\psi ^{2k}}{1-\psi ^{2k}}}-{\frac {\psi ^{4k}}{1-\psi ^{4k}}}\right)\!.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/745a38c05a7430bde48a6bb6d95b6bd7cf305bf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.979ex; height:6.176ex;" alt="{\displaystyle \textstyle {\frac {1}{F_{2k}}}={\sqrt {5}}\left({\frac {\psi ^{2k}}{1-\psi ^{2k}}}-{\frac {\psi ^{4k}}{1-\psi ^{4k}}}\right)\!.}"></span> </p><p>So the <a href="/wiki/Reciprocal_Fibonacci_constant" title="Reciprocal Fibonacci constant">reciprocal Fibonacci constant</a> is<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{F_{k}}}=\sum _{k=1}^{\infty }{\frac {1}{F_{2k-1}}}+\sum _{k=1}^{\infty }{\frac {1}{F_{2k}}}=3.359885666243\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mn>3.359885666243</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{F_{k}}}=\sum _{k=1}^{\infty }{\frac {1}{F_{2k-1}}}+\sum _{k=1}^{\infty }{\frac {1}{F_{2k}}}=3.359885666243\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/957ee1d557b8c180f202fc7dee46ab7c2021341b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:53.135ex; height:6.843ex;" alt="{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{F_{k}}}=\sum _{k=1}^{\infty }{\frac {1}{F_{2k-1}}}+\sum _{k=1}^{\infty }{\frac {1}{F_{2k}}}=3.359885666243\dots }"></span> </p><p>Moreover, this number has been proved <a href="/wiki/Irrational_number" title="Irrational number">irrational</a> by <a href="/w/index.php?title=Richard_Andr%C3%A9-Jeannin&action=edit&redlink=1" class="new" title="Richard André-Jeannin (page does not exist)">Richard André-Jeannin</a>.<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> </p><p><b>Millin's series</b> gives the identity<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=0}^{\infty }{\frac {1}{F_{2^{k}}}}={\frac {7-{\sqrt {5}}}{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>7</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{\infty }{\frac {1}{F_{2^{k}}}}={\frac {7-{\sqrt {5}}}{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c680173fdafa8cb3ec8b4a3915d6f3fdedfe139" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.669ex; height:7.176ex;" alt="{\displaystyle \sum _{k=0}^{\infty }{\frac {1}{F_{2^{k}}}}={\frac {7-{\sqrt {5}}}{2}},}"></span> which follows from the closed form for its partial sums as <span class="texhtml mvar" style="font-style:italic;">N</span> tends to infinity: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=0}^{N}{\frac {1}{F_{2^{k}}}}=3-{\frac {F_{2^{N}-1}}{F_{2^{N}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mn>3</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{N}{\frac {1}{F_{2^{k}}}}=3-{\frac {F_{2^{N}-1}}{F_{2^{N}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24df36d67580a0bac659dd5765d59fb08c18a4f6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.569ex; height:7.509ex;" alt="{\displaystyle \sum _{k=0}^{N}{\frac {1}{F_{2^{k}}}}=3-{\frac {F_{2^{N}-1}}{F_{2^{N}}}}.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Primes_and_divisibility">Primes and divisibility</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=23" title="Edit section: Primes and divisibility"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Divisibility_properties">Divisibility properties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=24" title="Edit section: Divisibility properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Every third number of the sequence is even (a multiple of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{3}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{3}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caf4ec3fa730d75d5508487f92ebe0a64307a4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{3}=2}"></span>) and, more generally, every <span class="texhtml mvar" style="font-style:italic;">k</span>-th number of the sequence is a multiple of <i>F<sub>k</sub></i>. Thus the Fibonacci sequence is an example of a <a href="/wiki/Divisibility_sequence" title="Divisibility sequence">divisibility sequence</a>. In fact, the Fibonacci sequence satisfies the stronger divisibility property<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gcd(F_{a},F_{b},F_{c},\ldots )=F_{\gcd(a,b,c,\ldots )}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gcd(F_{a},F_{b},F_{c},\ldots )=F_{\gcd(a,b,c,\ldots )}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4d40f8f75eed59cdacf83ed8c9cecd852e65835" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:33.132ex; height:3.176ex;" alt="{\displaystyle \gcd(F_{a},F_{b},F_{c},\ldots )=F_{\gcd(a,b,c,\ldots )}\,}"></span> where <span class="texhtml">gcd</span> is the <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> function. (This relation is different if a different indexing convention is used, such as the one that starts the sequence with <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{0}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{0}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f734b6be2347aa83cbad3145e254bb2b15d9c15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{0}=1}"></span>⁠</span> and <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c374ba08c140de90c6cbb4c9b9fcd26e3f99ef56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{1}=1}"></span>⁠</span>.) </p><p>In particular, any three consecutive Fibonacci numbers are pairwise <a href="/wiki/Coprime_integers" title="Coprime integers">coprime</a> because both <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c374ba08c140de90c6cbb4c9b9fcd26e3f99ef56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{1}=1}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf100a6879452aff05a6027ad4f36029f360dcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{2}=1}"></span>. That is, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gcd(F_{n},F_{n+1})=\gcd(F_{n},F_{n+2})=\gcd(F_{n+1},F_{n+2})=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gcd(F_{n},F_{n+1})=\gcd(F_{n},F_{n+2})=\gcd(F_{n+1},F_{n+2})=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ecd39398dcf2dacc9e8c15876f47b4f8379e3e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.13ex; height:2.843ex;" alt="{\displaystyle \gcd(F_{n},F_{n+1})=\gcd(F_{n},F_{n+2})=\gcd(F_{n+1},F_{n+2})=1}"></span></dd></dl> <p>for every <span class="texhtml mvar" style="font-style:italic;">n</span>. </p><p>Every <a href="/wiki/Prime_number" title="Prime number">prime number</a> <span class="texhtml mvar" style="font-style:italic;">p</span> divides a Fibonacci number that can be determined by the value of <span class="texhtml mvar" style="font-style:italic;">p</span> <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modulo</a> 5. If <span class="texhtml mvar" style="font-style:italic;">p</span> is congruent to 1 or 4 modulo 5, then <span class="texhtml mvar" style="font-style:italic;">p</span> divides <span class="texhtml"><i>F</i><sub><i>p</i>−1</sub></span>, and if <span class="texhtml mvar" style="font-style:italic;">p</span> is congruent to 2 or 3 modulo 5, then, <span class="texhtml mvar" style="font-style:italic;">p</span> divides <span class="texhtml"><i>F</i><sub><i>p</i>+1</sub></span>. The remaining case is that <span class="texhtml"><i>p</i> = 5</span>, and in this case <span class="texhtml mvar" style="font-style:italic;">p</span> divides <i>F<sub>p</sub></i>. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}p=5&\Rightarrow p\mid F_{p},\\p\equiv \pm 1{\pmod {5}}&\Rightarrow p\mid F_{p-1},\\p\equiv \pm 2{\pmod {5}}&\Rightarrow p\mid F_{p+1}.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>p</mi> <mo>=</mo> <mn>5</mn> </mtd> <mtd> <mo stretchy="false">⇒</mo> <mi>p</mi> <mo>∣</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>p</mi> <mo>≡</mo> <mo>±</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>5</mn> <mo stretchy="false">)</mo> </mrow> </mtd> <mtd> <mo stretchy="false">⇒</mo> <mi>p</mi> <mo>∣</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>p</mi> <mo>≡</mo> <mo>±</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>5</mn> <mo stretchy="false">)</mo> </mrow> </mtd> <mtd> <mo stretchy="false">⇒</mo> <mi>p</mi> <mo>∣</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}p=5&\Rightarrow p\mid F_{p},\\p\equiv \pm 1{\pmod {5}}&\Rightarrow p\mid F_{p-1},\\p\equiv \pm 2{\pmod {5}}&\Rightarrow p\mid F_{p+1}.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87652c32e3edec329520b4db9e5e6e05131ce05a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:33.311ex; height:9.176ex;" alt="{\displaystyle {\begin{cases}p=5&\Rightarrow p\mid F_{p},\\p\equiv \pm 1{\pmod {5}}&\Rightarrow p\mid F_{p-1},\\p\equiv \pm 2{\pmod {5}}&\Rightarrow p\mid F_{p+1}.\end{cases}}}"></span> </p><p>These cases can be combined into a single, non-<a href="/wiki/Piecewise" class="mw-redirect" title="Piecewise">piecewise</a> formula, using the <a href="/wiki/Legendre_symbol" title="Legendre symbol">Legendre symbol</a>:<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\mid F_{p\;-\,\left({\frac {5}{p}}\right)}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∣</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mspace width="thickmathspace" /> <mo>−</mo> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mi>p</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\mid F_{p\;-\,\left({\frac {5}{p}}\right)}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4621ed23ef47509950667d1f4e113f73bdd416e5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-left: -0.089ex; width:12.177ex; height:4.343ex;" alt="{\displaystyle p\mid F_{p\;-\,\left({\frac {5}{p}}\right)}.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Primality_testing">Primality testing</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=25" title="Edit section: Primality testing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The above formula can be used as a <a href="/wiki/Primality_test" title="Primality test">primality test</a> in the sense that if <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\mid F_{n\;-\,\left({\frac {5}{n}}\right)},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∣</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mspace width="thickmathspace" /> <mo>−</mo> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\mid F_{n\;-\,\left({\frac {5}{n}}\right)},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd01546ee964e87a4d9ea8306f06093af6908e93" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.602ex; height:4.343ex;" alt="{\displaystyle n\mid F_{n\;-\,\left({\frac {5}{n}}\right)},}"></span> where the Legendre symbol has been replaced by the <a href="/wiki/Jacobi_symbol" title="Jacobi symbol">Jacobi symbol</a>, then this is evidence that <span class="texhtml mvar" style="font-style:italic;">n</span> is a prime, and if it fails to hold, then <span class="texhtml mvar" style="font-style:italic;">n</span> is definitely not a prime. If <span class="texhtml mvar" style="font-style:italic;">n</span> is <a href="/wiki/Composite_number" title="Composite number">composite</a> and satisfies the formula, then <span class="texhtml mvar" style="font-style:italic;">n</span> is a <i>Fibonacci pseudoprime</i>. When <span class="texhtml mvar" style="font-style:italic;">m</span> is large – say a 500-<a href="/wiki/Bit" title="Bit">bit</a> number – then we can calculate <span class="texhtml"><i>F</i><sub><i>m</i></sub> (mod <i>n</i>)</span> efficiently using the matrix form. Thus </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}F_{m+1}&F_{m}\\F_{m}&F_{m-1}\end{pmatrix}}\equiv {\begin{pmatrix}1&1\\1&0\end{pmatrix}}^{m}{\pmod {n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>≡</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}F_{m+1}&F_{m}\\F_{m}&F_{m-1}\end{pmatrix}}\equiv {\begin{pmatrix}1&1\\1&0\end{pmatrix}}^{m}{\pmod {n}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39b584c6192d5356fd40279d94b43964b9f1872e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.227ex; margin-bottom: -0.278ex; width:42.355ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}F_{m+1}&F_{m}\\F_{m}&F_{m-1}\end{pmatrix}}\equiv {\begin{pmatrix}1&1\\1&0\end{pmatrix}}^{m}{\pmod {n}}.}"></span> Here the matrix power <span class="texhtml"><i>A</i><sup><i>m</i></sup></span> is calculated using <a href="/wiki/Modular_exponentiation" title="Modular exponentiation">modular exponentiation</a>, which can be <a href="/wiki/Modular_exponentiation#Matrices" title="Modular exponentiation">adapted to matrices</a>.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Fibonacci_primes">Fibonacci primes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=26" title="Edit section: Fibonacci primes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Fibonacci_prime" title="Fibonacci prime">Fibonacci prime</a></div> <p>A <i>Fibonacci prime</i> is a Fibonacci number that is <a href="/wiki/Prime_number" title="Prime number">prime</a>. The first few are:<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> </p> <dl><dd>2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ...</dd></dl> <p>Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many.<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> </p><p><span class="texhtml"><i>F</i><sub><i>kn</i></sub></span> is divisible by <span class="texhtml"><i>F</i><sub><i>n</i></sub></span>, so, apart from <span class="texhtml"><i>F</i><sub>4</sub> = 3</span>, any Fibonacci prime must have a prime index. As there are <a href="/wiki/Arbitrarily_large" title="Arbitrarily large">arbitrarily long</a> runs of <a href="/wiki/Composite_number" title="Composite number">composite numbers</a>, there are therefore also arbitrarily long runs of composite Fibonacci numbers. </p><p>No Fibonacci number greater than <span class="texhtml"><i>F</i><sub>6</sub> = 8</span> is one greater or one less than a prime number.<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> </p><p>The only nontrivial <a href="/wiki/Square_number" title="Square number">square</a> Fibonacci number is 144.<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> Attila Pethő proved in 2001 that there is only a finite number of <a href="/wiki/Perfect_power" title="Perfect power">perfect power</a> Fibonacci numbers.<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that 8 and 144 are the only such non-trivial perfect powers.<sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> </p><p>1, 3, 21, and 55 are the only <a href="/wiki/Triangular_number" title="Triangular number">triangular</a> Fibonacci numbers, which was <a href="/wiki/Conjecture" title="Conjecture">conjectured</a> by <a href="/wiki/Verner_Emil_Hoggatt_Jr." title="Verner Emil Hoggatt Jr.">Vern Hoggatt</a> and proved by Luo Ming.<sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> </p><p>No Fibonacci number can be a <a href="/wiki/Perfect_number" title="Perfect number">perfect number</a>.<sup id="cite_ref-Luca2000_54-0" class="reference"><a href="#cite_note-Luca2000-54"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> More generally, no Fibonacci number other than 1 can be <a href="/wiki/Multiply_perfect_number" title="Multiply perfect number">multiply perfect</a>,<sup id="cite_ref-BGLLHT2011_55-0" class="reference"><a href="#cite_note-BGLLHT2011-55"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> and no ratio of two Fibonacci numbers can be perfect.<sup id="cite_ref-LucaMH2010_56-0" class="reference"><a href="#cite_note-LucaMH2010-56"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Prime_divisors">Prime divisors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=27" title="Edit section: Prime divisors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>With the exceptions of 1, 8 and 144 (<span class="texhtml"><i>F</i><sub>1</sub> = <i>F</i><sub>2</sub></span>, <span class="texhtml"><i>F</i><sub>6</sub></span> and <span class="texhtml"><i>F</i><sub>12</sub></span>) every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number (<a href="/wiki/Carmichael%27s_theorem" title="Carmichael's theorem">Carmichael's theorem</a>).<sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> As a result, 8 and 144 (<span class="texhtml"><i>F</i><sub>6</sub></span> and <span class="texhtml"><i>F</i><sub>12</sub></span>) are the only Fibonacci numbers that are the product of other Fibonacci numbers.<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> </p><p>The divisibility of Fibonacci numbers by a prime <span class="texhtml mvar" style="font-style:italic;">p</span> is related to the <a href="/wiki/Legendre_symbol" title="Legendre symbol">Legendre symbol</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigl (}{\tfrac {p}{5}}{\bigr )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>p</mi> <mn>5</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl (}{\tfrac {p}{5}}{\bigr )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9b0e7b45bf9bb5f19aaa7a8ad12697c32ae79af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.793ex; height:3.676ex;" alt="{\displaystyle {\bigl (}{\tfrac {p}{5}}{\bigr )}}"></span> which is evaluated as follows: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {p}{5}}\right)={\begin{cases}0&{\text{if }}p=5\\1&{\text{if }}p\equiv \pm 1{\pmod {5}}\\-1&{\text{if }}p\equiv \pm 2{\pmod {5}}.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mn>5</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>p</mi> <mo>=</mo> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>p</mi> <mo>≡</mo> <mo>±</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>5</mn> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>p</mi> <mo>≡</mo> <mo>±</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>5</mn> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {p}{5}}\right)={\begin{cases}0&{\text{if }}p=5\\1&{\text{if }}p\equiv \pm 1{\pmod {5}}\\-1&{\text{if }}p\equiv \pm 2{\pmod {5}}.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0ba91d5e0f4f8a921b1825d4fda8cee261cd04" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:35.371ex; height:8.509ex;" alt="{\displaystyle \left({\frac {p}{5}}\right)={\begin{cases}0&{\text{if }}p=5\\1&{\text{if }}p\equiv \pm 1{\pmod {5}}\\-1&{\text{if }}p\equiv \pm 2{\pmod {5}}.\end{cases}}}"></span> </p><p>If <span class="texhtml mvar" style="font-style:italic;">p</span> is a prime number then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{p}\equiv \left({\frac {p}{5}}\right){\pmod {p}}\quad {\text{and}}\quad F_{p-\left({\frac {p}{5}}\right)}\equiv 0{\pmod {p}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>≡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mn>5</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mn>5</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </msub> <mo>≡</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{p}\equiv \left({\frac {p}{5}}\right){\pmod {p}}\quad {\text{and}}\quad F_{p-\left({\frac {p}{5}}\right)}\equiv 0{\pmod {p}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d8fc1f15634c511b2d7f2c06dca346d8b8e3acc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:53.13ex; height:5.343ex;" alt="{\displaystyle F_{p}\equiv \left({\frac {p}{5}}\right){\pmod {p}}\quad {\text{and}}\quad F_{p-\left({\frac {p}{5}}\right)}\equiv 0{\pmod {p}}.}"></span><sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTELemmermeyer200073–74ex._2.25–28_60-0" class="reference"><a href="#cite_note-FOOTNOTELemmermeyer200073–74ex._2.25–28-60"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup> </p><p>For example, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\bigl (}{\tfrac {2}{5}}{\bigr )}&=-1,&F_{3}&=2,&F_{2}&=1,\\{\bigl (}{\tfrac {3}{5}}{\bigr )}&=-1,&F_{4}&=3,&F_{3}&=2,\\{\bigl (}{\tfrac {5}{5}}{\bigr )}&=0,&F_{5}&=5,\\{\bigl (}{\tfrac {7}{5}}{\bigr )}&=-1,&F_{8}&=21,&F_{7}&=13,\\{\bigl (}{\tfrac {11}{5}}{\bigr )}&=+1,&F_{10}&=55,&F_{11}&=89.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> </mtd> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mtd> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> </mtd> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> </mtd> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>5</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>7</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> </mtd> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>21</mn> <mo>,</mo> </mtd> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>13</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>11</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>+</mo> <mn>1</mn> <mo>,</mo> </mtd> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>55</mn> <mo>,</mo> </mtd> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>89.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\bigl (}{\tfrac {2}{5}}{\bigr )}&=-1,&F_{3}&=2,&F_{2}&=1,\\{\bigl (}{\tfrac {3}{5}}{\bigr )}&=-1,&F_{4}&=3,&F_{3}&=2,\\{\bigl (}{\tfrac {5}{5}}{\bigr )}&=0,&F_{5}&=5,\\{\bigl (}{\tfrac {7}{5}}{\bigr )}&=-1,&F_{8}&=21,&F_{7}&=13,\\{\bigl (}{\tfrac {11}{5}}{\bigr )}&=+1,&F_{10}&=55,&F_{11}&=89.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afa5e389583187fdc06ddd4d67aa00353a6094de" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:40.25ex; height:19.176ex;" alt="{\displaystyle {\begin{aligned}{\bigl (}{\tfrac {2}{5}}{\bigr )}&=-1,&F_{3}&=2,&F_{2}&=1,\\{\bigl (}{\tfrac {3}{5}}{\bigr )}&=-1,&F_{4}&=3,&F_{3}&=2,\\{\bigl (}{\tfrac {5}{5}}{\bigr )}&=0,&F_{5}&=5,\\{\bigl (}{\tfrac {7}{5}}{\bigr )}&=-1,&F_{8}&=21,&F_{7}&=13,\\{\bigl (}{\tfrac {11}{5}}{\bigr )}&=+1,&F_{10}&=55,&F_{11}&=89.\end{aligned}}}"></span> </p><p>It is not known whether there exists a prime <span class="texhtml mvar" style="font-style:italic;">p</span> such that </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{p-\left({\frac {p}{5}}\right)}\equiv 0{\pmod {p^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mn>5</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </msub> <mo>≡</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{p-\left({\frac {p}{5}}\right)}\equiv 0{\pmod {p^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7008014b2a81eb82decbbeb094a117868cc404de" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.118ex; height:4.676ex;" alt="{\displaystyle F_{p-\left({\frac {p}{5}}\right)}\equiv 0{\pmod {p^{2}}}.}"></span> </p><p>Such primes (if there are any) would be called <a href="/wiki/Wall%E2%80%93Sun%E2%80%93Sun_prime" title="Wall–Sun–Sun prime">Wall–Sun–Sun primes</a>. </p><p>Also, if <span class="texhtml"><i>p</i> ≠ 5</span> is an odd prime number then:<sup id="cite_ref-FOOTNOTELemmermeyer200073–74ex._2.28_61-0" class="reference"><a href="#cite_note-FOOTNOTELemmermeyer200073–74ex._2.28-61"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5{F_{\frac {p\pm 1}{2}}}^{2}\equiv {\begin{cases}{\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {p}{5}}{\bigr )}\pm 5\right){\pmod {p}}&{\text{if }}p\equiv 1{\pmod {4}}\\{\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {p}{5}}{\bigr )}\mp 3\right){\pmod {p}}&{\text{if }}p\equiv 3{\pmod {4}}.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo>±</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>p</mi> <mn>5</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>±</mo> <mn>5</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>p</mi> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>p</mi> <mn>5</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>∓</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>p</mi> <mo>≡</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5{F_{\frac {p\pm 1}{2}}}^{2}\equiv {\begin{cases}{\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {p}{5}}{\bigr )}\pm 5\right){\pmod {p}}&{\text{if }}p\equiv 1{\pmod {4}}\\{\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {p}{5}}{\bigr )}\mp 3\right){\pmod {p}}&{\text{if }}p\equiv 3{\pmod {4}}.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e41245bd1d7999954a7f6278380b6df6d2fc897a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:56.697ex; height:9.343ex;" alt="{\displaystyle 5{F_{\frac {p\pm 1}{2}}}^{2}\equiv {\begin{cases}{\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {p}{5}}{\bigr )}\pm 5\right){\pmod {p}}&{\text{if }}p\equiv 1{\pmod {4}}\\{\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {p}{5}}{\bigr )}\mp 3\right){\pmod {p}}&{\text{if }}p\equiv 3{\pmod {4}}.\end{cases}}}"></span> </p><p><b>Example 1.</b> <span class="texhtml"><i>p</i> = 7</span>, in this case <span class="texhtml"><i>p</i> ≡ 3 (mod 4)</span> and we have: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigl (}{\tfrac {7}{5}}{\bigr )}=-1:\qquad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {7}{5}}{\bigr )}+3\right)=-1,\quad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {7}{5}}{\bigr )}-3\right)=-4.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>7</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>:</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>7</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>+</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>7</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>4.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl (}{\tfrac {7}{5}}{\bigr )}=-1:\qquad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {7}{5}}{\bigr )}+3\right)=-1,\quad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {7}{5}}{\bigr )}-3\right)=-4.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70d1097aa5138a9318df2db5c1ba98c6ffab5306" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:60.129ex; height:4.843ex;" alt="{\displaystyle {\bigl (}{\tfrac {7}{5}}{\bigr )}=-1:\qquad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {7}{5}}{\bigr )}+3\right)=-1,\quad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {7}{5}}{\bigr )}-3\right)=-4.}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{3}=2{\text{ and }}F_{4}=3.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mn>3.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{3}=2{\text{ and }}F_{4}=3.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aef29471dd6023b0d9a528827f29cdc69e8d30c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.175ex; height:2.509ex;" alt="{\displaystyle F_{3}=2{\text{ and }}F_{4}=3.}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5{F_{3}}^{2}=20\equiv -1{\pmod {7}}\;\;{\text{ and }}\;\;5{F_{4}}^{2}=45\equiv -4{\pmod {7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>20</mn> <mo>≡</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>7</mn> <mo stretchy="false">)</mo> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mn>5</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>45</mn> <mo>≡</mo> <mo>−</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>7</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5{F_{3}}^{2}=20\equiv -1{\pmod {7}}\;\;{\text{ and }}\;\;5{F_{4}}^{2}=45\equiv -4{\pmod {7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5e2a5490de87de5fef411ef4439118ecc3f0cd8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:61.698ex; height:3.176ex;" alt="{\displaystyle 5{F_{3}}^{2}=20\equiv -1{\pmod {7}}\;\;{\text{ and }}\;\;5{F_{4}}^{2}=45\equiv -4{\pmod {7}}}"></span> </p><p><b>Example 2.</b> <span class="texhtml"><i>p</i> = 11</span>, in this case <span class="texhtml"><i>p</i> ≡ 3 (mod 4)</span> and we have: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigl (}{\tfrac {11}{5}}{\bigr )}=+1:\qquad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {11}{5}}{\bigr )}+3\right)=4,\quad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {11}{5}}{\bigr )}-3\right)=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>11</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mo>+</mo> <mn>1</mn> <mo>:</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>11</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>+</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>4</mn> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>11</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl (}{\tfrac {11}{5}}{\bigr )}=+1:\qquad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {11}{5}}{\bigr )}+3\right)=4,\quad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {11}{5}}{\bigr )}-3\right)=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44c44154b42e7af8e966d085057dbb854b8c5a0a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:58.978ex; height:4.843ex;" alt="{\displaystyle {\bigl (}{\tfrac {11}{5}}{\bigr )}=+1:\qquad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {11}{5}}{\bigr )}+3\right)=4,\quad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {11}{5}}{\bigr )}-3\right)=1.}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{5}=5{\text{ and }}F_{6}=8.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>=</mo> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>=</mo> <mn>8.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{5}=5{\text{ and }}F_{6}=8.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4030ec71003561997d2d436b4521a9070971c37a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.175ex; height:2.509ex;" alt="{\displaystyle F_{5}=5{\text{ and }}F_{6}=8.}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5{F_{5}}^{2}=125\equiv 4{\pmod {11}}\;\;{\text{ and }}\;\;5{F_{6}}^{2}=320\equiv 1{\pmod {11}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>125</mn> <mo>≡</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>11</mn> <mo stretchy="false">)</mo> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mn>5</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>320</mn> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>11</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5{F_{5}}^{2}=125\equiv 4{\pmod {11}}\;\;{\text{ and }}\;\;5{F_{6}}^{2}=320\equiv 1{\pmod {11}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d234c5c864851bc639d0a280638974e5100d409c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:62.732ex; height:3.176ex;" alt="{\displaystyle 5{F_{5}}^{2}=125\equiv 4{\pmod {11}}\;\;{\text{ and }}\;\;5{F_{6}}^{2}=320\equiv 1{\pmod {11}}}"></span> </p><p><b>Example 3.</b> <span class="texhtml"><i>p</i> = 13</span>, in this case <span class="texhtml"><i>p</i> ≡ 1 (mod 4)</span> and we have: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigl (}{\tfrac {13}{5}}{\bigr )}=-1:\qquad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {13}{5}}{\bigr )}-5\right)=-5,\quad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {13}{5}}{\bigr )}+5\right)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>13</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>:</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>13</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>−</mo> <mn>5</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>5</mn> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>13</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>+</mo> <mn>5</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl (}{\tfrac {13}{5}}{\bigr )}=-1:\qquad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {13}{5}}{\bigr )}-5\right)=-5,\quad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {13}{5}}{\bigr )}+5\right)=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80be5a52ae5c7c575492f8ff64626ee4cab93b98" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:60.786ex; height:4.843ex;" alt="{\displaystyle {\bigl (}{\tfrac {13}{5}}{\bigr )}=-1:\qquad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {13}{5}}{\bigr )}-5\right)=-5,\quad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {13}{5}}{\bigr )}+5\right)=0.}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{6}=8{\text{ and }}F_{7}=13.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>=</mo> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo>=</mo> <mn>13.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{6}=8{\text{ and }}F_{7}=13.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95345f99232e7b318ac9982c73a2653ae09d72c4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.338ex; height:2.509ex;" alt="{\displaystyle F_{6}=8{\text{ and }}F_{7}=13.}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5{F_{6}}^{2}=320\equiv -5{\pmod {13}}\;\;{\text{ and }}\;\;5{F_{7}}^{2}=845\equiv 0{\pmod {13}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>320</mn> <mo>≡</mo> <mo>−</mo> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>13</mn> <mo stretchy="false">)</mo> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mn>5</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>845</mn> <mo>≡</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>13</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5{F_{6}}^{2}=320\equiv -5{\pmod {13}}\;\;{\text{ and }}\;\;5{F_{7}}^{2}=845\equiv 0{\pmod {13}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdd205bce63fb9497073c3ac39e1a032012ded66" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:64.54ex; height:3.176ex;" alt="{\displaystyle 5{F_{6}}^{2}=320\equiv -5{\pmod {13}}\;\;{\text{ and }}\;\;5{F_{7}}^{2}=845\equiv 0{\pmod {13}}}"></span> </p><p><b>Example 4.</b> <span class="texhtml"><i>p</i> = 29</span>, in this case <span class="texhtml"><i>p</i> ≡ 1 (mod 4)</span> and we have: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigl (}{\tfrac {29}{5}}{\bigr )}=+1:\qquad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {29}{5}}{\bigr )}-5\right)=0,\quad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {29}{5}}{\bigr )}+5\right)=5.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>29</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mo>+</mo> <mn>1</mn> <mo>:</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>29</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>−</mo> <mn>5</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>29</mn> <mn>5</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>+</mo> <mn>5</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>5.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl (}{\tfrac {29}{5}}{\bigr )}=+1:\qquad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {29}{5}}{\bigr )}-5\right)=0,\quad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {29}{5}}{\bigr )}+5\right)=5.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c4aecf3759ca7a7fe3718061ce41deca6b841c5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:58.978ex; height:4.843ex;" alt="{\displaystyle {\bigl (}{\tfrac {29}{5}}{\bigr )}=+1:\qquad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {29}{5}}{\bigr )}-5\right)=0,\quad {\tfrac {1}{2}}\left(5{\bigl (}{\tfrac {29}{5}}{\bigr )}+5\right)=5.}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{14}=377{\text{ and }}F_{15}=610.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>14</mn> </mrow> </msub> <mo>=</mo> <mn>377</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </msub> <mo>=</mo> <mn>610.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{14}=377{\text{ and }}F_{15}=610.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f91b7705f45c3b99f1b689a501ea1db4725a07d5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.469ex; height:2.509ex;" alt="{\displaystyle F_{14}=377{\text{ and }}F_{15}=610.}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5{F_{14}}^{2}=710645\equiv 0{\pmod {29}}\;\;{\text{ and }}\;\;5{F_{15}}^{2}=1860500\equiv 5{\pmod {29}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>14</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>710645</mn> <mo>≡</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>29</mn> <mo stretchy="false">)</mo> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mn>5</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1860500</mn> <mo>≡</mo> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>29</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5{F_{14}}^{2}=710645\equiv 0{\pmod {29}}\;\;{\text{ and }}\;\;5{F_{15}}^{2}=1860500\equiv 5{\pmod {29}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d22ee40b274abfc8563f3382672694e8148a4a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:72.513ex; height:3.176ex;" alt="{\displaystyle 5{F_{14}}^{2}=710645\equiv 0{\pmod {29}}\;\;{\text{ and }}\;\;5{F_{15}}^{2}=1860500\equiv 5{\pmod {29}}}"></span> </p><p>For odd <span class="texhtml mvar" style="font-style:italic;">n</span>, all odd prime divisors of <span class="texhtml"><i>F</i><sub><i>n</i></sub></span> are congruent to 1 modulo 4, implying that all odd divisors of <span class="texhtml"><i>F</i><sub><i>n</i></sub></span> (as the products of odd prime divisors) are congruent to 1 modulo 4.<sup id="cite_ref-FOOTNOTELemmermeyer200073ex._2.27_62-0" class="reference"><a href="#cite_note-FOOTNOTELemmermeyer200073ex._2.27-62"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> </p><p>For example, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}=1,\ F_{3}=2,\ F_{5}=5,\ F_{7}=13,\ F_{9}={\color {Red}34}=2\cdot 17,\ F_{11}=89,\ F_{13}=233,\ F_{15}={\color {Red}610}=2\cdot 5\cdot 61.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>=</mo> <mn>5</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo>=</mo> <mn>13</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#ED1B23"> <mn>34</mn> </mstyle> </mrow> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mn>17</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>=</mo> <mn>89</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <mo>=</mo> <mn>233</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#ED1B23"> <mn>610</mn> </mstyle> </mrow> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mn>5</mn> <mo>⋅</mo> <mn>61.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}=1,\ F_{3}=2,\ F_{5}=5,\ F_{7}=13,\ F_{9}={\color {Red}34}=2\cdot 17,\ F_{11}=89,\ F_{13}=233,\ F_{15}={\color {Red}610}=2\cdot 5\cdot 61.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4512e4871850b79adeca7b8296980ad334390d86" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:96.401ex; height:2.509ex;" alt="{\displaystyle F_{1}=1,\ F_{3}=2,\ F_{5}=5,\ F_{7}=13,\ F_{9}={\color {Red}34}=2\cdot 17,\ F_{11}=89,\ F_{13}=233,\ F_{15}={\color {Red}610}=2\cdot 5\cdot 61.}"></span> </p><p>All known factors of Fibonacci numbers <span class="texhtml"><i>F</i>(<i>i</i>)</span> for all <span class="texhtml"><i>i</i> < 50000</span> are collected at the relevant repositories.<sup id="cite_ref-63" class="reference"><a href="#cite_note-63"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-64" class="reference"><a href="#cite_note-64"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Periodicity_modulo_n">Periodicity modulo <i>n</i></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=28" title="Edit section: Periodicity modulo n"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Pisano_period" title="Pisano period">Pisano period</a></div> <p>If the members of the Fibonacci sequence are taken mod <span class="texhtml mvar" style="font-style:italic;">n</span>, the resulting sequence is <a href="/wiki/Periodic_sequence" title="Periodic sequence">periodic</a> with period at most <span class="texhtml">6<i>n</i></span>.<sup id="cite_ref-65" class="reference"><a href="#cite_note-65"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup> The lengths of the periods for various <span class="texhtml mvar" style="font-style:italic;">n</span> form the so-called <a href="/wiki/Pisano_period" title="Pisano period">Pisano periods</a>.<sup id="cite_ref-66" class="reference"><a href="#cite_note-66"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup> Determining a general formula for the Pisano periods is an <a href="/wiki/Open_problem" title="Open problem">open problem</a>, which includes as a subproblem a special instance of the problem of finding the <a href="/wiki/Multiplicative_order" title="Multiplicative order">multiplicative order</a> of a <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modular integer</a> or of an element in a <a href="/wiki/Finite_field" title="Finite field">finite field</a>. However, for any particular <span class="texhtml mvar" style="font-style:italic;">n</span>, the Pisano period may be found as an instance of <a href="/wiki/Cycle_detection" title="Cycle detection">cycle detection</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=29" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Generalizations_of_Fibonacci_numbers" title="Generalizations of Fibonacci numbers">Generalizations of Fibonacci numbers</a></div> <p>The Fibonacci sequence is one of the simplest and earliest known sequences defined by a <a href="/wiki/Recurrence_relation" title="Recurrence relation">recurrence relation</a>, and specifically by a linear <a href="/wiki/Difference_equation" class="mw-redirect" title="Difference equation">difference equation</a>. All these sequences may be viewed as generalizations of the Fibonacci sequence. In particular, Binet's formula may be generalized to any sequence that is a solution of a <a href="/wiki/Linear_recurrence_with_constant_coefficients" title="Linear recurrence with constant coefficients">homogeneous linear difference equation with constant coefficients</a>. </p><p>Some specific examples that are close, in some sense, to the Fibonacci sequence include: </p> <ul><li>Generalizing the index to negative integers to produce the <a href="/wiki/Negafibonacci" class="mw-redirect" title="Negafibonacci">negafibonacci</a> numbers.</li> <li>Generalizing the index to <a href="/wiki/Real_number" title="Real number">real numbers</a> using a modification of Binet's formula.<sup id="cite_ref-MathWorld_37-2" class="reference"><a href="#cite_note-MathWorld-37"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup></li> <li>Starting with other integers. <a href="/wiki/Lucas_number" title="Lucas number">Lucas numbers</a> have <span class="texhtml"><i>L</i><sub>1</sub> = 1</span>, <span class="texhtml"><i>L</i><sub>2</sub> = 3</span>, and <span class="texhtml"><i>L<sub>n</sub></i> = <i>L</i><sub><i>n</i>−1</sub> + <i>L</i><sub><i>n</i>−2</sub></span>. <a href="/wiki/Primefree_sequence" title="Primefree sequence">Primefree sequences</a> use the Fibonacci recursion with other starting points to generate sequences in which all numbers are composite.</li> <li>Letting a number be a linear function (other than the sum) of the 2 preceding numbers. The <a href="/wiki/Pell_number" title="Pell number">Pell numbers</a> have <span class="texhtml"><i>P<sub>n</sub></i> = 2<i>P</i><sub><i>n</i>−1</sub> + <i>P</i><sub><i>n</i>−2</sub></span>. If the coefficient of the preceding value is assigned a variable value <span class="texhtml mvar" style="font-style:italic;">x</span>, the result is the sequence of <a href="/wiki/Fibonacci_polynomials" title="Fibonacci polynomials">Fibonacci polynomials</a>.</li> <li>Not adding the immediately preceding numbers. The <a href="/wiki/Padovan_sequence" title="Padovan sequence">Padovan sequence</a> and <a href="/wiki/Perrin_number" title="Perrin number">Perrin numbers</a> have <span class="texhtml"><i>P</i>(<i>n</i>) = <i>P</i>(<i>n</i> − 2) + <i>P</i>(<i>n</i> − 3)</span>.</li> <li>Generating the next number by adding 3 numbers (tribonacci numbers), 4 numbers (tetranacci numbers), or more. The resulting sequences are known as <i>n-Step Fibonacci numbers</i>.<sup id="cite_ref-67" class="reference"><a href="#cite_note-67"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=30" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Mathematics">Mathematics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=31" title="Edit section: Mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Pascal_triangle_fibonacci.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Pascal_triangle_fibonacci.svg/260px-Pascal_triangle_fibonacci.svg.png" decoding="async" width="260" height="260" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Pascal_triangle_fibonacci.svg/390px-Pascal_triangle_fibonacci.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Pascal_triangle_fibonacci.svg/520px-Pascal_triangle_fibonacci.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>The Fibonacci numbers are the sums of the diagonals (shown in red) of a left-justified <a href="/wiki/Pascal%27s_triangle" title="Pascal's triangle">Pascal's triangle</a>.</figcaption></figure> <p>The Fibonacci numbers occur as the sums of <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficients</a> in the "shallow" diagonals of <a href="/wiki/Pascal%27s_triangle" title="Pascal's triangle">Pascal's triangle</a>:<sup id="cite_ref-FOOTNOTELucas18917_68-0" class="reference"><a href="#cite_note-FOOTNOTELucas18917-68"><span class="cite-bracket">[</span>67<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=\sum _{k=0}^{\left\lfloor {\frac {n-1}{2}}\right\rfloor }{\binom {n-k-1}{k}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>⌋</mo> </mrow> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=\sum _{k=0}^{\left\lfloor {\frac {n-1}{2}}\right\rfloor }{\binom {n-k-1}{k}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aa5f7416715b0a56008b0df9c6e2c3978352154" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.794ex; height:9.009ex;" alt="{\displaystyle F_{n}=\sum _{k=0}^{\left\lfloor {\frac {n-1}{2}}\right\rfloor }{\binom {n-k-1}{k}}.}"></span> This can be proved by expanding the generating function <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x}{1-x-x^{2}}}=x+x^{2}(1+x)+x^{3}(1+x)^{2}+\dots +x^{k+1}(1+x)^{k}+\dots =\sum \limits _{n=0}^{\infty }F_{n}x^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>x</mi> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x}{1-x-x^{2}}}=x+x^{2}(1+x)+x^{3}(1+x)^{2}+\dots +x^{k+1}(1+x)^{k}+\dots =\sum \limits _{n=0}^{\infty }F_{n}x^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8abb17653b5a397e55ad72e31296b93a2063f15b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:80.427ex; height:6.843ex;" alt="{\displaystyle {\frac {x}{1-x-x^{2}}}=x+x^{2}(1+x)+x^{3}(1+x)^{2}+\dots +x^{k+1}(1+x)^{k}+\dots =\sum \limits _{n=0}^{\infty }F_{n}x^{n}}"></span> and collecting like terms of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/150d38e238991bc4d0689ffc9d2a852547d2658d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.548ex; height:2.343ex;" alt="{\displaystyle x^{n}}"></span>. </p><p>To see how the formula is used, we can arrange the sums by the number of terms present: </p> <dl><dd><table> <tbody><tr> <td><span class="texhtml">5</span> </td> <td><span class="texhtml">= 1+1+1+1+1</span> </td></tr> <tr> <td> </td> <td><span class="texhtml">= 2+1+1+1</span> </td> <td><span class="texhtml">= 1+2+1+1</span> </td> <td><span class="texhtml">= 1+1+2+1</span> </td> <td><span class="texhtml">= 1+1+1+2</span> </td></tr> <tr> <td> </td> <td><span class="texhtml">= 2+2+1</span> </td> <td><span class="texhtml">= 2+1+2</span> </td> <td><span class="texhtml">= 1+2+2</span> </td></tr></tbody></table></dd></dl> <p>which is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\binom {5}{0}}+{\binom {4}{1}}+{\binom {3}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>5</mn> <mn>0</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>4</mn> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>3</mn> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\binom {5}{0}}+{\binom {4}{1}}+{\binom {3}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3b75215ce19fbf8985ab2740df1ea50bebbf3db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.536ex; height:3.509ex;" alt="{\displaystyle \textstyle {\binom {5}{0}}+{\binom {4}{1}}+{\binom {3}{2}}}"></span>, where we are choosing the positions of <span class="texhtml mvar" style="font-style:italic;">k</span> twos from <span class="texhtml"><i>n</i>−<i>k</i>−1</span> terms. </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Fibonacci_climbing_stairs.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Fibonacci_climbing_stairs.svg/220px-Fibonacci_climbing_stairs.svg.png" decoding="async" width="220" height="293" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Fibonacci_climbing_stairs.svg/330px-Fibonacci_climbing_stairs.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/Fibonacci_climbing_stairs.svg/440px-Fibonacci_climbing_stairs.svg.png 2x" data-file-width="512" data-file-height="683" /></a><figcaption>Use of the Fibonacci sequence to count <span class="nowrap">{1, 2}-restricted</span> compositions</figcaption></figure> <p>These numbers also give the solution to certain enumerative problems,<sup id="cite_ref-69" class="reference"><a href="#cite_note-69"><span class="cite-bracket">[</span>68<span class="cite-bracket">]</span></a></sup> the most common of which is that of counting the number of ways of writing a given number <span class="texhtml mvar" style="font-style:italic;">n</span> as an ordered sum of 1s and 2s (called <a href="/wiki/Composition_(combinatorics)#Number_of_compositions" title="Composition (combinatorics)">compositions</a>); there are <span class="texhtml"><i>F</i><sub><i>n</i>+1</sub></span> ways to do this (equivalently, it's also the number of <a href="/wiki/Domino_tiling" title="Domino tiling">domino tilings</a> of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5155392314bbbc3e6304ec9dd8f7f633317fb87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.398ex; height:2.176ex;" alt="{\displaystyle 2\times n}"></span> rectangle). For example, there are <span class="texhtml"><i>F</i><sub>5+1</sub> = <i>F</i><sub>6</sub> = 8</span> ways one can climb a staircase of 5 steps, taking one or two steps at a time: </p> <dl><dd><table> <tbody><tr> <td><span class="texhtml">5</span> </td> <td><span class="texhtml">= 1+1+1+1+1</span> </td> <td><span class="texhtml">= 2+1+1+1</span> </td> <td><span class="texhtml">= 1+2+1+1</span> </td> <td><span class="texhtml">= 1+1+2+1</span> </td> <td><span class="texhtml">= 2+2+1</span> </td></tr> <tr> <td> </td> <td><span class="texhtml">= 1+1+1+2</span> </td> <td><span class="texhtml">= 2+1+2</span> </td> <td><span class="texhtml">= 1+2+2</span> </td></tr></tbody></table></dd></dl> <p>The figure shows that 8 can be decomposed into 5 (the number of ways to climb 4 steps, followed by a single-step) plus 3 (the number of ways to climb 3 steps, followed by a double-step). The same reasoning is applied <a href="/wiki/Recursion" title="Recursion">recursively</a> until a single step, of which there is only one way to climb. </p><p>The Fibonacci numbers can be found in different ways among the set of <a href="/wiki/Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary</a> <a href="/wiki/String_(computer_science)" title="String (computer science)">strings</a>, or equivalently, among the <a href="/wiki/Subset" title="Subset">subsets</a> of a given set. </p> <ul><li>The number of binary strings of length <span class="texhtml mvar" style="font-style:italic;">n</span> without consecutive <span class="texhtml">1</span>s is the Fibonacci number <span class="texhtml"><i>F</i><sub><i>n</i>+2</sub></span>. For example, out of the 16 binary strings of length 4, there are <span class="texhtml"><i>F</i><sub>6</sub> = 8</span> without consecutive <span class="texhtml">1</span>s—they are 0000, 0001, 0010, 0100, 0101, 1000, 1001, and 1010. Such strings are the binary representations of <a href="/wiki/Fibbinary_number" title="Fibbinary number">Fibbinary numbers</a>. Equivalently, <span class="texhtml"><i>F</i><sub><i>n</i>+2</sub></span> is the number of subsets <span class="texhtml mvar" style="font-style:italic;">S</span> of <span class="texhtml">{1, ..., <i>n</i>}</span> without consecutive integers, that is, those <span class="texhtml mvar" style="font-style:italic;">S</span> for which <span class="texhtml">{<i>i</i>, <i>i</i> + 1} ⊈ <i>S</i></span> for every <span class="texhtml mvar" style="font-style:italic;">i</span>. A <a href="/wiki/Bijection" title="Bijection">bijection</a> with the sums to <span class="texhtml"><i>n</i>+1</span> is to replace 1 with 0 and 2 with 10, and drop the last zero.</li> <li>The number of binary strings of length <span class="texhtml mvar" style="font-style:italic;">n</span> without an odd number of consecutive <span class="texhtml">1</span>s is the Fibonacci number <span class="texhtml"><i>F</i><sub><i>n</i>+1</sub></span>. For example, out of the 16 binary strings of length 4, there are <span class="texhtml"><i>F</i><sub>5</sub> = 5</span> without an odd number of consecutive <span class="texhtml">1</span>s—they are 0000, 0011, 0110, 1100, 1111. Equivalently, the number of subsets <span class="texhtml mvar" style="font-style:italic;">S</span> of <span class="texhtml">{1, ..., <i>n</i>}</span> without an odd number of consecutive integers is <span class="texhtml"><i>F</i><sub><i>n</i>+1</sub></span>. A bijection with the sums to <span class="texhtml mvar" style="font-style:italic;">n</span> is to replace 1 with 0 and 2 with 11.</li> <li>The number of binary strings of length <span class="texhtml mvar" style="font-style:italic;">n</span> without an even number of consecutive <span class="texhtml">0</span>s or <span class="texhtml">1</span>s is <span class="texhtml">2<i>F</i><sub><i>n</i></sub></span>. For example, out of the 16 binary strings of length 4, there are <span class="texhtml">2<i>F</i><sub>4</sub> = 6</span> without an even number of consecutive <span class="texhtml">0</span>s or <span class="texhtml">1</span>s—they are 0001, 0111, 0101, 1000, 1010, 1110. There is an equivalent statement about subsets.</li> <li><a href="/wiki/Yuri_Matiyasevich" title="Yuri Matiyasevich">Yuri Matiyasevich</a> was able to show that the Fibonacci numbers can be defined by a <a href="/wiki/Diophantine_equation" title="Diophantine equation">Diophantine equation</a>, which led to <a href="/wiki/Matiyasevich%27s_theorem" class="mw-redirect" title="Matiyasevich's theorem">his solving</a> <a href="/wiki/Hilbert%27s_tenth_problem" title="Hilbert's tenth problem">Hilbert's tenth problem</a>.<sup id="cite_ref-70" class="reference"><a href="#cite_note-70"><span class="cite-bracket">[</span>69<span class="cite-bracket">]</span></a></sup></li> <li>The Fibonacci numbers are also an example of a <a href="/wiki/Complete_sequence" title="Complete sequence">complete sequence</a>. This means that every positive integer can be written as a sum of Fibonacci numbers, where any one number is used once at most.</li> <li>Moreover, every positive integer can be written in a unique way as the sum of <i>one or more</i> distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. This is known as <a href="/wiki/Zeckendorf%27s_theorem" title="Zeckendorf's theorem">Zeckendorf's theorem</a>, and a sum of Fibonacci numbers that satisfies these conditions is called a Zeckendorf representation. The Zeckendorf representation of a number can be used to derive its <a href="/wiki/Fibonacci_coding" title="Fibonacci coding">Fibonacci coding</a>.</li> <li>Starting with 5, every second Fibonacci number is the length of the <a href="/wiki/Hypotenuse" title="Hypotenuse">hypotenuse</a> of a <a href="/wiki/Right_triangle" title="Right triangle">right triangle</a> with integer sides, or in other words, the largest number in a <a href="/wiki/Pythagorean_triple" title="Pythagorean triple">Pythagorean triple</a>, obtained from the formula <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (F_{n}F_{n+3})^{2}+(2F_{n+1}F_{n+2})^{2}={F_{2n+3}}^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>3</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (F_{n}F_{n+3})^{2}+(2F_{n+1}F_{n+2})^{2}={F_{2n+3}}^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93801f4a2fa8417786c3caf08a9aa147cdb9edbb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.319ex; height:3.176ex;" alt="{\displaystyle (F_{n}F_{n+3})^{2}+(2F_{n+1}F_{n+2})^{2}={F_{2n+3}}^{2}.}"></span> The sequence of Pythagorean triangles obtained from this formula has sides of lengths (3,4,5), (5,12,13), (16,30,34), (39,80,89), ... . The middle side of each of these triangles is the sum of the three sides of the preceding triangle.<sup id="cite_ref-71" class="reference"><a href="#cite_note-71"><span class="cite-bracket">[</span>70<span class="cite-bracket">]</span></a></sup></li> <li>The <a href="/wiki/Fibonacci_cube" title="Fibonacci cube">Fibonacci cube</a> is an <a href="/wiki/Undirected_graph" class="mw-redirect" title="Undirected graph">undirected graph</a> with a Fibonacci number of nodes that has been proposed as a <a href="/wiki/Network_topology" title="Network topology">network topology</a> for <a href="/wiki/Parallel_computing" title="Parallel computing">parallel computing</a>.</li> <li>Fibonacci numbers appear in the <a href="/wiki/Ring_lemma" title="Ring lemma">ring lemma</a>, used to prove connections between the <a href="/wiki/Circle_packing_theorem" title="Circle packing theorem">circle packing theorem</a> and <a href="/wiki/Conformal_map" title="Conformal map">conformal maps</a>.<sup id="cite_ref-72" class="reference"><a href="#cite_note-72"><span class="cite-bracket">[</span>71<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Computer_science">Computer science</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=32" title="Edit section: Computer science"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Fibonacci_Tree_6.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Fibonacci_Tree_6.svg/260px-Fibonacci_Tree_6.svg.png" decoding="async" width="260" height="135" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Fibonacci_Tree_6.svg/390px-Fibonacci_Tree_6.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Fibonacci_Tree_6.svg/520px-Fibonacci_Tree_6.svg.png 2x" data-file-width="634" data-file-height="330" /></a><figcaption>Fibonacci tree of height 6. <a href="/wiki/AVL_tree#Balance_factor" title="AVL tree">Balance factors</a> green; heights red.<br />The keys in the left spine are Fibonacci numbers.</figcaption></figure> <ul><li>The Fibonacci numbers are important in <a href="/wiki/Analysis_of_algorithms" title="Analysis of algorithms">computational run-time analysis</a> of <a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclid's algorithm</a> to determine the <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> of two integers: the worst case input for this algorithm is a pair of consecutive Fibonacci numbers.<sup id="cite_ref-73" class="reference"><a href="#cite_note-73"><span class="cite-bracket">[</span>72<span class="cite-bracket">]</span></a></sup></li> <li>Fibonacci numbers are used in a polyphase version of the <a href="/wiki/Merge_sort" title="Merge sort">merge sort</a> algorithm in which an unsorted list is divided into two lists whose lengths correspond to sequential Fibonacci numbers—by dividing the list so that the two parts have lengths in the approximate proportion <span class="texhtml mvar" style="font-style:italic;">φ</span>. A tape-drive implementation of the <a href="/wiki/Polyphase_merge_sort" title="Polyphase merge sort">polyphase merge sort</a> was described in <i><a href="/wiki/The_Art_of_Computer_Programming" title="The Art of Computer Programming">The Art of Computer Programming</a></i>.</li> <li><span class="anchor" id="Fibonacci_Tree"></span>A Fibonacci tree is a <a href="/wiki/Binary_tree" title="Binary tree">binary tree</a> whose child trees (recursively) differ in <a href="/wiki/Tree_height" class="mw-redirect" title="Tree height">height</a> by exactly 1. So it is an <a href="/wiki/AVL_tree" title="AVL tree">AVL tree</a>, and one with the fewest nodes for a given height—the "thinnest" AVL tree. These trees have a number of vertices that is a Fibonacci number minus one, an important fact in the analysis of AVL trees.<sup id="cite_ref-74" class="reference"><a href="#cite_note-74"><span class="cite-bracket">[</span>73<span class="cite-bracket">]</span></a></sup></li> <li>Fibonacci numbers are used by some <a href="/wiki/Pseudorandom_number_generator" title="Pseudorandom number generator">pseudorandom number generators</a>.</li> <li>Fibonacci numbers arise in the analysis of the <a href="/wiki/Fibonacci_heap" title="Fibonacci heap">Fibonacci heap</a> data structure.</li> <li>A one-dimensional optimization method, called the <a href="/wiki/Fibonacci_search_technique" title="Fibonacci search technique">Fibonacci search technique</a>, uses Fibonacci numbers.<sup id="cite_ref-75" class="reference"><a href="#cite_note-75"><span class="cite-bracket">[</span>74<span class="cite-bracket">]</span></a></sup></li> <li>The Fibonacci number series is used for optional <a href="/wiki/Lossy_compression" title="Lossy compression">lossy compression</a> in the <a href="/wiki/Interchange_File_Format" title="Interchange File Format">IFF</a> <a href="/wiki/8SVX" title="8SVX">8SVX</a> audio file format used on <a href="/wiki/Amiga" title="Amiga">Amiga</a> computers. The number series <a href="/wiki/Companding" title="Companding">compands</a> the original audio wave similar to logarithmic methods such as <a href="/wiki/%CE%9C-law" class="mw-redirect" title="Μ-law">μ-law</a>.<sup id="cite_ref-76" class="reference"><a href="#cite_note-76"><span class="cite-bracket">[</span>75<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-77" class="reference"><a href="#cite_note-77"><span class="cite-bracket">[</span>76<span class="cite-bracket">]</span></a></sup></li> <li>Some Agile teams use a modified series called the "Modified Fibonacci Series" in <a href="/wiki/Planning_poker" title="Planning poker">planning poker</a>, as an estimation tool. Planning Poker is a formal part of the <a href="/wiki/Scaled_agile_framework" title="Scaled agile framework">Scaled Agile Framework</a>.<sup id="cite_ref-78" class="reference"><a href="#cite_note-78"><span class="cite-bracket">[</span>77<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Fibonacci_coding" title="Fibonacci coding">Fibonacci coding</a></li> <li><a href="/wiki/Negafibonacci_coding" title="Negafibonacci coding">Negafibonacci coding</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Nature">Nature</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=33" title="Edit section: Nature"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Patterns_in_nature" title="Patterns in nature">Patterns in nature</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Golden_ratio#Nature" title="Golden ratio">Golden ratio § Nature</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:FibonacciChamomile.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/FibonacciChamomile.PNG/220px-FibonacciChamomile.PNG" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/5/5a/FibonacciChamomile.PNG 1.5x" data-file-width="300" data-file-height="300" /></a><figcaption><a href="/wiki/Yellow_chamomile" class="mw-redirect" title="Yellow chamomile">Yellow chamomile</a> head showing the arrangement in 21 (blue) and 13 (cyan) spirals. Such arrangements involving consecutive Fibonacci numbers appear in a wide variety of plants.</figcaption></figure> <p>Fibonacci sequences appear in biological settings,<sup id="cite_ref-79" class="reference"><a href="#cite_note-79"><span class="cite-bracket">[</span>78<span class="cite-bracket">]</span></a></sup> such as branching in trees, <a href="/wiki/Phyllotaxis" title="Phyllotaxis">arrangement of leaves on a stem</a>, the fruitlets of a <a href="/wiki/Pineapple" title="Pineapple">pineapple</a>,<sup id="cite_ref-80" class="reference"><a href="#cite_note-80"><span class="cite-bracket">[</span>79<span class="cite-bracket">]</span></a></sup> the flowering of <a href="/wiki/Artichoke" title="Artichoke">artichoke</a>, the arrangement of a <a href="/wiki/Pine_cone" class="mw-redirect" title="Pine cone">pine cone</a>,<sup id="cite_ref-81" class="reference"><a href="#cite_note-81"><span class="cite-bracket">[</span>80<span class="cite-bracket">]</span></a></sup> and the family tree of <a href="/wiki/Honeybee" class="mw-redirect" title="Honeybee">honeybees</a>.<sup id="cite_ref-82" class="reference"><a href="#cite_note-82"><span class="cite-bracket">[</span>81<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-83" class="reference"><a href="#cite_note-83"><span class="cite-bracket">[</span>82<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Kepler" class="mw-redirect" title="Kepler">Kepler</a> pointed out the presence of the Fibonacci sequence in nature, using it to explain the (<a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</a>-related) <a href="/wiki/Pentagon" title="Pentagon">pentagonal</a> form of some flowers.<sup id="cite_ref-FOOTNOTELivio2003110_84-0" class="reference"><a href="#cite_note-FOOTNOTELivio2003110-84"><span class="cite-bracket">[</span>83<span class="cite-bracket">]</span></a></sup> Field <a href="/wiki/Leucanthemum_vulgare" title="Leucanthemum vulgare">daisies</a> most often have petals in counts of Fibonacci numbers.<sup id="cite_ref-FOOTNOTELivio2003112–13_85-0" class="reference"><a href="#cite_note-FOOTNOTELivio2003112–13-85"><span class="cite-bracket">[</span>84<span class="cite-bracket">]</span></a></sup> In 1830, <a href="/wiki/Karl_Friedrich_Schimper" title="Karl Friedrich Schimper">Karl Friedrich Schimper</a> and <a href="/wiki/Alexander_Braun" title="Alexander Braun">Alexander Braun</a> discovered that the <a href="/wiki/Parastichy" title="Parastichy">parastichies</a> (spiral <a href="/wiki/Phyllotaxis" title="Phyllotaxis">phyllotaxis</a>) of plants were frequently expressed as fractions involving Fibonacci numbers.<sup id="cite_ref-86" class="reference"><a href="#cite_note-86"><span class="cite-bracket">[</span>85<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Przemys%C5%82aw_Prusinkiewicz" title="Przemysław Prusinkiewicz">Przemysław Prusinkiewicz</a> advanced the idea that real instances can in part be understood as the expression of certain algebraic constraints on <a href="/wiki/Free_group" title="Free group">free groups</a>, specifically as certain <a href="/wiki/L-system" title="L-system">Lindenmayer grammars</a>.<sup id="cite_ref-87" class="reference"><a href="#cite_note-87"><span class="cite-bracket">[</span>86<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:SunflowerModel.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/SunflowerModel.svg/220px-SunflowerModel.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/SunflowerModel.svg/330px-SunflowerModel.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/SunflowerModel.svg/440px-SunflowerModel.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>Illustration of Vogel's model for <span class="texhtml"><i>n</i> = 1 ... 500</span></figcaption></figure> <p>A model for the pattern of <a href="/wiki/Floret" class="mw-redirect" title="Floret">florets</a> in the head of a <a href="/wiki/Sunflower" class="mw-redirect" title="Sunflower">sunflower</a> was proposed by <a href="/w/index.php?title=Helmut_Vogel&action=edit&redlink=1" class="new" title="Helmut Vogel (page does not exist)">Helmut Vogel</a><span class="noprint" style="font-size:85%; font-style: normal;"> [<a href="https://de.wikipedia.org/wiki/Helmut_Vogel_(Physiker)" class="extiw" title="de:Helmut Vogel (Physiker)">de</a>]</span> in 1979.<sup id="cite_ref-88" class="reference"><a href="#cite_note-88"><span class="cite-bracket">[</span>87<span class="cite-bracket">]</span></a></sup> This has the form </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ={\frac {2\pi }{\varphi ^{2}}}n,\ r=c{\sqrt {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mi>n</mi> <mo>,</mo> <mtext> </mtext> <mi>r</mi> <mo>=</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ={\frac {2\pi }{\varphi ^{2}}}n,\ r=c{\sqrt {n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38c3e214263b257efb33a0accc6f505c5e827341" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.093ex; height:5.843ex;" alt="{\displaystyle \theta ={\frac {2\pi }{\varphi ^{2}}}n,\ r=c{\sqrt {n}}}"></span> </p><p>where <span class="texhtml mvar" style="font-style:italic;">n</span> is the index number of the floret and <span class="texhtml mvar" style="font-style:italic;">c</span> is a constant scaling factor; the florets thus lie on <a href="/wiki/Fermat%27s_spiral" title="Fermat's spiral">Fermat's spiral</a>. The divergence <a href="/wiki/Angle" title="Angle">angle</a>, approximately 137.51°, is the <a href="/wiki/Golden_angle" title="Golden angle">golden angle</a>, dividing the circle in the golden ratio. Because this ratio is irrational, no floret has a neighbor at exactly the same angle from the center, so the florets pack efficiently. Because the rational approximations to the golden ratio are of the form <span class="texhtml"><i>F</i>( <i>j</i>):<i>F</i>( <i>j</i> + 1)</span>, the nearest neighbors of floret number <span class="texhtml mvar" style="font-style:italic;">n</span> are those at <span class="texhtml"><i>n</i> ± <i>F</i>( <i>j</i>)</span> for some index <span class="texhtml mvar" style="font-style:italic;">j</span>, which depends on <span class="texhtml mvar" style="font-style:italic;">r</span>, the distance from the center. Sunflowers and similar flowers most commonly have spirals of florets in clockwise and counter-clockwise directions in the amount of adjacent Fibonacci numbers,<sup id="cite_ref-FOOTNOTELivio2003112_89-0" class="reference"><a href="#cite_note-FOOTNOTELivio2003112-89"><span class="cite-bracket">[</span>88<span class="cite-bracket">]</span></a></sup> typically counted by the outermost range of radii.<sup id="cite_ref-90" class="reference"><a href="#cite_note-90"><span class="cite-bracket">[</span>89<span class="cite-bracket">]</span></a></sup> </p><p>Fibonacci numbers also appear in the ancestral pedigrees of <a href="/wiki/Bee" title="Bee">bees</a> (which are <a href="/wiki/Haplodiploid" class="mw-redirect" title="Haplodiploid">haplodiploids</a>), according to the following rules: </p> <ul><li>If an egg is laid but not fertilized, it produces a male (or <a href="/wiki/Drone_(bee)" title="Drone (bee)">drone bee</a> in honeybees).</li> <li>If, however, an egg is fertilized, it produces a female.</li></ul> <p>Thus, a male bee always has one parent, and a female bee has two. If one traces the pedigree of any male bee (1 bee), he has 1 parent (1 bee), 2 grandparents, 3 great-grandparents, 5 great-great-grandparents, and so on. This sequence of numbers of parents is the Fibonacci sequence. The number of ancestors at each level, <span class="texhtml"><i>F</i><sub><i>n</i></sub></span>, is the number of female ancestors, which is <span class="texhtml"><i>F</i><sub><i>n</i>−1</sub></span>, plus the number of male ancestors, which is <span class="texhtml"><i>F</i><sub><i>n</i>−2</sub></span>.<sup id="cite_ref-91" class="reference"><a href="#cite_note-91"><span class="cite-bracket">[</span>90<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-92" class="reference"><a href="#cite_note-92"><span class="cite-bracket">[</span>91<span class="cite-bracket">]</span></a></sup> This is under the unrealistic assumption that the ancestors at each level are otherwise unrelated. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:X_chromosome_ancestral_line_Fibonacci_sequence.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/X_chromosome_ancestral_line_Fibonacci_sequence.svg/260px-X_chromosome_ancestral_line_Fibonacci_sequence.svg.png" decoding="async" width="260" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/X_chromosome_ancestral_line_Fibonacci_sequence.svg/390px-X_chromosome_ancestral_line_Fibonacci_sequence.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/X_chromosome_ancestral_line_Fibonacci_sequence.svg/520px-X_chromosome_ancestral_line_Fibonacci_sequence.svg.png 2x" data-file-width="1100" data-file-height="676" /></a><figcaption>The number of possible ancestors on the X chromosome inheritance line at a given ancestral generation follows the Fibonacci sequence. (After Hutchison, L. "Growing the Family Tree: The Power of DNA in Reconstructing Family Relationships".<sup id="cite_ref-xcs_93-0" class="reference"><a href="#cite_note-xcs-93"><span class="cite-bracket">[</span>92<span class="cite-bracket">]</span></a></sup>)</figcaption></figure> <p>It has similarly been noticed that the number of possible ancestors on the human <a href="/wiki/X_chromosome" title="X chromosome">X chromosome</a> inheritance line at a given ancestral generation also follows the Fibonacci sequence.<sup id="cite_ref-xcs_93-1" class="reference"><a href="#cite_note-xcs-93"><span class="cite-bracket">[</span>92<span class="cite-bracket">]</span></a></sup> A male individual has an X chromosome, which he received from his mother, and a <a href="/wiki/Y_chromosome" title="Y chromosome">Y chromosome</a>, which he received from his father. The male counts as the "origin" of his own X chromosome (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c374ba08c140de90c6cbb4c9b9fcd26e3f99ef56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{1}=1}"></span>), and at his parents' generation, his X chromosome came from a single parent <span class="nowrap">(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf100a6879452aff05a6027ad4f36029f360dcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{2}=1}"></span>)</span>. The male's mother received one X chromosome from her mother (the son's maternal grandmother), and one from her father (the son's maternal grandfather), so two grandparents contributed to the male descendant's X chromosome <span class="nowrap">(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{3}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{3}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1caf4ec3fa730d75d5508487f92ebe0a64307a4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{3}=2}"></span>)</span>. The maternal grandfather received his X chromosome from his mother, and the maternal grandmother received X chromosomes from both of her parents, so three great-grandparents contributed to the male descendant's X chromosome <span class="nowrap">(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{4}=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{4}=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fcb7ddc635c556db90a8c277ce6b45e6d4aa185" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{4}=3}"></span>)</span>. Five great-great-grandparents contributed to the male descendant's X chromosome <span class="nowrap">(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{5}=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{5}=5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/767c16dde6476b8fd6618435d70266a3747e426f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.81ex; height:2.509ex;" alt="{\displaystyle F_{5}=5}"></span>)</span>, etc. (This assumes that all ancestors of a given descendant are independent, but if any genealogy is traced far enough back in time, ancestors begin to appear on multiple lines of the genealogy, until eventually a <a href="/wiki/Founder_effect" title="Founder effect">population founder</a> appears on all lines of the genealogy.) </p> <div class="mw-heading mw-heading3"><h3 id="Other">Other</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=34" title="Edit section: Other"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>In <a href="/wiki/Optics" title="Optics">optics</a>, when a beam of light shines at an angle through two stacked transparent plates of different materials of different <a href="/wiki/Refractive_index" title="Refractive index">refractive indexes</a>, it may reflect off three surfaces: the top, middle, and bottom surfaces of the two plates. The number of different beam paths that have <span class="texhtml mvar" style="font-style:italic;">k</span> reflections, for <span class="texhtml"><i>k</i> > 1</span>, is the <span class="texhtml mvar" style="font-style:italic;">k</span>-th Fibonacci number. (However, when <span class="texhtml"><i>k</i> = 1</span>, there are three reflection paths, not two, one for each of the three surfaces.)<sup id="cite_ref-FOOTNOTELivio200398–99_94-0" class="reference"><a href="#cite_note-FOOTNOTELivio200398–99-94"><span class="cite-bracket">[</span>93<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Fibonacci_retracement" title="Fibonacci retracement">Fibonacci retracement</a> levels are widely used in <a href="/wiki/Technical_analysis" title="Technical analysis">technical analysis</a> for financial market trading.</li> <li>Since the <a href="/wiki/Conversion_of_units" title="Conversion of units">conversion</a> factor 1.609344 for miles to kilometers is close to the golden ratio, the decomposition of distance in miles into a sum of Fibonacci numbers becomes nearly the kilometer sum when the Fibonacci numbers are replaced by their successors. This method amounts to a <a href="/wiki/Radix" title="Radix">radix</a> 2 number <a href="/wiki/Processor_register" title="Processor register">register</a> in <a href="/wiki/Golden_ratio_base" title="Golden ratio base">golden ratio base</a> <span class="texhtml mvar" style="font-style:italic;">φ</span> being shifted. To convert from kilometers to miles, shift the register down the Fibonacci sequence instead.<sup id="cite_ref-95" class="reference"><a href="#cite_note-95"><span class="cite-bracket">[</span>94<span class="cite-bracket">]</span></a></sup></li> <li>The measured values of voltages and currents in the infinite resistor chain circuit (also called the <a href="/wiki/Resistor_ladder" title="Resistor ladder">resistor ladder</a> or infinite series-parallel circuit) follow the Fibonacci sequence. The intermediate results of adding the alternating series and parallel resistances yields fractions composed of consecutive Fibonacci numbers. The equivalent resistance of the entire circuit equals the golden ratio.<sup id="cite_ref-96" class="reference"><a href="#cite_note-96"><span class="cite-bracket">[</span>95<span class="cite-bracket">]</span></a></sup></li> <li>Brasch et al. 2012 show how a generalized Fibonacci sequence also can be connected to the field of <a href="/wiki/Economics" title="Economics">economics</a>.<sup id="cite_ref-Brasch_et_al._2012_97-0" class="reference"><a href="#cite_note-Brasch_et_al._2012-97"><span class="cite-bracket">[</span>96<span class="cite-bracket">]</span></a></sup> In particular, it is shown how a generalized Fibonacci sequence enters the control function of finite-horizon dynamic optimisation problems with one state and one control variable. The procedure is illustrated in an example often referred to as the Brock–Mirman economic growth model.</li> <li><a href="/wiki/Mario_Merz" title="Mario Merz">Mario Merz</a> included the Fibonacci sequence in some of his artworks beginning in 1970.<sup id="cite_ref-FOOTNOTELivio2003176_98-0" class="reference"><a href="#cite_note-FOOTNOTELivio2003176-98"><span class="cite-bracket">[</span>97<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Joseph_Schillinger" title="Joseph Schillinger">Joseph Schillinger</a> (1895–1943) developed <a href="/wiki/Schillinger_System" class="mw-redirect" title="Schillinger System">a system of composition</a> which uses Fibonacci intervals in some of its melodies; he viewed these as the musical counterpart to the elaborate harmony evident within nature.<sup id="cite_ref-FOOTNOTELivio2003193_99-0" class="reference"><a href="#cite_note-FOOTNOTELivio2003193-99"><span class="cite-bracket">[</span>98<span class="cite-bracket">]</span></a></sup> See also <a href="/wiki/Golden_ratio#Music" title="Golden ratio">Golden ratio § Music</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=35" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/The_Fibonacci_Association" title="The Fibonacci Association">The Fibonacci Association</a> – Organization for research on Fibonacci numbers</li> <li><a href="/wiki/Fibonacci_numbers_in_popular_culture" title="Fibonacci numbers in popular culture">Fibonacci numbers in popular culture</a></li> <li><a href="/wiki/Fibonacci_word" title="Fibonacci word">Fibonacci word</a> – Binary sequence from Fibonacci recurrence</li> <li><a href="/wiki/Random_Fibonacci_sequence" title="Random Fibonacci sequence">Random Fibonacci sequence</a> – Randomized mathematical sequence based upon the Fibonacci sequence</li> <li><a href="/wiki/Wythoff_array" title="Wythoff array">Wythoff array</a> – Infinite matrix of integers derived from the Fibonacci sequence</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=36" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Explanatory_footnotes">Explanatory footnotes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=37" title="Edit section: Explanatory footnotes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">"For four, variations of meters of two [and] three being mixed, five happens. For five, variations of two earlier—three [and] four, being mixed, eight is obtained. In this way, for six, [variations] of four [and] of five being mixed, thirteen happens. And like that, variations of two earlier meters being mixed, seven <a href="/wiki/Mora_(linguistics)" title="Mora (linguistics)">morae</a> [is] twenty-one. In this way, the process should be followed in all mātrā-vṛttas" <sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=38" title="Edit section: Citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Richard A. Brualdi, <i>Introductory Combinatorics</i>, Fifth edition, Pearson, 2005</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Peter Cameron, <i>Combinatorics: Topics, Techniques, Algorithms</i>, Cambridge University Press, 1994</span> </li> <li id="cite_note-GlobalScience-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-GlobalScience_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-GlobalScience_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-GlobalScience_3-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFGoonatilake1998" class="citation cs2">Goonatilake, Susantha (1998), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SI5ip95BbgEC&pg=PA126"><i>Toward a Global Science</i></a>, Indiana University Press, p. 126, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-253-33388-9" title="Special:BookSources/978-0-253-33388-9"><bdi>978-0-253-33388-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Toward+a+Global+Science&rft.pages=126&rft.pub=Indiana+University+Press&rft.date=1998&rft.isbn=978-0-253-33388-9&rft.aulast=Goonatilake&rft.aufirst=Susantha&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DSI5ip95BbgEC%26pg%3DPA126&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-HistoriaMathematica-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-HistoriaMathematica_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-HistoriaMathematica_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-HistoriaMathematica_4-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSingh1985" class="citation cs2">Singh, Parmanand (1985), "The So-called Fibonacci numbers in ancient and medieval India", <i>Historia Mathematica</i>, <b>12</b> (3): <span class="nowrap">229–</span>244, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0315-0860%2885%2990021-7">10.1016/0315-0860(85)90021-7</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Historia+Mathematica&rft.atitle=The+So-called+Fibonacci+numbers+in+ancient+and+medieval+India&rft.volume=12&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E229-%3C%2Fspan%3E244&rft.date=1985&rft_id=info%3Adoi%2F10.1016%2F0315-0860%2885%2990021-7&rft.aulast=Singh&rft.aufirst=Parmanand&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-Donald_Knuth_2006_50-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Donald_Knuth_2006_50_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Donald_Knuth_2006_50_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnuth2006" class="citation cs2 cs1-prop-long-vol"><a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, Donald</a> (2006), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=56LNfE2QGtYC&q=rhythms&pg=PA50"><i>The Art of Computer Programming</i></a>, vol. 4. Generating All Trees – History of Combinatorial Generation, Addison–Wesley, p. 50, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-33570-8" title="Special:BookSources/978-0-321-33570-8"><bdi>978-0-321-33570-8</bdi></a>, <q>it was natural to consider the set of all sequences of [L] and [S] that have exactly m beats. ... there are exactly Fm+1 of them. For example the 21 sequences when <span class="texhtml"><i>m</i> = 7</span> are: [gives list]. In this way Indian prosodists were led to discover the Fibonacci sequence, as we have observed in Section 1.2.8 (from v.1)</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Art+of+Computer+Programming&rft.pages=50&rft.pub=Addison%E2%80%93Wesley&rft.date=2006&rft.isbn=978-0-321-33570-8&rft.aulast=Knuth&rft.aufirst=Donald&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D56LNfE2QGtYC%26q%3Drhythms%26pg%3DPA50&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTESigler2002404–05-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESigler2002404–05_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSigler2002">Sigler 2002</a>, pp. 404–05.</span> </li> <li id="cite_note-FOOTNOTELucas18913-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELucas18913_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLucas1891">Lucas 1891</a>, p. 3.</span> </li> <li id="cite_note-FOOTNOTEBeckGeoghegan2010-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBeckGeoghegan2010_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBeckGeoghegan2010">Beck & Geoghegan 2010</a>.</span> </li> <li id="cite_note-FOOTNOTEBóna2011180-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBóna2011180_9-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBóna2011">Bóna 2011</a>, p. 180.</span> </li> <li id="cite_note-knuth-v1-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-knuth-v1_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnuth1968" class="citation cs2"><a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, Donald</a> (1968), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=MooMkK6ERuYC&pg=PA100"><i>The Art of Computer Programming</i></a>, vol. 1, Addison Wesley, p. 100, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-7758-754-8" title="Special:BookSources/978-81-7758-754-8"><bdi>978-81-7758-754-8</bdi></a>, <q>Before Fibonacci wrote his work, the sequence Fn had already been discussed by Indian scholars, who had long been interested in rhythmic patterns ... both Gopala (before 1135 AD) and Hemachandra (c. 1150) mentioned the numbers 1,2,3,5,8,13,21 explicitly [see P. Singh Historia Math 12 (1985) 229–44]" p. 100 (3d ed) ...</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Art+of+Computer+Programming&rft.pages=100&rft.pub=Addison+Wesley&rft.date=1968&rft.isbn=978-81-7758-754-8&rft.aulast=Knuth&rft.aufirst=Donald&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DMooMkK6ERuYC%26pg%3DPA100&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTELivio2003197-11"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTELivio2003197_11-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTELivio2003197_11-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFLivio2003">Livio 2003</a>, p. 197.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAgrawala1969" class="citation cs2">Agrawala, VS (1969), <i><span></span></i>Pāṇinikālīna Bhāratavarṣa<i> (Hn.). Varanasi-I: TheChowkhamba Vidyabhawan</i>, <q>SadgurushiShya writes that Pingala was a younger brother of Pāṇini [Agrawala 1969, lb]. There is an alternative opinion that he was a maternal uncle of Pāṇini [Vinayasagar 1965, Preface, 121]. ... Agrawala [1969, 463–76], after a careful investigation, in which he considered the views of earlier scholars, has concluded that Pāṇini lived between 480 and 410 BC</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=P%C4%81%E1%B9%87inik%C4%81l%C4%ABna+Bh%C4%81ratavar%E1%B9%A3a+%28Hn.%29.+Varanasi-I%3A+TheChowkhamba+Vidyabhawan&rft.date=1969&rft.aulast=Agrawala&rft.aufirst=VS&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVelankar1962" class="citation cs2">Velankar, HD (1962), <i><span></span>'Vṛttajātisamuccaya' of kavi Virahanka</i>, Jodhpur: Rajasthan Oriental Research Institute, p. 101</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%27V%E1%B9%9Bttaj%C4%81tisamuccaya%27+of+kavi+Virahanka&rft.place=Jodhpur&rft.pages=101&rft.pub=Rajasthan+Oriental+Research+Institute&rft.date=1962&rft.aulast=Velankar&rft.aufirst=HD&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTELivio2003197–198-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELivio2003197–198_15-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLivio2003">Livio 2003</a>, p. 197–198.</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShah1991" class="citation cs2">Shah, Jayant (1991), <a rel="nofollow" class="external text" href="https://web.northeastern.edu/shah/papers/Pingala.pdf"><i>A History of Piṅgala's Combinatorics</i></a> <span class="cs1-format">(PDF)</span>, <a href="/wiki/Northeastern_University" title="Northeastern University">Northeastern University</a>, p. 41<span class="reference-accessdate">, retrieved <span class="nowrap">4 January</span> 2019</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Pi%E1%B9%85gala%27s+Combinatorics&rft.pages=41&rft.pub=Northeastern+University&rft.date=1991&rft.aulast=Shah&rft.aufirst=Jayant&rft_id=https%3A%2F%2Fweb.northeastern.edu%2Fshah%2Fpapers%2FPingala.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTESigler2002404–405-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESigler2002404–405_17-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSigler2002">Sigler 2002</a>, pp. 404–405.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.math.utah.edu/~beebe/software/java/fibonacci/liber-abaci.html">"Fibonacci's Liber Abaci (Book of Calculation)"</a>, <i><a href="/wiki/The_University_of_Utah" class="mw-redirect" title="The University of Utah">The University of Utah</a></i>, 13 December 2009<span class="reference-accessdate">, retrieved <span class="nowrap">28 November</span> 2018</span></cite><span 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title="Special:BookSources/1-4027-3522-7"><bdi>1-4027-3522-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Divine+Proportion%3A+Phi+In+Art%2C+Nature%2C+and+Science&rft.place=New+York&rft.pages=%3Cspan+class%3D%22nowrap%22%3E20-%3C%2Fspan%3E21&rft.pub=Sterling&rft.date=2005&rft.isbn=1-4027-3522-7&rft.aulast=Hemenway&rft.aufirst=Priya&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnott2016" class="citation cs2">Knott, Ron (25 September 2016), <a rel="nofollow" class="external text" href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html#Rabbits">"The Fibonacci Numbers and Golden section in Nature – 1"</a>, <i><a href="/wiki/University_of_Surrey" title="University of Surrey">University of Surrey</a></i><span class="reference-accessdate">, retrieved <span class="nowrap">27 November</span> 2018</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=University+of+Surrey&rft.atitle=The+Fibonacci+Numbers+and+Golden+section+in+Nature+%E2%80%93+1&rft.date=2016-09-25&rft.aulast=Knott&rft.aufirst=Ron&rft_id=http%3A%2F%2Fwww.maths.surrey.ac.uk%2Fhosted-sites%2FR.Knott%2FFibonacci%2Ffibnat.html%23Rabbits&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnott" class="citation cs2">Knott, Ron, <a rel="nofollow" class="external text" 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Gardner">Gardner, Martin</a> (1996), <i>Mathematical Circus</i>, The Mathematical Association of America, p. 153, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-506-5" title="Special:BookSources/978-0-88385-506-5"><bdi>978-0-88385-506-5</bdi></a>, <q>It is ironic that Leonardo, who made valuable contributions to mathematics, is remembered today mainly because a 19th-century French number theorist, Édouard Lucas... attached the name Fibonacci to a number sequence that appears in a trivial problem in Liber abaci</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Circus&rft.pages=153&rft.pub=The+Mathematical+Association+of+America&rft.date=1996&rft.isbn=978-0-88385-506-5&rft.aulast=Gardner&rft.aufirst=Martin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSarah-Marie_Belcastro2018" class="citation book cs1">Sarah-Marie Belcastro (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xoqADwAAQBAJ"><i>Discrete Mathematics with Ducks</i></a> (2nd, illustrated ed.). CRC Press. p. 260. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-351-68369-2" title="Special:BookSources/978-1-351-68369-2"><bdi>978-1-351-68369-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+Mathematics+with+Ducks&rft.pages=260&rft.edition=2nd%2C+illustrated&rft.pub=CRC+Press&rft.date=2018&rft.isbn=978-1-351-68369-2&rft.au=Sarah-Marie+Belcastro&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DxoqADwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xoqADwAAQBAJ&pg=PA260">Extract of page 260</a></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" 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title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Fibonacci-Zahlen&rft.btitle=Der+Goldene+Schnitt&rft.series=Einblick+in+die+Wissenschaft&rft.pages=%3Cspan+class%3D%22nowrap%22%3E87-%3C%2Fspan%3E98&rft.pub=Vieweg%2BTeubner+Verlag&rft.date=1996&rft_id=info%3Adoi%2F10.1007%2F978-3-322-85165-9_6&rft.isbn=978-3-8154-2511-4&rft.aulast=Beutelspacher&rft.aufirst=Albrecht&rft.au=Petri%2C+Bernhard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEBall2003156-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBall2003156_25-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBall2003">Ball 2003</a>, p. 156.</span> </li> <li id="cite_note-FOOTNOTEBall2003155–156-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBall2003155–156_26-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBall2003">Ball 2003</a>, pp. 155–156.</span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSloane_"A002390"" class="citation web cs2"><a href="/wiki/Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.), <a rel="nofollow" class="external text" href="https://oeis.org/A002390">"Sequence A002390 (Decimal expansion of natural logarithm of golden ratio)"</a>, <i>The <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a></i>, OEIS Foundation</cite><span 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Williams calls this property "well known".</span> </li> <li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text"><i>Prime Numbers</i>, Richard Crandall, Carl Pomerance, Springer, second edition, 2005, p. 142.</span> </li> <li id="cite_note-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-47">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSloane_"A005478"" class="citation web cs2"><a href="/wiki/Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.), <a rel="nofollow" class="external text" href="https://oeis.org/A005478">"Sequence A005478 (Prime Fibonacci numbers)"</a>, <i>The <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a></i>, OEIS Foundation</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+On-Line+Encyclopedia+of+Integer+Sequences&rft.atitle=Sequence%26%23x20%3BA005478%26%23x20%3B%28Prime+Fibonacci+numbers%29&rft_id=https%3A%2F%2Foeis.org%2FA005478&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-48">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDiaconis2018" class="citation cs2"><a href="/wiki/Persi_Diaconis" title="Persi Diaconis">Diaconis, Persi</a> (2018), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20231118192225/https://statweb.stanford.edu/~cgates/PERSI/papers/probabilizing-fibonacci.pdf">"Probabilizing Fibonacci numbers"</a> <span class="cs1-format">(PDF)</span>, in <a href="/wiki/Steve_Butler_(mathematician)" title="Steve Butler (mathematician)">Butler, Steve</a>; Cooper, Joshua; Hurlbert, Glenn (eds.), <i>Connections in Discrete Mathematics: A Celebration of the Work of Ron Graham</i>, Cambridge University Press, pp. <span class="nowrap">1–</span>12, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-15398-1" title="Special:BookSources/978-1-107-15398-1"><bdi>978-1-107-15398-1</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=3821829">3821829</a>, archived from <a rel="nofollow" class="external text" href="https://statweb.stanford.edu/~cgates/PERSI/papers/probabilizing-fibonacci.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2023-11-18<span class="reference-accessdate">, retrieved <span class="nowrap">2022-11-23</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Probabilizing+Fibonacci+numbers&rft.btitle=Connections+in+Discrete+Mathematics%3A+A+Celebration+of+the+Work+of+Ron+Graham&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E12&rft.pub=Cambridge+University+Press&rft.date=2018&rft.isbn=978-1-107-15398-1&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3821829%23id-name%3DMR&rft.aulast=Diaconis&rft.aufirst=Persi&rft_id=https%3A%2F%2Fstatweb.stanford.edu%2F~cgates%2FPERSI%2Fpapers%2Fprobabilizing-fibonacci.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-49">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHonsberger1985" class="citation cs2">Honsberger, Ross (1985), "Mathematical Gems III", <i>AMS Dolciani Mathematical Expositions</i> (9): 133, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-318-4" title="Special:BookSources/978-0-88385-318-4"><bdi>978-0-88385-318-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=AMS+Dolciani+Mathematical+Expositions&rft.atitle=Mathematical+Gems+III&rft.issue=9&rft.pages=133&rft.date=1985&rft.isbn=978-0-88385-318-4&rft.aulast=Honsberger&rft.aufirst=Ross&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-50">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCohn1964" class="citation cs2">Cohn, J. 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title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ann.+Math.&rft.atitle=Classical+and+modular+approaches+to+exponential+Diophantine+equations.+I.+Fibonacci+and+Lucas+perfect+powers&rft.volume=2&rft.issue=163&rft.pages=%3Cspan+class%3D%22nowrap%22%3E969-%3C%2Fspan%3E1018&rft.date=2006&rft_id=info%3Aarxiv%2Fmath%2F0403046&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A10266596%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.4007%2Fannals.2006.163.969&rft_id=info%3Abibcode%2F2004math......3046B&rft.aulast=Bugeaud&rft.aufirst=Y&rft.au=Mignotte%2C+M&rft.au=Siksek%2C+S&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-53">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLuo1989" class="citation cs2">Luo, Ming (1989), <a 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href="https://mathscinet.ams.org/mathscinet-getitem?mr=1765401">1765401</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121789033">121789033</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Rendiconti+del+Circolo+Matematico+di+Palermo&rft.atitle=Perfect+Fibonacci+and+Lucas+numbers&rft.volume=49&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E313-%3C%2Fspan%3E18&rft.date=2000&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121789033%23id-name%3DS2CID&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1765401%23id-name%3DMR&rft.issn=1973-4409&rft_id=info%3Adoi%2F10.1007%2FBF02904236&rft.aulast=Luca&rft.aufirst=Florian&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-BGLLHT2011-55"><span 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title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annales+Mathematicae+at+Informaticae&rft.atitle=On+Perfect+numbers+which+are+ratios+of+two+Fibonacci+numbers&rft.volume=37&rft.pages=%3Cspan+class%3D%22nowrap%22%3E107-%3C%2Fspan%3E24&rft.date=2010&rft.issn=1787-6117&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2753031%23id-name%3DMR&rft.aulast=Luca&rft.aufirst=Florian&rft.au=Mej%C3%ADa+Huguet%2C+V.+Janitzio&rft_id=http%3A%2F%2Fami.ektf.hu%2Findex.php%3Fvol%3D37&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-57"><span class="mw-cite-backlink"><b><a href="#cite_ref-57">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnott" class="citation cs2">Knott, Ron, <a rel="nofollow" class="external text" href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibtable.html"><i>The Fibonacci numbers</i></a>, UK: Surrey</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Fibonacci+numbers&rft.place=UK&rft.pub=Surrey&rft.aulast=Knott&rft.aufirst=Ron&rft_id=http%3A%2F%2Fwww.maths.surrey.ac.uk%2Fhosted-sites%2FR.Knott%2FFibonacci%2Ffibtable.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-58"><span class="mw-cite-backlink"><b><a href="#cite_ref-58">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSloane_"A235383"" class="citation web cs2"><a href="/wiki/Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.), <a rel="nofollow" class="external text" href="https://oeis.org/A235383">"Sequence A235383 (Fibonacci numbers that are the product of other Fibonacci numbers)"</a>, <i>The <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a></i>, OEIS Foundation</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+On-Line+Encyclopedia+of+Integer+Sequences&rft.atitle=Sequence%26%23x20%3BA235383%26%23x20%3B%28Fibonacci+numbers+that+are+the+product+of+other+Fibonacci+numbers%29&rft_id=https%3A%2F%2Foeis.org%2FA235383&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-59"><span class="mw-cite-backlink"><b><a href="#cite_ref-59">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRibenboim1996" class="citation cs2"><a href="/wiki/Paulo_Ribenboim" title="Paulo Ribenboim">Ribenboim, Paulo</a> (1996), <i>The New Book of Prime Number Records</i>, New York: Springer, p. 64, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-94457-9" title="Special:BookSources/978-0-387-94457-9"><bdi>978-0-387-94457-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+New+Book+of+Prime+Number+Records&rft.place=New+York&rft.pages=64&rft.pub=Springer&rft.date=1996&rft.isbn=978-0-387-94457-9&rft.aulast=Ribenboim&rft.aufirst=Paulo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTELemmermeyer200073–74ex._2.25–28-60"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELemmermeyer200073–74ex._2.25–28_60-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLemmermeyer2000">Lemmermeyer 2000</a>, pp. 73–74, ex. 2.25–28.</span> </li> <li id="cite_note-FOOTNOTELemmermeyer200073–74ex._2.28-61"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELemmermeyer200073–74ex._2.28_61-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLemmermeyer2000">Lemmermeyer 2000</a>, pp. 73–74, ex. 2.28.</span> </li> <li id="cite_note-FOOTNOTELemmermeyer200073ex._2.27-62"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELemmermeyer200073ex._2.27_62-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLemmermeyer2000">Lemmermeyer 2000</a>, p. 73, ex. 2.27.</span> </li> <li id="cite_note-63"><span class="mw-cite-backlink"><b><a href="#cite_ref-63">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="http://mersennus.net/fibonacci/"><i>Fibonacci and Lucas factorizations</i></a>, Mersennus</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fibonacci+and+Lucas+factorizations&rft.pub=Mersennus&rft_id=http%3A%2F%2Fmersennus.net%2Ffibonacci%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span> collects all known factors of <span class="texhtml"><i>F</i>(<i>i</i>)</span> with <span class="texhtml"><i>i</i> < 10000</span>.</span> </li> <li id="cite_note-64"><span class="mw-cite-backlink"><b><a href="#cite_ref-64">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="http://fibonacci.redgolpe.com/"><i>Factors of Fibonacci and Lucas numbers</i></a>, Red golpe</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Factors+of+Fibonacci+and+Lucas+numbers&rft.pub=Red+golpe&rft_id=http%3A%2F%2Ffibonacci.redgolpe.com%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span> collects all known factors of <span class="texhtml"><i>F</i>(<i>i</i>)</span> with <span class="texhtml">10000 < <i>i</i> < 50000</span>.</span> </li> <li id="cite_note-65"><span class="mw-cite-backlink"><b><a href="#cite_ref-65">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFreydBrown1993" class="citation cs2">Freyd, Peter; Brown, Kevin S. (1993), "Problems and Solutions: Solutions: E3410", <i>The American Mathematical Monthly</i>, <b>99</b> (3): <span class="nowrap">278–</span>79, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2325076">10.2307/2325076</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2325076">2325076</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=Problems+and+Solutions%3A+Solutions%3A+E3410&rft.volume=99&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E278-%3C%2Fspan%3E79&rft.date=1993&rft_id=info%3Adoi%2F10.2307%2F2325076&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2325076%23id-name%3DJSTOR&rft.aulast=Freyd&rft.aufirst=Peter&rft.au=Brown%2C+Kevin+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-66"><span class="mw-cite-backlink"><b><a href="#cite_ref-66">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSloane_"A001175"" class="citation web cs2"><a href="/wiki/Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.), <a rel="nofollow" class="external text" href="https://oeis.org/A001175">"Sequence A001175 (Pisano periods (or Pisano numbers): period of Fibonacci numbers mod n)"</a>, <i>The <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a></i>, OEIS Foundation</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+On-Line+Encyclopedia+of+Integer+Sequences&rft.atitle=Sequence%26%23x20%3BA001175%26%23x20%3B%28Pisano+periods+%28or+Pisano+numbers%29%3A+period+of+Fibonacci+numbers+mod+n%29&rft_id=https%3A%2F%2Foeis.org%2FA001175&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-67"><span class="mw-cite-backlink"><b><a href="#cite_ref-67">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLüWang2006" class="citation cs2">Lü, Kebo; Wang, Jun (2006), <a rel="nofollow" class="external text" href="http://utilitasmathematica.com/index.php/Index/article/view/410">"<span class="texhtml mvar" style="font-style:italic;">k</span>-step Fibonacci sequence modulo <span class="texhtml mvar" style="font-style:italic;">m</span>"</a>, <i>Utilitas Mathematica</i>, <b>71</b>: <span class="nowrap">169–</span>177, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2278830">2278830</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Utilitas+Mathematica&rft.atitle=%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3Ek%3C%2Fspan%3E-step+Fibonacci+sequence+modulo+%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3Em%3C%2Fspan%3E&rft.volume=71&rft.pages=%3Cspan+class%3D%22nowrap%22%3E169-%3C%2Fspan%3E177&rft.date=2006&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2278830%23id-name%3DMR&rft.aulast=L%C3%BC&rft.aufirst=Kebo&rft.au=Wang%2C+Jun&rft_id=http%3A%2F%2Futilitasmathematica.com%2Findex.php%2FIndex%2Farticle%2Fview%2F410&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTELucas18917-68"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELucas18917_68-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLucas1891">Lucas 1891</a>, p. 7.</span> </li> <li id="cite_note-69"><span class="mw-cite-backlink"><b><a href="#cite_ref-69">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStanley2011" class="citation cs2">Stanley, Richard (2011), <i>Enumerative Combinatorics I (2nd ed.)</i>, Cambridge Univ. Press, p. 121, Ex 1.35, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-60262-5" title="Special:BookSources/978-1-107-60262-5"><bdi>978-1-107-60262-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Enumerative+Combinatorics+I+%282nd+ed.%29&rft.pages=121%2C+Ex+1.35&rft.pub=Cambridge+Univ.+Press&rft.date=2011&rft.isbn=978-1-107-60262-5&rft.aulast=Stanley&rft.aufirst=Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-70"><span class="mw-cite-backlink"><b><a href="#cite_ref-70">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarizanov1995" class="citation cs2">Harizanov, Valentina (1995), <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.rml/1204900767">"Review of Yuri V. Matiyasevich, <i>Hibert's Tenth Problem</i>"</a>, <i>Modern Logic</i>, <b>5</b> (3): <span class="nowrap">345–</span>55</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Modern+Logic&rft.atitle=Review+of+Yuri+V.+Matiyasevich%2C+Hibert%27s+Tenth+Problem&rft.volume=5&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E345-%3C%2Fspan%3E55&rft.date=1995&rft.aulast=Harizanov&rft.aufirst=Valentina&rft_id=http%3A%2F%2Fprojecteuclid.org%2Feuclid.rml%2F1204900767&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-71"><span class="mw-cite-backlink"><b><a href="#cite_ref-71">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPagni2001" class="citation cs2">Pagni, David (September 2001), "Fibonacci Meets Pythagoras", <i>Mathematics in School</i>, <b>30</b> (4): <span class="nowrap">39–</span>40, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/30215477">30215477</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+in+School&rft.atitle=Fibonacci+Meets+Pythagoras&rft.volume=30&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E39-%3C%2Fspan%3E40&rft.date=2001-09&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F30215477%23id-name%3DJSTOR&rft.aulast=Pagni&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-72"><span class="mw-cite-backlink"><b><a href="#cite_ref-72">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStephenson2005" class="citation cs2">Stephenson, Kenneth (2005), <a href="/wiki/Introduction_to_Circle_Packing" title="Introduction to Circle Packing"><i>Introduction to Circle Packing: The Theory of Discrete Analytic Functions</i></a>, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-82356-2" title="Special:BookSources/978-0-521-82356-2"><bdi>978-0-521-82356-2</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2131318">2131318</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Circle+Packing%3A+The+Theory+of+Discrete+Analytic+Functions&rft.pub=Cambridge+University+Press&rft.date=2005&rft.isbn=978-0-521-82356-2&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2131318%23id-name%3DMR&rft.aulast=Stephenson&rft.aufirst=Kenneth&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span>; see especially Lemma 8.2 (Ring Lemma), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=38PxEmKKhysC&pg=PA73">pp. 73–74</a>, and Appendix B, The Ring Lemma, pp. 318–321.</span> </li> <li id="cite_note-73"><span class="mw-cite-backlink"><b><a href="#cite_ref-73">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnuth1997" class="citation cs2 cs1-prop-long-vol"><a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, Donald E</a> (1997), <i>The Art of Computer Programming</i>, vol. 1: Fundamental Algorithms (3rd ed.), Addison–Wesley, p. 343, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-89683-1" title="Special:BookSources/978-0-201-89683-1"><bdi>978-0-201-89683-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Art+of+Computer+Programming&rft.pages=343&rft.edition=3rd&rft.pub=Addison%E2%80%93Wesley&rft.date=1997&rft.isbn=978-0-201-89683-1&rft.aulast=Knuth&rft.aufirst=Donald+E&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-74"><span class="mw-cite-backlink"><b><a href="#cite_ref-74">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAdelson-VelskyLandis1962" class="citation cs2 cs1-prop-foreign-lang-source">Adelson-Velsky, Georgy; Landis, Evgenii (1962), "An algorithm for the organization of information", <i><a href="/wiki/Proceedings_of_the_USSR_Academy_of_Sciences" title="Proceedings of the USSR Academy of Sciences">Proceedings of the USSR Academy of Sciences</a></i> (in Russian), <b>146</b>: <span class="nowrap">263–</span>266</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+USSR+Academy+of+Sciences&rft.atitle=An+algorithm+for+the+organization+of+information&rft.volume=146&rft.pages=%3Cspan+class%3D%22nowrap%22%3E263-%3C%2Fspan%3E266&rft.date=1962&rft.aulast=Adelson-Velsky&rft.aufirst=Georgy&rft.au=Landis%2C+Evgenii&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://zhjwpku.com/assets/pdf/AED2-10-avl-paper.pdf">English translation</a> by Myron J. 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Soc. 69 Suppl.: 98-115.</span> </li> <li id="cite_note-xcs-93"><span class="mw-cite-backlink">^ <a href="#cite_ref-xcs_93-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-xcs_93-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHutchison2004" class="citation cs2">Hutchison, Luke (September 2004), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200925132536/https://fhtw.byu.edu/static/conf/2005/hutchison-growing-fhtw2005.pdf">"Growing the Family Tree: The Power of DNA in Reconstructing Family Relationships"</a> <span class="cs1-format">(PDF)</span>, <i>Proceedings of the First Symposium on Bioinformatics and Biotechnology (BIOT-04)</i>, archived from <a rel="nofollow" class="external text" href="http://fhtw.byu.edu/static/conf/2005/hutchison-growing-fhtw2005.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2020-09-25<span class="reference-accessdate">, retrieved <span class="nowrap">2016-09-03</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+First+Symposium+on+Bioinformatics+and+Biotechnology+%28BIOT-04%29&rft.atitle=Growing+the+Family+Tree%3A+The+Power+of+DNA+in+Reconstructing+Family+Relationships&rft.date=2004-09&rft.aulast=Hutchison&rft.aufirst=Luke&rft_id=http%3A%2F%2Ffhtw.byu.edu%2Fstatic%2Fconf%2F2005%2Fhutchison-growing-fhtw2005.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTELivio200398–99-94"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELivio200398–99_94-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLivio2003">Livio 2003</a>, pp. 98–99.</span> </li> <li id="cite_note-95"><span class="mw-cite-backlink"><b><a href="#cite_ref-95">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2">"Zeckendorf representation", <a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php/Zeckendorf_representation"><i>Encyclopedia of Math</i></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Zeckendorf+representation&rft.btitle=Encyclopedia+of+Math&rft_id=http%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%2FZeckendorf_representation&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-96"><span class="mw-cite-backlink"><b><a href="#cite_ref-96">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPatranabisDana1985" class="citation cs2 cs1-prop-long-vol">Patranabis, D.; Dana, S. K. (December 1985), "Single-shunt fault diagnosis through terminal attenuation measurement and using Fibonacci numbers", <i><a href="/wiki/IEEE_Transactions_on_Instrumentation_and_Measurement" title="IEEE Transactions on Instrumentation and Measurement">IEEE Transactions on Instrumentation and Measurement</a></i>, IM-34 (4): <span class="nowrap">650–</span>653, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1985ITIM...34..650P">1985ITIM...34..650P</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Ftim.1985.4315428">10.1109/tim.1985.4315428</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:35413237">35413237</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Instrumentation+and+Measurement&rft.atitle=Single-shunt+fault+diagnosis+through+terminal+attenuation+measurement+and+using+Fibonacci+numbers&rft.volume=IM-34&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E650-%3C%2Fspan%3E653&rft.date=1985-12&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A35413237%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1109%2Ftim.1985.4315428&rft_id=info%3Abibcode%2F1985ITIM...34..650P&rft.aulast=Patranabis&rft.aufirst=D.&rft.au=Dana%2C+S.+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-Brasch_et_al._2012-97"><span class="mw-cite-backlink"><b><a href="#cite_ref-Brasch_et_al._2012_97-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBraschByströmLystad2012" class="citation cs2">Brasch, T. von; Byström, J.; Lystad, L.P. (2012), <a rel="nofollow" class="external text" href="http://urn.kb.se/resolve?urn=urn:nbn:se:ltu:diva-24073">"Optimal Control and the Fibonacci Sequence"</a>, <i>Journal of Optimization Theory and Applications</i>, <b>154</b> (3): <span class="nowrap">857–</span>78, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10957-012-0061-2">10.1007/s10957-012-0061-2</a>, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/11250%2F180781">11250/180781</a></span>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:8550726">8550726</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Optimization+Theory+and+Applications&rft.atitle=Optimal+Control+and+the+Fibonacci+Sequence&rft.volume=154&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E857-%3C%2Fspan%3E78&rft.date=2012&rft_id=info%3Ahdl%2F11250%2F180781&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A8550726%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs10957-012-0061-2&rft.aulast=Brasch&rft.aufirst=T.+von&rft.au=Bystr%C3%B6m%2C+J.&rft.au=Lystad%2C+L.P.&rft_id=http%3A%2F%2Furn.kb.se%2Fresolve%3Furn%3Durn%3Anbn%3Ase%3Altu%3Adiva-24073&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTELivio2003176-98"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELivio2003176_98-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLivio2003">Livio 2003</a>, p. 176.</span> </li> <li id="cite_note-FOOTNOTELivio2003193-99"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELivio2003193_99-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLivio2003">Livio 2003</a>, p. 193.</span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Works_cited">Works cited</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fibonacci_sequence&action=edit&section=39" title="Edit section: Works cited"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBall2003" class="citation cs2">Ball, Keith M (2003), "8: Fibonacci's Rabbits Revisited", <i>Strange Curves, Counting Rabbits, and Other Mathematical Explorations</i>, Princeton, NJ: <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-11321-0" title="Special:BookSources/978-0-691-11321-0"><bdi>978-0-691-11321-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=8%3A+Fibonacci%27s+Rabbits+Revisited&rft.btitle=Strange+Curves%2C+Counting+Rabbits%2C+and+Other+Mathematical+Explorations&rft.place=Princeton%2C+NJ&rft.pub=Princeton+University+Press&rft.date=2003&rft.isbn=978-0-691-11321-0&rft.aulast=Ball&rft.aufirst=Keith+M&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeckGeoghegan2010" class="citation cs2">Beck, Matthias; Geoghegan, Ross (2010), <i>The Art of Proof: Basic Training for Deeper Mathematics</i>, New York: Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4419-7022-0" title="Special:BookSources/978-1-4419-7022-0"><bdi>978-1-4419-7022-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Art+of+Proof%3A+Basic+Training+for+Deeper+Mathematics&rft.place=New+York&rft.pub=Springer&rft.date=2010&rft.isbn=978-1-4419-7022-0&rft.aulast=Beck&rft.aufirst=Matthias&rft.au=Geoghegan%2C+Ross&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBóna2011" class="citation cs2"><a href="/wiki/Mikl%C3%B3s_B%C3%B3na" title="Miklós Bóna">Bóna, Miklós</a> (2011), <i>A Walk Through Combinatorics</i> (3rd ed.), New Jersey: World Scientific, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4335-23-2" title="Special:BookSources/978-981-4335-23-2"><bdi>978-981-4335-23-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Walk+Through+Combinatorics&rft.place=New+Jersey&rft.edition=3rd&rft.pub=World+Scientific&rft.date=2011&rft.isbn=978-981-4335-23-2&rft.aulast=B%C3%B3na&rft.aufirst=Mikl%C3%B3s&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span>.</li> <li><span class="anchor" id="Borwein"></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBorweinBorwein1998" class="citation cs2"><a href="/wiki/Jonathan_Borwein" title="Jonathan Borwein">Borwein, Jonathan M.</a>; <a href="/wiki/Peter_Borwein" title="Peter Borwein">Borwein, Peter B.</a> (July 1998), <a rel="nofollow" class="external text" href="http://www.wiley.com/WileyCDA/WileyTitle/productCd-047131515X.html"><i>Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity</i></a>, Wiley, pp. <span class="nowrap">91–</span>101, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-31515-5" title="Special:BookSources/978-0-471-31515-5"><bdi>978-0-471-31515-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Pi+and+the+AGM%3A+A+Study+in+Analytic+Number+Theory+and+Computational+Complexity&rft.pages=%3Cspan+class%3D%22nowrap%22%3E91-%3C%2Fspan%3E101&rft.pub=Wiley&rft.date=1998-07&rft.isbn=978-0-471-31515-5&rft.aulast=Borwein&rft.aufirst=Jonathan+M.&rft.au=Borwein%2C+Peter+B.&rft_id=http%3A%2F%2Fwww.wiley.com%2FWileyCDA%2FWileyTitle%2FproductCd-047131515X.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLemmermeyer2000" class="citation cs2">Lemmermeyer, Franz (2000), <i>Reciprocity Laws: From Euler to Eisenstein</i>, Springer Monographs in Mathematics, New York: Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-66957-9" title="Special:BookSources/978-3-540-66957-9"><bdi>978-3-540-66957-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Reciprocity+Laws%3A+From+Euler+to+Eisenstein&rft.place=New+York&rft.series=Springer+Monographs+in+Mathematics&rft.pub=Springer&rft.date=2000&rft.isbn=978-3-540-66957-9&rft.aulast=Lemmermeyer&rft.aufirst=Franz&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLivio2003" class="citation cs2"><a href="/wiki/Mario_Livio" title="Mario Livio">Livio, Mario</a> (2003) [2002], <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bUARfgWRH14C"><i>The Golden Ratio: The Story of Phi, the World's Most Astonishing Number</i></a> (First trade paperback ed.), New York City: <a href="/wiki/Random_House" title="Random House">Broadway Books</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7679-0816-3" title="Special:BookSources/0-7679-0816-3"><bdi>0-7679-0816-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Golden+Ratio%3A+The+Story+of+Phi%2C+the+World%27s+Most+Astonishing+Number&rft.place=New+York+City&rft.edition=First+trade+paperback&rft.pub=Broadway+Books&rft.date=2003&rft.isbn=0-7679-0816-3&rft.aulast=Livio&rft.aufirst=Mario&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DbUARfgWRH14C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLucas1891" class="citation cs2 cs1-prop-foreign-lang-source">Lucas, Édouard (1891), <a rel="nofollow" class="external text" href="https://archive.org/details/thoriedesnombr01lucauoft"><i>Théorie des nombres</i></a> (in French), vol. 1, Paris: Gauthier-Villars</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Th%C3%A9orie+des+nombres&rft.place=Paris&rft.pub=Gauthier-Villars&rft.date=1891&rft.aulast=Lucas&rft.aufirst=%C3%89douard&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fthoriedesnombr01lucauoft&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSigler2002" class="citation cs2">Sigler, L. E. (2002), <i>Fibonacci's Liber Abaci: A Translation into Modern English of Leonardo Pisano's Book of Calculation</i>, Sources and Studies in the History of Mathematics and Physical Sciences, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-95419-6" title="Special:BookSources/978-0-387-95419-6"><bdi>978-0-387-95419-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fibonacci%27s+Liber+Abaci%3A+A+Translation+into+Modern+English+of+Leonardo+Pisano%27s+Book+of+Calculation&rft.series=Sources+and+Studies+in+the+History+of+Mathematics+and+Physical+Sciences&rft.pub=Springer&rft.date=2002&rft.isbn=978-0-387-95419-6&rft.aulast=Sigler&rft.aufirst=L.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFibonacci+sequence" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span 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title="wikibooks:Fibonacci number program">Fibonacci number program</a></b></i></div></div> </div> <ul><li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=hbUQlrLDAgw"><span class="plainlinks">Fibonacci Sequence and Golden Ratio: Mathematics in the Modern World - Mathuklasan with Sir Ram</span></a> on <a href="/wiki/YouTube_video_(identifier)" class="mw-redirect" title="YouTube video (identifier)">YouTube</a> - animation of sequence, spiral, golden ratio, rabbit pair growth. Examples in art, music, architecture, nature, and astronomy</li> <li><a rel="nofollow" class="external text" href="http://www.mathpages.com/home/kmath078/kmath078.htm">Periods of Fibonacci Sequences Mod m</a> at MathPages</li> <li><a rel="nofollow" class="external text" href="http://www.physorg.com/news97227410.html">Scientists find clues to the formation of Fibonacci spirals in nature</a></li> <li><a rel="nofollow" class="external text" href="https://www.bbc.co.uk/programmes/b008ct2j">Fibonacci Sequence</a> on <a href="/wiki/In_Our_Time_(radio_series)" title="In Our Time (radio series)"><i>In Our Time</i></a> at the <a href="/wiki/BBC" title="BBC">BBC</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Fibonacci_numbers">"Fibonacci numbers"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of 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title="Exponentiation">Powers</a> and related numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Achilles_number" title="Achilles number">Achilles</a></li> <li><a href="/wiki/Power_of_two" title="Power of two">Power of 2</a></li> <li><a href="/wiki/Power_of_three" title="Power of three">Power of 3</a></li> <li><a href="/wiki/Power_of_10" title="Power of 10">Power of 10</a></li> <li><a href="/wiki/Square_number" title="Square number">Square</a></li> <li><a href="/wiki/Cube_(algebra)" title="Cube (algebra)">Cube</a></li> <li><a href="/wiki/Fourth_power" title="Fourth power">Fourth power</a></li> <li><a href="/wiki/Fifth_power_(algebra)" title="Fifth power (algebra)">Fifth power</a></li> <li><a href="/wiki/Sixth_power" title="Sixth power">Sixth power</a></li> <li><a href="/wiki/Seventh_power" title="Seventh power">Seventh power</a></li> <li><a href="/wiki/Eighth_power" title="Eighth power">Eighth power</a></li> <li><a href="/wiki/Perfect_power" title="Perfect power">Perfect power</a></li> <li><a href="/wiki/Powerful_number" title="Powerful number">Powerful</a></li> <li><a href="/wiki/Prime_power" title="Prime power">Prime power</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Of_the_form_a_×_2b_±_1743" style="font-size:114%;margin:0 4em">Of the form <i>a</i> × 2<sup><i>b</i></sup> ± 1</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cullen_number" title="Cullen number">Cullen</a></li> <li><a href="/wiki/Double_Mersenne_number" title="Double Mersenne number">Double Mersenne</a></li> <li><a href="/wiki/Fermat_number" title="Fermat number">Fermat</a></li> <li><a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne</a></li> <li><a href="/wiki/Proth_number" class="mw-redirect" title="Proth number">Proth</a></li> <li><a href="/wiki/Thabit_number" title="Thabit number">Thabit</a></li> <li><a href="/wiki/Woodall_number" title="Woodall number">Woodall</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_polynomial_numbers743" style="font-size:114%;margin:0 4em">Other polynomial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hilbert_number" title="Hilbert number">Hilbert</a></li> <li><a href="/wiki/Idoneal_number" title="Idoneal number">Idoneal</a></li> <li><a href="/wiki/Leyland_number" title="Leyland number">Leyland</a></li> <li><a href="/wiki/Loeschian_number" class="mw-redirect" title="Loeschian number">Loeschian</a></li> <li><a href="/wiki/Lucky_numbers_of_Euler" title="Lucky numbers of Euler">Lucky numbers of Euler</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Recursively_defined_numbers743" style="font-size:114%;margin:0 4em"><a href="/wiki/Recursion" title="Recursion">Recursively</a> defined numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Fibonacci</a></li> <li><a href="/wiki/Jacobsthal_number" title="Jacobsthal number">Jacobsthal</a></li> <li><a href="/wiki/Leonardo_number" title="Leonardo number">Leonardo</a></li> <li><a href="/wiki/Lucas_number" title="Lucas number">Lucas</a></li> <li><a href="/wiki/Supergolden_ratio#Narayana_sequence" title="Supergolden ratio">Narayana</a></li> <li><a href="/wiki/Padovan_sequence" title="Padovan sequence">Padovan</a></li> <li><a href="/wiki/Pell_number" title="Pell number">Pell</a></li> <li><a href="/wiki/Perrin_number" title="Perrin number">Perrin</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Possessing_a_specific_set_of_other_numbers743" style="font-size:114%;margin:0 4em">Possessing a specific set of other numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amenable_number" title="Amenable number">Amenable</a></li> <li><a href="/wiki/Congruent_number" title="Congruent number">Congruent</a></li> <li><a href="/wiki/Kn%C3%B6del_number" title="Knödel number">Knödel</a></li> <li><a href="/wiki/Riesel_number" title="Riesel number">Riesel</a></li> <li><a href="/wiki/Sierpi%C5%84ski_number" title="Sierpiński number">Sierpiński</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Expressible_via_specific_sums743" style="font-size:114%;margin:0 4em">Expressible via specific sums</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nonhypotenuse_number" title="Nonhypotenuse number">Nonhypotenuse</a></li> <li><a href="/wiki/Polite_number" title="Polite number">Polite</a></li> <li><a href="/wiki/Practical_number" title="Practical number">Practical</a></li> <li><a href="/wiki/Primary_pseudoperfect_number" title="Primary pseudoperfect number">Primary pseudoperfect</a></li> <li><a href="/wiki/Ulam_number" title="Ulam number">Ulam</a></li> <li><a href="/wiki/Wolstenholme_number" title="Wolstenholme number">Wolstenholme</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Figurate_numbers743" style="font-size:114%;margin:0 4em"><a href="/wiki/Figurate_number" title="Figurate number">Figurate numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">2-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Centered_polygonal_number" title="Centered polygonal number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Centered_triangular_number" title="Centered triangular number">Centered triangular</a></li> <li><a href="/wiki/Centered_square_number" title="Centered square number">Centered square</a></li> <li><a href="/wiki/Centered_pentagonal_number" title="Centered pentagonal number">Centered pentagonal</a></li> <li><a href="/wiki/Centered_hexagonal_number" title="Centered hexagonal number">Centered hexagonal</a></li> <li><a href="/wiki/Centered_heptagonal_number" title="Centered heptagonal number">Centered heptagonal</a></li> <li><a href="/wiki/Centered_octagonal_number" title="Centered octagonal number">Centered octagonal</a></li> <li><a href="/wiki/Centered_nonagonal_number" title="Centered nonagonal number">Centered nonagonal</a></li> <li><a href="/wiki/Centered_decagonal_number" title="Centered decagonal number">Centered decagonal</a></li> <li><a href="/wiki/Star_number" title="Star number">Star</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Polygonal_number" title="Polygonal number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Triangular_number" title="Triangular number">Triangular</a></li> <li><a href="/wiki/Square_number" title="Square number">Square</a></li> <li><a href="/wiki/Square_triangular_number" title="Square triangular number">Square triangular</a></li> <li><a href="/wiki/Pentagonal_number" title="Pentagonal number">Pentagonal</a></li> <li><a href="/wiki/Hexagonal_number" title="Hexagonal number">Hexagonal</a></li> <li><a href="/wiki/Heptagonal_number" title="Heptagonal number">Heptagonal</a></li> <li><a href="/wiki/Octagonal_number" title="Octagonal number">Octagonal</a></li> <li><a href="/wiki/Nonagonal_number" title="Nonagonal number">Nonagonal</a></li> <li><a href="/wiki/Decagonal_number" title="Decagonal number">Decagonal</a></li> <li><a href="/wiki/Dodecagonal_number" title="Dodecagonal number">Dodecagonal</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Three-dimensional_space" title="Three-dimensional space">3-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Centered_polyhedral_number" title="Centered polyhedral number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Centered_tetrahedral_number" title="Centered tetrahedral number">Centered tetrahedral</a></li> <li><a href="/wiki/Centered_cube_number" title="Centered cube number">Centered cube</a></li> <li><a href="/wiki/Centered_octahedral_number" title="Centered octahedral number">Centered octahedral</a></li> <li><a href="/wiki/Centered_dodecahedral_number" title="Centered dodecahedral number">Centered dodecahedral</a></li> <li><a href="/wiki/Centered_icosahedral_number" title="Centered icosahedral number">Centered icosahedral</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Polyhedral_number" class="mw-redirect" title="Polyhedral number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tetrahedral_number" title="Tetrahedral number">Tetrahedral</a></li> <li><a href="/wiki/Cube_(algebra)" title="Cube (algebra)">Cubic</a></li> <li><a href="/wiki/Octahedral_number" title="Octahedral number">Octahedral</a></li> <li><a href="/wiki/Dodecahedral_number" title="Dodecahedral number">Dodecahedral</a></li> <li><a href="/wiki/Icosahedral_number" title="Icosahedral number">Icosahedral</a></li> <li><a href="/wiki/Stella_octangula_number" title="Stella octangula number">Stella octangula</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Pyramidal_number" title="Pyramidal number">pyramidal</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Square_pyramidal_number" title="Square pyramidal number">Square pyramidal</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Four-dimensional_space" title="Four-dimensional space">4-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">non-centered</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pentatope_number" title="Pentatope number">Pentatope</a></li> <li><a href="/wiki/Squared_triangular_number" title="Squared triangular number">Squared triangular</a></li> <li><a href="/wiki/Fourth_power" title="Fourth power">Tesseractic</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Combinatorial_numbers743" style="font-size:114%;margin:0 4em">Combinatorial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell_number" title="Bell number">Bell</a></li> <li><a href="/wiki/Cake_number" title="Cake number">Cake</a></li> <li><a href="/wiki/Catalan_number" title="Catalan number">Catalan</a></li> <li><a href="/wiki/Dedekind_number" title="Dedekind number">Dedekind</a></li> <li><a href="/wiki/Delannoy_number" title="Delannoy number">Delannoy</a></li> <li><a href="/wiki/Euler_number" class="mw-redirect" title="Euler number">Euler</a></li> <li><a href="/wiki/Eulerian_number" title="Eulerian number">Eulerian</a></li> <li><a href="/wiki/Fuss%E2%80%93Catalan_number" title="Fuss–Catalan number">Fuss–Catalan</a></li> <li><a href="/wiki/Lah_number" title="Lah number">Lah</a></li> <li><a href="/wiki/Lazy_caterer%27s_sequence" title="Lazy caterer's sequence">Lazy caterer's sequence</a></li> <li><a href="/wiki/Lobb_number" title="Lobb number">Lobb</a></li> <li><a href="/wiki/Motzkin_number" title="Motzkin number">Motzkin</a></li> <li><a href="/wiki/Narayana_number" title="Narayana number">Narayana</a></li> <li><a href="/wiki/Ordered_Bell_number" title="Ordered Bell number">Ordered Bell</a></li> <li><a href="/wiki/Schr%C3%B6der_number" title="Schröder number">Schröder</a></li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Hipparchus_number" title="Schröder–Hipparchus number">Schröder–Hipparchus</a></li> <li><a href="/wiki/Stirling_numbers_of_the_first_kind" title="Stirling numbers of the first kind">Stirling first</a></li> <li><a href="/wiki/Stirling_numbers_of_the_second_kind" title="Stirling numbers of the second kind">Stirling second</a></li> <li><a href="/wiki/Telephone_number_(mathematics)" title="Telephone number (mathematics)">Telephone number</a></li> <li><a href="/wiki/Wedderburn%E2%80%93Etherington_number" title="Wedderburn–Etherington number">Wedderburn–Etherington</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Primes743" style="font-size:114%;margin:0 4em"><a href="/wiki/Prime_number" title="Prime number">Primes</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Wieferich_prime#Wieferich_numbers" title="Wieferich prime">Wieferich</a></li> <li><a href="/wiki/Wall%E2%80%93Sun%E2%80%93Sun_prime" title="Wall–Sun–Sun prime">Wall–Sun–Sun</a></li> <li><a href="/wiki/Wolstenholme_prime" title="Wolstenholme prime">Wolstenholme prime</a></li> <li><a href="/wiki/Wilson_prime#Wilson_numbers" title="Wilson prime">Wilson</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Pseudoprimes743" style="font-size:114%;margin:0 4em"><a href="/wiki/Pseudoprime" title="Pseudoprime">Pseudoprimes</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Carmichael_number" title="Carmichael number">Carmichael number</a></li> <li><a href="/wiki/Catalan_pseudoprime" title="Catalan pseudoprime">Catalan pseudoprime</a></li> <li><a href="/wiki/Elliptic_pseudoprime" title="Elliptic pseudoprime">Elliptic pseudoprime</a></li> <li><a href="/wiki/Euler_pseudoprime" title="Euler pseudoprime">Euler pseudoprime</a></li> <li><a href="/wiki/Euler%E2%80%93Jacobi_pseudoprime" title="Euler–Jacobi pseudoprime">Euler–Jacobi pseudoprime</a></li> <li><a href="/wiki/Fermat_pseudoprime" title="Fermat pseudoprime">Fermat pseudoprime</a></li> <li><a href="/wiki/Frobenius_pseudoprime" title="Frobenius pseudoprime">Frobenius pseudoprime</a></li> <li><a href="/wiki/Lucas_pseudoprime" title="Lucas pseudoprime">Lucas pseudoprime</a></li> <li><a href="/wiki/Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael number</a></li> <li><a href="/wiki/Perrin_number#Perrin_primality_test" title="Perrin number">Perrin pseudoprime</a></li> <li><a href="/wiki/Somer%E2%80%93Lucas_pseudoprime" title="Somer–Lucas pseudoprime">Somer–Lucas pseudoprime</a></li> <li><a href="/wiki/Strong_pseudoprime" title="Strong pseudoprime">Strong pseudoprime</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Arithmetic_functions_and_dynamics743" style="font-size:114%;margin:0 4em"><a href="/wiki/Arithmetic_function" title="Arithmetic function">Arithmetic functions</a> and <a href="/wiki/Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Divisor_function" title="Divisor function">Divisor functions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abundant_number" title="Abundant number">Abundant</a></li> <li><a href="/wiki/Almost_perfect_number" title="Almost perfect number">Almost perfect</a></li> <li><a href="/wiki/Arithmetic_number" title="Arithmetic number">Arithmetic</a></li> <li><a href="/wiki/Betrothed_numbers" title="Betrothed numbers">Betrothed</a></li> <li><a href="/wiki/Colossally_abundant_number" title="Colossally abundant number">Colossally abundant</a></li> <li><a href="/wiki/Deficient_number" title="Deficient number">Deficient</a></li> <li><a href="/wiki/Descartes_number" title="Descartes number">Descartes</a></li> <li><a href="/wiki/Hemiperfect_number" title="Hemiperfect number">Hemiperfect</a></li> <li><a href="/wiki/Highly_abundant_number" title="Highly abundant number">Highly abundant</a></li> <li><a href="/wiki/Highly_composite_number" title="Highly composite number">Highly composite</a></li> <li><a href="/wiki/Hyperperfect_number" title="Hyperperfect number">Hyperperfect</a></li> <li><a href="/wiki/Multiply_perfect_number" title="Multiply perfect number">Multiply perfect</a></li> <li><a href="/wiki/Perfect_number" title="Perfect number">Perfect</a></li> <li><a href="/wiki/Practical_number" title="Practical number">Practical</a></li> <li><a href="/wiki/Primitive_abundant_number" title="Primitive abundant number">Primitive abundant</a></li> <li><a href="/wiki/Quasiperfect_number" title="Quasiperfect number">Quasiperfect</a></li> <li><a href="/wiki/Refactorable_number" title="Refactorable number">Refactorable</a></li> <li><a href="/wiki/Semiperfect_number" title="Semiperfect number">Semiperfect</a></li> <li><a href="/wiki/Sublime_number" title="Sublime number">Sublime</a></li> <li><a href="/wiki/Superabundant_number" title="Superabundant number">Superabundant</a></li> <li><a href="/wiki/Superior_highly_composite_number" title="Superior highly composite number">Superior highly composite</a></li> <li><a href="/wiki/Superperfect_number" title="Superperfect number">Superperfect</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Prime_omega_function" title="Prime omega function">Prime omega functions</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Almost_prime" title="Almost prime">Almost prime</a></li> <li><a href="/wiki/Semiprime" title="Semiprime">Semiprime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Euler%27s_totient_function" title="Euler's totient function">Euler's totient function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Highly_cototient_number" title="Highly cototient number">Highly cototient</a></li> <li><a href="/wiki/Highly_totient_number" title="Highly totient number">Highly totient</a></li> <li><a href="/wiki/Noncototient" title="Noncototient">Noncototient</a></li> <li><a href="/wiki/Nontotient" title="Nontotient">Nontotient</a></li> <li><a href="/wiki/Perfect_totient_number" title="Perfect totient number">Perfect totient</a></li> <li><a href="/wiki/Sparsely_totient_number" title="Sparsely totient number">Sparsely totient</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Aliquot_sequence" title="Aliquot sequence">Aliquot sequences</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amicable_numbers" title="Amicable numbers">Amicable</a></li> <li><a href="/wiki/Perfect_number" title="Perfect number">Perfect</a></li> <li><a href="/wiki/Sociable_numbers" class="mw-redirect" title="Sociable numbers">Sociable</a></li> <li><a href="/wiki/Untouchable_number" title="Untouchable number">Untouchable</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Primorial" title="Primorial">Primorial</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Euclid_number" title="Euclid number">Euclid</a></li> <li><a href="/wiki/Fortunate_number" title="Fortunate number">Fortunate</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_prime_factor_or_divisor_related_numbers743" style="font-size:114%;margin:0 4em">Other <a href="/wiki/Prime_factor" class="mw-redirect" title="Prime factor">prime factor</a> or <a href="/wiki/Divisor" title="Divisor">divisor</a> related numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Blum_integer" title="Blum integer">Blum</a></li> <li><a href="/wiki/Cyclic_number_(group_theory)" title="Cyclic number (group theory)">Cyclic</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Nicolas_number" title="Erdős–Nicolas number">Erdős–Nicolas</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Woods_number" title="Erdős–Woods number">Erdős–Woods</a></li> <li><a href="/wiki/Friendly_number" title="Friendly number">Friendly</a></li> <li><a href="/wiki/Giuga_number" title="Giuga number">Giuga</a></li> <li><a href="/wiki/Harmonic_divisor_number" title="Harmonic divisor number">Harmonic divisor</a></li> <li><a href="/wiki/Jordan%E2%80%93P%C3%B3lya_number" title="Jordan–Pólya number">Jordan–Pólya</a></li> <li><a href="/wiki/Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael</a></li> <li><a href="/wiki/Pronic_number" title="Pronic number">Pronic</a></li> <li><a href="/wiki/Regular_number" title="Regular number">Regular</a></li> <li><a href="/wiki/Rough_number" title="Rough number">Rough</a></li> <li><a href="/wiki/Smooth_number" title="Smooth number">Smooth</a></li> <li><a href="/wiki/Sphenic_number" title="Sphenic number">Sphenic</a></li> <li><a href="/wiki/St%C3%B8rmer_number" title="Størmer number">Størmer</a></li> <li><a href="/wiki/Super-Poulet_number" title="Super-Poulet number">Super-Poulet</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Numeral_system-dependent_numbers743" style="font-size:114%;margin:0 4em"><a href="/wiki/Numeral_system" title="Numeral system">Numeral system</a>-dependent numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Arithmetic_function" title="Arithmetic function">Arithmetic functions</a> <br />and <a href="/wiki/Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Persistence_of_a_number" title="Persistence of a number">Persistence</a> <ul><li><a href="/wiki/Additive_persistence" class="mw-redirect" title="Additive persistence">Additive</a></li> <li><a href="/wiki/Multiplicative_persistence" class="mw-redirect" title="Multiplicative persistence">Multiplicative</a></li></ul></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Digit_sum" title="Digit sum">Digit sum</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Digit_sum" title="Digit sum">Digit sum</a></li> <li><a href="/wiki/Digital_root" title="Digital root">Digital root</a></li> <li><a href="/wiki/Self_number" title="Self number">Self</a></li> <li><a href="/wiki/Sum-product_number" title="Sum-product number">Sum-product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digit product</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Multiplicative_digital_root" title="Multiplicative digital root">Multiplicative digital root</a></li> <li><a href="/wiki/Sum-product_number" title="Sum-product number">Sum-product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Coding-related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Meertens_number" title="Meertens number">Meertens</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dudeney_number" title="Dudeney number">Dudeney</a></li> <li><a href="/wiki/Factorion" title="Factorion">Factorion</a></li> <li><a href="/wiki/Kaprekar_number" title="Kaprekar number">Kaprekar</a></li> <li><a href="/wiki/Kaprekar%27s_routine" title="Kaprekar's routine">Kaprekar's constant</a></li> <li><a href="/wiki/Keith_number" title="Keith number">Keith</a></li> <li><a href="/wiki/Lychrel_number" title="Lychrel number">Lychrel</a></li> <li><a href="/wiki/Narcissistic_number" title="Narcissistic number">Narcissistic</a></li> <li><a href="/wiki/Perfect_digit-to-digit_invariant" title="Perfect digit-to-digit invariant">Perfect digit-to-digit invariant</a></li> <li><a href="/wiki/Perfect_digital_invariant" title="Perfect digital invariant">Perfect digital invariant</a> <ul><li><a href="/wiki/Happy_number" title="Happy number">Happy</a></li></ul></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/P-adic_numbers" class="mw-redirect" title="P-adic numbers">P-adic numbers</a>-related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Automorphic_number" title="Automorphic number">Automorphic</a> <ul><li><a href="/wiki/Trimorphic_number" class="mw-redirect" title="Trimorphic number">Trimorphic</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Numerical_digit" title="Numerical digit">Digit</a>-composition related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Palindromic_number" title="Palindromic number">Palindromic</a></li> <li><a href="/wiki/Pandigital_number" title="Pandigital number">Pandigital</a></li> <li><a href="/wiki/Repdigit" title="Repdigit">Repdigit</a></li> <li><a href="/wiki/Repunit" title="Repunit">Repunit</a></li> <li><a href="/wiki/Self-descriptive_number" title="Self-descriptive number">Self-descriptive</a></li> <li><a href="/wiki/Smarandache%E2%80%93Wellin_number" title="Smarandache–Wellin number">Smarandache–Wellin</a></li> <li><a href="/wiki/Undulating_number" title="Undulating number">Undulating</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digit-<a href="/wiki/Permutation" title="Permutation">permutation</a> related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cyclic_number" title="Cyclic number">Cyclic</a></li> <li><a href="/wiki/Digit-reassembly_number" title="Digit-reassembly number">Digit-reassembly</a></li> <li><a href="/wiki/Parasitic_number" title="Parasitic number">Parasitic</a></li> <li><a href="/wiki/Primeval_number" title="Primeval number">Primeval</a></li> <li><a href="/wiki/Transposable_integer" title="Transposable integer">Transposable</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Divisor-related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Equidigital_number" title="Equidigital number">Equidigital</a></li> <li><a href="/wiki/Extravagant_number" title="Extravagant number">Extravagant</a></li> <li><a href="/wiki/Frugal_number" title="Frugal number">Frugal</a></li> <li><a href="/wiki/Harshad_number" title="Harshad number">Harshad</a></li> <li><a href="/wiki/Polydivisible_number" title="Polydivisible number">Polydivisible</a></li> <li><a href="/wiki/Smith_number" title="Smith number">Smith</a></li> <li><a href="/wiki/Vampire_number" title="Vampire number">Vampire</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Friedman_number" title="Friedman number">Friedman</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Binary_numbers743" style="font-size:114%;margin:0 4em"><a href="/wiki/Binary_number" title="Binary number">Binary numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Evil_number" title="Evil number">Evil</a></li> <li><a href="/wiki/Odious_number" title="Odious number">Odious</a></li> <li><a href="/wiki/Pernicious_number" title="Pernicious number">Pernicious</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Generated_via_a_sieve743" style="font-size:114%;margin:0 4em">Generated via a <a href="/wiki/Sieve_theory" title="Sieve theory">sieve</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Lucky_number" title="Lucky number">Lucky</a></li> <li><a href="/wiki/Generation_of_primes" title="Generation of primes">Prime</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Sorting_related743" style="font-size:114%;margin:0 4em"><a href="/wiki/Sorting_algorithm" title="Sorting algorithm">Sorting</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pancake_sorting" title="Pancake sorting">Pancake number</a></li> <li><a href="/wiki/Sorting_number" title="Sorting number">Sorting number</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Natural_language_related743" style="font-size:114%;margin:0 4em"><a href="/wiki/Natural_language" title="Natural language">Natural language</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Aronson%27s_sequence" title="Aronson's sequence">Aronson's sequence</a></li> <li><a href="/wiki/Ban_number" title="Ban number">Ban</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Graphemics_related743" style="font-size:114%;margin:0 4em"><a href="/wiki/Graphemics" title="Graphemics">Graphemics</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Strobogrammatic_number" title="Strobogrammatic number">Strobogrammatic</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2" style="font-weight:bold;"><div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Symbol_portal_class.svg" class="mw-file-description" title="Portal"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/16px-Symbol_portal_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/23px-Symbol_portal_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/31px-Symbol_portal_class.svg.png 2x" data-file-width="180" data-file-height="185" /></a></span> <a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Metallic_means18" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Metallic_ratios" title="Template:Metallic ratios"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Metallic_ratios" title="Template talk:Metallic ratios"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Metallic_ratios" title="Special:EditPage/Template:Metallic ratios"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Metallic_means18" style="font-size:114%;margin:0 4em"><a href="/wiki/Metallic_mean" title="Metallic mean">Metallic means</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pisot%E2%80%93Vijayaraghavan_number" title="Pisot–Vijayaraghavan number">Pisot number</a></li> <li><a href="/wiki/Golden_ratio" title="Golden ratio">Gold</a> <ul><li><a href="/wiki/Golden_angle" title="Golden angle">Angle</a></li> <li><a href="/wiki/Golden_ratio_base" title="Golden ratio base">Base</a></li> <li><a class="mw-selflink selflink">Fibonacci sequence</a></li> <li><a href="/wiki/Kepler_triangle" title="Kepler triangle">Kepler triangle</a></li> <li><a href="/wiki/Golden_rectangle" title="Golden rectangle">Rectangle</a></li> <li><a href="/wiki/Golden_rhombus" title="Golden rhombus">Rhombus</a></li> <li><a href="/wiki/Golden-section_search" title="Golden-section search">Section search</a></li> <li><a href="/wiki/Golden_spiral" title="Golden spiral">Spiral</a></li> <li><a href="/wiki/Golden_triangle_(mathematics)" title="Golden triangle (mathematics)">Triangle</a></li> <li><a href="/wiki/Supergolden_ratio" title="Supergolden ratio">Supergolden ratio</a></li></ul></li></ul> <ul><li><a href="/wiki/Silver_ratio" title="Silver ratio">Silver</a> <ul><li><a href="/wiki/Pell_number" title="Pell number">Pell number</a></li> <li><a href="/wiki/Supersilver_ratio" title="Supersilver ratio">Supersilver ratio</a></li></ul></li> <li>Bronze</li> <li>Copper</li> <li>Nickel</li> <li>etc...</li></ul> </div></td><td class="noviewer navbox-image" rowspan="1" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Gold,_silver,_and_bronze_rectangles_vertical.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Gold%2C_silver%2C_and_bronze_rectangles_vertical.svg/50px-Gold%2C_silver%2C_and_bronze_rectangles_vertical.svg.png" decoding="async" width="50" height="71" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Gold%2C_silver%2C_and_bronze_rectangles_vertical.svg/75px-Gold%2C_silver%2C_and_bronze_rectangles_vertical.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Gold%2C_silver%2C_and_bronze_rectangles_vertical.svg/100px-Gold%2C_silver%2C_and_bronze_rectangles_vertical.svg.png 2x" data-file-width="756" data-file-height="1074" /></a></span></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Sequences_and_series331" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Series_(mathematics)" title="Template:Series (mathematics)"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Series_(mathematics)" title="Template talk:Series (mathematics)"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Series_(mathematics)" title="Special:EditPage/Template:Series (mathematics)"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Sequences_and_series331" style="font-size:114%;margin:0 4em"><a href="/wiki/Sequence" title="Sequence">Sequences</a> and <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Integer_sequence" title="Integer sequence">Integer sequences</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Basic</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetic_progression" title="Arithmetic progression">Arithmetic progression</a></li> <li><a href="/wiki/Geometric_progression" title="Geometric progression">Geometric progression</a></li> <li><a href="/wiki/Harmonic_progression_(mathematics)" title="Harmonic progression (mathematics)">Harmonic progression</a></li> <li><a href="/wiki/Square_number" title="Square number">Square number</a></li> <li><a href="/wiki/Cube_(algebra)" title="Cube (algebra)">Cubic number</a></li> <li><a href="/wiki/Factorial" title="Factorial">Factorial</a></li> <li><a href="/wiki/Power_of_two" title="Power of two">Powers of two</a></li> <li><a href="/wiki/Power_of_three" title="Power of three">Powers of three</a></li> <li><a href="/wiki/Power_of_10" title="Power of 10">Powers of 10</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Advanced <span class="nobold">(<a href="/wiki/List_of_OEIS_sequences" class="mw-redirect" title="List of OEIS sequences">list</a>)</span></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Complete_sequence" title="Complete sequence">Complete sequence</a></li> <li><a class="mw-selflink selflink">Fibonacci sequence</a></li> <li><a href="/wiki/Figurate_number" title="Figurate number">Figurate number</a></li> <li><a href="/wiki/Heptagonal_number" title="Heptagonal number">Heptagonal number</a></li> <li><a href="/wiki/Hexagonal_number" title="Hexagonal number">Hexagonal number</a></li> <li><a href="/wiki/Lucas_number" title="Lucas number">Lucas number</a></li> <li><a href="/wiki/Pell_number" title="Pell number">Pell number</a></li> <li><a href="/wiki/Pentagonal_number" title="Pentagonal number">Pentagonal number</a></li> <li><a href="/wiki/Polygonal_number" title="Polygonal number">Polygonal number</a></li> <li><a href="/wiki/Triangular_number" title="Triangular number">Triangular number</a> <ul><li><a href="/wiki/Triangular_array" title="Triangular array">array</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/Fibonacci_sequence" title="Fibonacci sequence"><img alt="Fibonacci spiral with square sizes up to 34." src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Fibonacci_spiral_34.svg/80px-Fibonacci_spiral_34.svg.png" decoding="async" width="80" height="51" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Fibonacci_spiral_34.svg/120px-Fibonacci_spiral_34.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Fibonacci_spiral_34.svg/160px-Fibonacci_spiral_34.svg.png 2x" data-file-width="915" data-file-height="579" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties of sequences</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cauchy_sequence" title="Cauchy sequence">Cauchy sequence</a></li> <li><a href="/wiki/Monotonic_function" title="Monotonic function">Monotonic function</a></li> <li><a href="/wiki/Periodic_sequence" title="Periodic sequence">Periodic sequence</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties of series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Series</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alternating_series" title="Alternating series">Alternating</a></li> <li><a href="/wiki/Convergent_series" title="Convergent series">Convergent</a></li> <li><a href="/wiki/Divergent_series" title="Divergent series">Divergent</a></li> <li><a href="/wiki/Telescoping_series" title="Telescoping series">Telescoping</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Convergence</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Absolute_convergence" title="Absolute convergence">Absolute</a></li> <li><a href="/wiki/Conditional_convergence" title="Conditional convergence">Conditional</a></li> <li><a href="/wiki/Uniform_convergence" title="Uniform convergence">Uniform</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Explicit series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Convergent</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/1/2_%E2%88%92_1/4_%2B_1/8_%E2%88%92_1/16_%2B_%E2%8B%AF" title="1/2 − 1/4 + 1/8 − 1/16 + ⋯">1/2 − 1/4 + 1/8 − 1/16 + ⋯</a></li> <li><a href="/wiki/1/2_%2B_1/4_%2B_1/8_%2B_1/16_%2B_%E2%8B%AF" title="1/2 + 1/4 + 1/8 + 1/16 + ⋯">1/2 + 1/4 + 1/8 + 1/16 + ⋯</a></li> <li><a href="/wiki/1/4_%2B_1/16_%2B_1/64_%2B_1/256_%2B_%E2%8B%AF" title="1/4 + 1/16 + 1/64 + 1/256 + ⋯">1/4 + 1/16 + 1/64 + 1/256 + ⋯</a></li> <li><a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">1 + 1/2<sup><i>s</i></sup> + 1/3<sup><i>s</i></sup> + ... (Riemann zeta function)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Divergent</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/1_%2B_1_%2B_1_%2B_1_%2B_%E2%8B%AF" title="1 + 1 + 1 + 1 + ⋯">1 + 1 + 1 + 1 + ⋯</a></li> <li><a href="/wiki/Grandi%27s_series" title="Grandi's series">1 − 1 + 1 − 1 + ⋯ (Grandi's series)</a></li> <li><a href="/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF" title="1 + 2 + 3 + 4 + ⋯">1 + 2 + 3 + 4 + ⋯</a></li> <li><a href="/wiki/1_%E2%88%92_2_%2B_3_%E2%88%92_4_%2B_%E2%8B%AF" title="1 − 2 + 3 − 4 + ⋯">1 − 2 + 3 − 4 + ⋯</a></li> <li><a href="/wiki/1_%2B_2_%2B_4_%2B_8_%2B_%E2%8B%AF" title="1 + 2 + 4 + 8 + ⋯">1 + 2 + 4 + 8 + ⋯</a></li> <li><a href="/wiki/1_%E2%88%92_2_%2B_4_%E2%88%92_8_%2B_%E2%8B%AF" title="1 − 2 + 4 − 8 + ⋯">1 − 2 + 4 − 8 + ⋯</a></li> <li><a href="/wiki/Infinite_arithmetic_series" class="mw-redirect" title="Infinite arithmetic series">Infinite arithmetic series</a></li> <li><a href="/wiki/1_%E2%88%92_1_%2B_2_%E2%88%92_6_%2B_24_%E2%88%92_120_%2B_%E2%8B%AF" title="1 − 1 + 2 − 6 + 24 − 120 + ⋯">1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)</a></li> <li><a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)</a></li> <li><a href="/wiki/Divergence_of_the_sum_of_the_reciprocals_of_the_primes" title="Divergence of the sum of the reciprocals of the primes">1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Kinds of series</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a></li> <li><a href="/wiki/Power_series" title="Power series">Power series</a></li> <li><a href="/wiki/Formal_power_series" title="Formal power series">Formal power series</a></li> <li><a href="/wiki/Laurent_series" title="Laurent series">Laurent series</a></li> <li><a href="/wiki/Puiseux_series" title="Puiseux series">Puiseux series</a></li> <li><a href="/wiki/Dirichlet_series" title="Dirichlet series">Dirichlet series</a></li> <li><a href="/wiki/Trigonometric_series" title="Trigonometric series">Trigonometric series</a></li> <li><a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a></li> <li><a href="/wiki/Generating_series" class="mw-redirect" title="Generating series">Generating series</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Hypergeometric_function" title="Hypergeometric function">Hypergeometric series</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Generalized_hypergeometric_series" class="mw-redirect" title="Generalized hypergeometric series">Generalized hypergeometric series</a></li> <li><a href="/wiki/Hypergeometric_function_of_a_matrix_argument" title="Hypergeometric function of a matrix argument">Hypergeometric function of a matrix argument</a></li> <li><a href="/wiki/Lauricella_hypergeometric_series" title="Lauricella hypergeometric series">Lauricella hypergeometric series</a></li> <li><a href="/wiki/Modular_hypergeometric_series" class="mw-redirect" title="Modular hypergeometric series">Modular hypergeometric series</a></li> <li><a href="/wiki/Riemann%27s_differential_equation" title="Riemann's differential equation">Riemann's differential equation</a></li> <li><a href="/wiki/Theta_hypergeometric_series" class="mw-redirect" title="Theta hypergeometric series">Theta hypergeometric series</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="3"><div> <ul><li><span class="noviewer" 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autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Fibonacci" title="Template:Fibonacci"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Fibonacci" title="Template talk:Fibonacci"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Fibonacci" title="Special:EditPage/Template:Fibonacci"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Fibonacci33" style="font-size:114%;margin:0 4em"><a href="/wiki/Fibonacci" title="Fibonacci">Fibonacci</a></div></th></tr><tr><th scope="row" class="navbox-group" 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0.25em"> <ul><li><a href="/wiki/Fibonacci_numbers_in_popular_culture" title="Fibonacci numbers in popular culture">Fibonacci numbers in popular culture</a></li> <li><a href="/wiki/List_of_things_named_after_Fibonacci" title="List of things named after Fibonacci">List of things named after Fibonacci</a></li> <li><a href="/wiki/Generalizations_of_Fibonacci_numbers" title="Generalizations of Fibonacci numbers">Generalizations of Fibonacci numbers</a></li> <li><a href="/wiki/The_Fibonacci_Association" title="The Fibonacci Association">The Fibonacci Association</a> <ul><li><i><a href="/wiki/Fibonacci_Quarterly" title="Fibonacci Quarterly">Fibonacci Quarterly</a></i></li></ul></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output 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[\"CITEREFFreydBrown1993\"] = 1,\n [\"CITEREFGardner1996\"] = 1,\n [\"CITEREFGessel1972\"] = 1,\n [\"CITEREFGlaister1995\"] = 1,\n [\"CITEREFGoonatilake1998\"] = 1,\n [\"CITEREFHarizanov1995\"] = 1,\n [\"CITEREFHemenway2005\"] = 1,\n [\"CITEREFHonsberger1985\"] = 2,\n [\"CITEREFHutchison2004\"] = 1,\n [\"CITEREFJonesWilson2006\"] = 1,\n [\"CITEREFKepler1966\"] = 1,\n [\"CITEREFKnott\"] = 2,\n [\"CITEREFKnott2016\"] = 1,\n [\"CITEREFKnuth1968\"] = 1,\n [\"CITEREFKnuth1997\"] = 1,\n [\"CITEREFKnuth2006\"] = 1,\n [\"CITEREFLemmermeyer2000\"] = 1,\n [\"CITEREFLivio2003\"] = 1,\n [\"CITEREFLuca2000\"] = 1,\n [\"CITEREFLucaMejía_Huguet2010\"] = 1,\n [\"CITEREFLucas1891\"] = 1,\n [\"CITEREFLuo1989\"] = 1,\n [\"CITEREFLüWang2006\"] = 1,\n [\"CITEREFPagni2001\"] = 1,\n [\"CITEREFPatranabisDana1985\"] = 1,\n [\"CITEREFPethő2001\"] = 1,\n [\"CITEREFPrusinkiewiczHanan1989\"] = 1,\n [\"CITEREFPrusinkiewiczLindenmayer1990\"] = 1,\n [\"CITEREFRibenboim1996\"] = 1,\n [\"CITEREFRibenboim2000\"] = 1,\n [\"CITEREFSarah-Marie_Belcastro2018\"] = 1,\n [\"CITEREFScottMarketos2014\"] = 1,\n [\"CITEREFShah1991\"] = 1,\n [\"CITEREFSigler2002\"] = 1,\n [\"CITEREFSingh1985\"] = 1,\n [\"CITEREFStanley2011\"] = 1,\n [\"CITEREFStephenson2005\"] = 1,\n [\"CITEREFSu2000\"] = 1,\n [\"CITEREFVarenne2010\"] = 1,\n [\"CITEREFVelankar1962\"] = 1,\n [\"CITEREFVogel1979\"] = 1,\n [\"CITEREFVorobievMartin2002\"] = 1,\n [\"CITEREFWilliams1982\"] = 1,\n [\"Fibonacci_Tree\"] = 1,\n}\ntemplate_list = table#1 {\n [\"\"] = 1,\n [\"=\"] = 2,\n [\"Anchor\"] = 2,\n [\"Annotated link\"] = 5,\n [\"Authority control\"] = 1,\n [\"Circa\"] = 1,\n [\"Citation\"] = 72,\n [\"Cite OEIS\"] = 6,\n [\"Cite book\"] = 1,\n [\"Cite web\"] = 1,\n [\"Classes of natural numbers\"] = 1,\n [\"Clear\"] = 2,\n [\"Efn\"] = 1,\n [\"F_k\"] = 1,\n [\"Fibonacci\"] = 1,\n [\"For\"] = 1,\n [\"Further\"] = 1,\n [\"Ill\"] = 1,\n [\"In Our Time\"] = 1,\n [\"Interwiki extra\"] = 1,\n [\"Lang\"] = 3,\n [\"Main\"] = 5,\n [\"Math\"] = 153,\n 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Fibonacci sequence - Wikipedia