Successione di Fibonacci

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laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">nascondi</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Inizio</div> </a> </li> <li id="toc-Storia" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Storia"> <div class="vector-toc-text"> 1 Storia </div> </a> <ul id="toc-Storia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proprietà" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Proprietà"> <div class="vector-toc-text"> 2 Proprietà </div> </a> <ul id="toc-Proprietà-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proprietà_di_una_qualunque_successione_definita_come_quella_di_Fibonacci" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Proprietà_di_una_qualunque_successione_definita_come_quella_di_Fibonacci"> <div class="vector-toc-text"> 3 Proprietà di una qualunque successione definita come quella di Fibonacci </div> </a> <ul id="toc-Proprietà_di_una_qualunque_successione_definita_come_quella_di_Fibonacci-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relazioni_con_il_triangolo_di_Tartaglia_e_i_coefficienti_binomiali" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Relazioni_con_il_triangolo_di_Tartaglia_e_i_coefficienti_binomiali"> <div class="vector-toc-text"> 4 Relazioni con il triangolo di Tartaglia e i coefficienti binomiali </div> </a> <button aria-controls="toc-Relazioni_con_il_triangolo_di_Tartaglia_e_i_coefficienti_binomiali-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Attiva/disattiva la sottosezione Relazioni con il triangolo di Tartaglia e i coefficienti binomiali </button> <ul id="toc-Relazioni_con_il_triangolo_di_Tartaglia_e_i_coefficienti_binomiali-sublist" class="vector-toc-list"> <li id="toc-Numeri_di_Fibonacci_e_fattori_comuni" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Numeri_di_Fibonacci_e_fattori_comuni"> <div class="vector-toc-text"> 4.1 Numeri di Fibonacci e fattori comuni </div> </a> <ul id="toc-Numeri_di_Fibonacci_e_fattori_comuni-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Numeri_di_Fibonacci_vicini" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Numeri_di_Fibonacci_vicini"> <div class="vector-toc-text"> 4.2 Numeri di Fibonacci vicini </div> </a> <ul id="toc-Numeri_di_Fibonacci_vicini-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Numeri_di_Fibonacci_primi" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Numeri_di_Fibonacci_primi"> <div class="vector-toc-text"> 4.3 Numeri di Fibonacci primi </div> </a> <ul id="toc-Numeri_di_Fibonacci_primi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Teorema_di_Carmichael_e_fattori_primi_caratteristici" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Teorema_di_Carmichael_e_fattori_primi_caratteristici"> <div class="vector-toc-text"> 4.4 Teorema di Carmichael e fattori primi caratteristici </div> </a> <ul id="toc-Teorema_di_Carmichael_e_fattori_primi_caratteristici-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proprietà_di_divisibilità" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Proprietà_di_divisibilità"> <div class="vector-toc-text"> 4.5 Proprietà di divisibilità </div> </a> <ul id="toc-Proprietà_di_divisibilità-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Primalità" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Primalità"> <div class="vector-toc-text"> 4.6 Primalità </div> </a> <ul id="toc-Primalità-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relazioni_con_il_massimo_comun_divisore_e_la_divisibilità" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relazioni_con_il_massimo_comun_divisore_e_la_divisibilità"> <div class="vector-toc-text"> 4.7 Relazioni con il massimo comun divisore e la divisibilità </div> </a> <ul id="toc-Relazioni_con_il_massimo_comun_divisore_e_la_divisibilità-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Altre_proprietà" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Altre_proprietà"> <div class="vector-toc-text"> 5 Altre proprietà </div> </a> <ul id="toc-Altre_proprietà-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algoritmo_di_Euclide_con_ciclo_più_lungo" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Algoritmo_di_Euclide_con_ciclo_più_lungo"> <div class="vector-toc-text"> 6 Algoritmo di Euclide con ciclo più lungo </div> </a> <ul id="toc-Algoritmo_di_Euclide_con_ciclo_più_lungo-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Frazioni_continue" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Frazioni_continue"> <div class="vector-toc-text"> 7 Frazioni continue </div> </a> <ul id="toc-Frazioni_continue-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalizzazioni" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalizzazioni"> <div class="vector-toc-text"> 8 Generalizzazioni </div> </a> <button aria-controls="toc-Generalizzazioni-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Attiva/disattiva la sottosezione Generalizzazioni </button> <ul id="toc-Generalizzazioni-sublist" class="vector-toc-list"> <li id="toc-Calcolo_con_le_matrici" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Calcolo_con_le_matrici"> <div class="vector-toc-text"> 8.1 Calcolo con le matrici </div> </a> <ul id="toc-Calcolo_con_le_matrici-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Successioni_Tribonacci_e_Tetranacci" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Successioni_Tribonacci_e_Tetranacci"> <div class="vector-toc-text"> 8.2 Successioni Tribonacci e Tetranacci </div> </a> <ul id="toc-Successioni_Tribonacci_e_Tetranacci-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Numeri_complessi_di_Fibonacci" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Numeri_complessi_di_Fibonacci"> <div class="vector-toc-text"> 9 Numeri complessi di Fibonacci </div> </a> <ul id="toc-Numeri_complessi_di_Fibonacci-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sequenza_casuale_di_Fibonacci" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Sequenza_casuale_di_Fibonacci"> <div class="vector-toc-text"> 10 Sequenza casuale di Fibonacci </div> </a> <ul id="toc-Sequenza_casuale_di_Fibonacci-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sequenze_Repfigit" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Sequenze_Repfigit"> <div class="vector-toc-text"> 11 Sequenze Repfigit </div> </a> <button aria-controls="toc-Sequenze_Repfigit-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Attiva/disattiva la sottosezione Sequenze Repfigit </button> <ul id="toc-Sequenze_Repfigit-sublist" class="vector-toc-list"> <li id="toc-Numeri_Repfigit" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Numeri_Repfigit"> <div class="vector-toc-text"> 11.1 Numeri Repfigit </div> </a> <ul id="toc-Numeri_Repfigit-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Numeri_Repfigit_inversi" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Numeri_Repfigit_inversi"> <div class="vector-toc-text"> 11.2 Numeri Repfigit inversi </div> </a> <ul id="toc-Numeri_Repfigit_inversi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Congetture" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Congetture"> <div class="vector-toc-text"> 11.3 Congetture </div> </a> <ul id="toc-Congetture-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Numeri_di_Fibonacci_e_legami_con_altri_settori" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Numeri_di_Fibonacci_e_legami_con_altri_settori"> <div class="vector-toc-text"> 12 Numeri di Fibonacci e legami con altri settori </div> </a> <button aria-controls="toc-Numeri_di_Fibonacci_e_legami_con_altri_settori-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Attiva/disattiva la sottosezione Numeri di Fibonacci e legami con altri settori </button> <ul id="toc-Numeri_di_Fibonacci_e_legami_con_altri_settori-sublist" class="vector-toc-list"> <li id="toc-In_chimica" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_chimica"> <div class="vector-toc-text"> 12.1 In chimica </div> </a> <ul id="toc-In_chimica-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nella_musica" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nella_musica"> <div class="vector-toc-text"> 12.2 Nella musica </div> </a> <ul id="toc-Nella_musica-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_botanica" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_botanica"> <div class="vector-toc-text"> 12.3 In botanica </div> </a> <ul id="toc-In_botanica-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nel_corpo_umano" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nel_corpo_umano"> <div class="vector-toc-text"> 12.4 Nel corpo umano </div> </a> <ul id="toc-Nel_corpo_umano-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_geometria_e_in_natura" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_geometria_e_in_natura"> <div class="vector-toc-text"> 12.5 In geometria e in natura </div> </a> <ul id="toc-In_geometria_e_in_natura-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nell'arte" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nell'arte"> <div class="vector-toc-text"> 12.6 Nell'arte </div> </a> <ul id="toc-Nell'arte-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nell'economia" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nell'economia"> <div class="vector-toc-text"> 12.7 Nell'economia </div> </a> <ul id="toc-Nell'economia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_informatica" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_informatica"> <div class="vector-toc-text"> 12.8 In informatica </div> </a> <ul id="toc-In_informatica-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nei_frattali" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nei_frattali"> <div class="vector-toc-text"> 12.9 Nei frattali </div> </a> <ul id="toc-Nei_frattali-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_elettrotecnica" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_elettrotecnica"> <div class="vector-toc-text"> 12.10 In elettrotecnica </div> </a> <ul id="toc-In_elettrotecnica-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nei_giochi_sistemici" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nei_giochi_sistemici"> <div class="vector-toc-text"> 12.11 Nei giochi sistemici </div> </a> <ul id="toc-Nei_giochi_sistemici-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Note" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Note"> <div class="vector-toc-text"> 13 Note </div> </a> <ul id="toc-Note-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> 14 Bibliografia </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voci_correlate" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Voci_correlate"> <div class="vector-toc-text"> 15 Voci correlate </div> </a> <ul id="toc-Voci_correlate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Altri_progetti" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Altri_progetti"> <div class="vector-toc-text"> 16 Altri progetti </div> </a> <ul id="toc-Altri_progetti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Collegamenti_esterni" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Collegamenti_esterni"> <div class="vector-toc-text"> 17 Collegamenti esterni </div> </a> <ul id="toc-Collegamenti_esterni-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="Indice" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <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="Mostra/Nascondi l'indice" > <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" > Mostra/Nascondi l'indice </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">Successione di Fibonacci</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="Vai a una voce in un'altra lingua. Disponibile in 38 lingue" > <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-38" aria-hidden="true" > 38 lingue </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">Afrikaans</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 - arabo" lang="ar" hreflang="ar" data-title="متتالية فيبوناتشي" data-language-autonym="العربية" data-language-local-name="arabo" class="interlanguage-link-target">العربية</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 - bengalese" lang="bn" hreflang="bn" data-title="ফিবোনাচ্চি রাশিমালা" data-language-autonym="বাংলা" data-language-local-name="bengalese" class="interlanguage-link-target">বাংলা</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 - catalano" lang="ca" hreflang="ca" data-title="Successió de Fibonacci" data-language-autonym="Català" data-language-local-name="catalano" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Fibonacciho_posloupnost" title="Fibonacciho posloupnost - ceco" lang="cs" hreflang="cs" data-title="Fibonacciho posloupnost" data-language-autonym="Čeština" data-language-local-name="ceco" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Fibonacci-Folge" title="Fibonacci-Folge - tedesco" lang="de" hreflang="de" data-title="Fibonacci-Folge" data-language-autonym="Deutsch" data-language-local-name="tedesco" class="interlanguage-link-target">Deutsch</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 - greco" lang="el" hreflang="el" data-title="Ακολουθία Φιμπονάτσι" data-language-autonym="Ελληνικά" data-language-local-name="greco" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Fibonacci_sequence" title="Fibonacci sequence - inglese" lang="en" hreflang="en" data-title="Fibonacci sequence" data-language-autonym="English" data-language-local-name="inglese" class="interlanguage-link-target">English</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 - spagnolo" lang="es" hreflang="es" data-title="Sucesión de Fibonacci" data-language-autonym="Español" data-language-local-name="spagnolo" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Fibonacciren_segida" title="Fibonacciren segida - basco" lang="eu" hreflang="eu" data-title="Fibonacciren segida" data-language-autonym="Euskara" data-language-local-name="basco" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Fibonaccin_lukujono" title="Fibonaccin lukujono - finlandese" lang="fi" hreflang="fi" data-title="Fibonaccin lukujono" data-language-autonym="Suomi" data-language-local-name="finlandese" class="interlanguage-link-target">Suomi</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 - francese" lang="fr" hreflang="fr" data-title="Suite de Fibonacci" data-language-autonym="Français" data-language-local-name="francese" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Fibonaccitaalen" title="Fibonaccitaalen - frisone settentrionale" lang="frr" hreflang="frr" data-title="Fibonaccitaalen" data-language-autonym="Nordfriisk" data-language-local-name="frisone settentrionale" class="interlanguage-link-target">Nordfriisk</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Seicheamh_Fibonacci" title="Seicheamh Fibonacci - irlandese" lang="ga" hreflang="ga" data-title="Seicheamh Fibonacci" data-language-autonym="Gaeilge" data-language-local-name="irlandese" class="interlanguage-link-target">Gaeilge</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 - galiziano" lang="gl" hreflang="gl" data-title="Sucesión de Fibonacci" data-language-autonym="Galego" data-language-local-name="galiziano" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/Straih_Fibonacci" title="Straih Fibonacci - mannese" lang="gv" hreflang="gv" data-title="Straih Fibonacci" data-language-autonym="Gaelg" data-language-local-name="mannese" class="interlanguage-link-target">Gaelg</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 - ebraico" lang="he" hreflang="he" data-title="סדרת פיבונאצ'י" data-language-autonym="עברית" data-language-local-name="ebraico" class="interlanguage-link-target">עברית</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">हिन्दी</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 - armeno" lang="hy" hreflang="hy" data-title="Ֆիբոնաչիի հաջորդականություն" data-language-autonym="Հայերեն" data-language-local-name="armeno" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Barisan_Fibonacci" title="Barisan Fibonacci - indonesiano" lang="id" hreflang="id" data-title="Barisan Fibonacci" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiano" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a 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srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Emblem-important.svg/60px-Emblem-important.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Emblem-important.svg/80px-Emblem-important.svg.png 2x" data-file-width="48" data-file-height="48" /></a></div> <div class="avviso-testo mbox-text"> <div class="mbox-text-div">Questa voce o sezione sull'argomento matematica è ritenuta <a href="/wiki/Aiuto:Voci_da_controllare" title="Aiuto:Voci da controllare">da controllare</a>. <div class="hide-when-compact">Motivo: È stato inserito parecchio materiale mal formattato, con notazioni diverse rispetto al resto della voce, alcuni passi sono poco contestualizzati e di altri non si coglie la rilevanza <div class="noprint"><hr />Partecipa alla <a href="/wiki/Discussione:Successione_di_Fibonacci" title="Discussione:Successione di Fibonacci">discussione</a> e/o <a class="external text" href="https://it.wikipedia.org/w/index.php?title=Successione_di_Fibonacci&action=edit">correggi</a> la voce. Segui i suggerimenti del <a href="/wiki/Progetto:Matematica" title="Progetto:Matematica">progetto di riferimento</a>.</div> </div> </div> </div> </div> In <a href="/wiki/Matematica" title="Matematica">matematica</a>, la successione di Fibonacci (detta anche successione aurea) è una <a href="/wiki/Successione_(matematica)" title="Successione (matematica)">successione</a> di <a href="/wiki/Numero_naturale" title="Numero naturale">numeri interi</a> in cui ciascun numero è la somma dei due precedenti, eccetto i primi due che sono, per definizione, 0 e 1.<a href="#cite_note-1">[1]</a> Questa successione, indicata con <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><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}}"> o con <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Fib} (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">b</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Fib} (n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b9963d1b0342837b5527f0a004e53e2d4d53ee2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.661ex; height:2.843ex;" alt="{\displaystyle \mathrm {Fib} (n)}">, è <a href="/wiki/Definizione_ricorsiva" title="Definizione ricorsiva">definita ricorsivamente</a>: partendo dai primi due elementi, <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><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}"> e <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><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}">, ogni altro elemento della successione sarà dato dalla relazione: <dl><dd><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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=F_{n-1}+F_{n-2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfb921b6316dbe7775ab46645807276b9cc2299e" 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-1}+F_{n-2}.}"></dd></dl> Gli elementi <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><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}}"> sono anche detti numeri di Fibonacci. I primi termini della successione di Fibonacci, che prende il nome dal <a href="/wiki/Matematico" title="Matematico">matematico</a> <a href="/wiki/Pisa" title="Pisa">pisano</a> del <a href="/wiki/XIII_secolo" title="XIII secolo">XIII secolo</a> <a href="/wiki/Leonardo_Fibonacci" title="Leonardo Fibonacci">Leonardo Fibonacci</a>, sono: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mn>13</mn> <mo>,</mo> <mn>21</mn> <mo>,</mo> <mn>34</mn> <mo>,</mo> <mn>55</mn> <mo>,</mo> <mn>89</mn> <mo>,</mo> <mn>144</mn> <mo>,</mo> <mn>233</mn> <mo>,</mo> <mn>377</mn> <mo>,</mo> <mn>610</mn> <mo>,</mo> <mn>987</mn> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a25eb2ec6dccc4a395257825c6ec4bada129476" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:57.499ex; height:2.509ex;" alt="{\displaystyle 0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,\ldots }"> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Storia">Storia</h2>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=1" title="Modifica la sezione Storia" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=1" title="Edit section's source code: Storia">modifica wikitesto</a>]</div> Nel <a href="/wiki/1202" title="1202">1202</a> il matematico <a href="/wiki/Pisa" title="Pisa">pisano</a> <a href="/wiki/Leonardo_Fibonacci" title="Leonardo Fibonacci">Leonardo Fibonacci</a> pubblicò il Liber abbaci, un <a href="/wiki/Trattato_(opera)" class="mw-redirect" title="Trattato (opera)">trattato</a> di <a href="/wiki/Aritmetica" title="Aritmetica">aritmetica</a> e <a href="/wiki/Algebra" title="Algebra">algebra</a> con il quale voleva introdurre in <a href="/wiki/Europa" title="Europa">Europa</a> il <a href="/wiki/Sistema_numerico_decimale" title="Sistema numerico decimale">sistema numerico decimale</a> indo-arabico e i principali metodi di calcolo ad esso relativi. All'interno del trattato portò diversi problemi aritmetici con relativa soluzione. Uno di questi riguardava la crescita di una popolazione di <a href="/wiki/Conigli" class="mw-redirect" title="Conigli">conigli</a><a href="#cite_note-2">[2]</a><a href="#cite_note-3">[3]</a>. Assumendo per ipotesi che: <ul><li>si disponga di una coppia di conigli appena nati</li> <li>questa prima coppia diventi fertile al compimento del primo mese e dia alla luce una nuova coppia al compimento del secondo mese;</li> <li>le nuove coppie nate si comportino in modo analogo;</li> <li>le coppie fertili dal secondo mese di vita in poi diano alla luce una coppia di figli al mese;</li></ul> si verifica quanto segue: <ul><li>dopo un mese una coppia di conigli sarà fertile,</li> <li>dopo due mesi ci saranno due coppie di cui una sola fertile,</li> <li>nel mese seguente, terzo mese dal momento iniziale, ci saranno <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2+1=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2+1=3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdca0e15ed01800bb3f8a2c4725d8a1bd98429c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;" alt="{\displaystyle 2+1=3}"> coppie perché solo la coppia fertile avrà generato; di queste tre, due saranno le coppie fertili, quindi</li> <li>nel mese seguente (quarto mese dal momento iniziale) ci saranno <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3+2=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3+2=5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12a0ae799cf5892cdf28c1b8c2502078ce6ada38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.426ex; height:2.343ex;" alt="{\displaystyle 3+2=5}"> coppie</li></ul> <table class="wikitable"> <tbody><tr> <th></th> <th>In principio</th> <th>Al termine del primo mese</th> <th>Al termine del secondo mese</th> <th>Al termine del terzo mese</th> <th>Al termine del quarto mese</th> <th>Al termine del quinto mese</th> <th>Al termine del sesto mese</th> <th>Al termine del settimo mese</th> <th>Al termine dell'ottavo mese</th> <th>Al termine del nono mese</th> <th>Al termine del decimo mese</th> <th>Al termine dell'undicesimo mese</th> <th>Al termine del dodicesimo mese </th></tr> <tr> <th style="text-align:left;">Coppie di conigli </th> <td style="text-align:center;">1</td> <td style="text-align:center;">1</td> <td style="text-align:center;">2</td> <td style="text-align:center;">3</td> <td style="text-align:center;">5</td> <td style="text-align:center;">8</td> <td style="text-align:center;">13</td> <td style="text-align:center;">21</td> <td style="text-align:center;">34</td> <td style="text-align:center;">55</td> <td style="text-align:center;">89</td> <td style="text-align:center;">144</td> <td style="text-align:center;">233 </td></tr></tbody></table> In questo esempio, il numero di coppie di conigli di ogni mese esprime la successione di Fibonacci. <div class="mw-heading mw-heading2"><h2 id="Proprietà">Proprietà</h2>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=2" title="Modifica la sezione Proprietà" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=2" title="Edit section's source code: Proprietà">modifica wikitesto</a>]</div> Il rapporto <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {F_{n}}{F_{n-1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <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> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {F_{n}}{F_{n-1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc9c05f15ed7915a26b3230479559b356dd20ad2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:5.65ex; height:5.676ex;" alt="{\displaystyle {\frac {F_{n}}{F_{n-1}}}}">, per <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><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}"> tendente all'infinito, tende al <a href="/wiki/Numero_algebrico" title="Numero algebrico">numero algebrico</a> irrazionale <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><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 }"> chiamato <a href="/wiki/Sezione_aurea" title="Sezione aurea">sezione aurea</a> o numero di <a href="/wiki/Fidia" title="Fidia">Fidia</a>. In termini matematici: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }{F_{n} \over F_{n-1}}=\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> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }{F_{n} \over F_{n-1}}=\varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffb340aeeb16f99fe4816dd9da105f770b890664" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.928ex; height:5.676ex;" alt="{\displaystyle \lim _{n\to \infty }{F_{n} \over F_{n-1}}=\varphi }"></dd></dl> dove <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ={1+{\sqrt {5}} \over 2}=1{,}618033988749895\dots }"> <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>1,618</mn> <mn>033988749895</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ={1+{\sqrt {5}} \over 2}=1{,}618033988749895\dots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb918a8d923dd1c3b37a75a77e515da972d5b74e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.011ex; height:5.843ex;" alt="{\displaystyle \varphi ={1+{\sqrt {5}} \over 2}=1{,}618033988749895\dots }"></dd></dl> Infatti, se poniamo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x:=\lim _{n\rightarrow \infty }{\dfrac {F_{n}}{F_{n-1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>:=</mo> <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"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <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> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x:=\lim _{n\rightarrow \infty }{\dfrac {F_{n}}{F_{n-1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21ffc8dca8db71866b336dd6301b2ed85800b2f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.384ex; height:5.676ex;" alt="{\displaystyle x:=\lim _{n\rightarrow \infty }{\dfrac {F_{n}}{F_{n-1}}}}"> risulta <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\lim _{n\rightarrow \infty }{\dfrac {F_{n}}{F_{n-1}}}=\lim _{n\rightarrow \infty }{\dfrac {F_{n-1}+F_{n-2}}{F_{n-1}}}=\lim _{n\rightarrow \infty }\left(1+{\dfrac {F_{n-2}}{F_{n-1}}}\right)=1+\lim _{n\rightarrow \infty }{\dfrac {F_{n-1}}{F_{n}}}=1+{\dfrac {1}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <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"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <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> </mfrac> </mstyle> </mrow> <mo>=</mo> <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"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <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>2</mn> </mrow> </msub> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mfrac> </mstyle> </mrow> <mo>=</mo> <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> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <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"> <mstyle displaystyle="true" scriptlevel="0"> <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> </mstyle> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\lim _{n\rightarrow \infty }{\dfrac {F_{n}}{F_{n-1}}}=\lim _{n\rightarrow \infty }{\dfrac {F_{n-1}+F_{n-2}}{F_{n-1}}}=\lim _{n\rightarrow \infty }\left(1+{\dfrac {F_{n-2}}{F_{n-1}}}\right)=1+\lim _{n\rightarrow \infty }{\dfrac {F_{n-1}}{F_{n}}}=1+{\dfrac {1}{x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dc7ba0a88429403cfa908d00f275940df80d9ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:83.309ex; height:6.176ex;" alt="{\displaystyle x=\lim _{n\rightarrow \infty }{\dfrac {F_{n}}{F_{n-1}}}=\lim _{n\rightarrow \infty }{\dfrac {F_{n-1}+F_{n-2}}{F_{n-1}}}=\lim _{n\rightarrow \infty }\left(1+{\dfrac {F_{n-2}}{F_{n-1}}}\right)=1+\lim _{n\rightarrow \infty }{\dfrac {F_{n-1}}{F_{n}}}=1+{\dfrac {1}{x}}}"></dd></dl> da cui segue che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=1+{\dfrac {1}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=1+{\dfrac {1}{x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf767ea21648964e968c4c5300faf949188a0117" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.597ex; height:5.176ex;" alt="{\displaystyle x=1+{\dfrac {1}{x}}}">, ossia <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}-x-1=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" 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> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}-x-1=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e22ea5278827fbb7e09fb5fbeb5f50b234410f84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.818ex; height:2.843ex;" alt="{\displaystyle x^{2}-x-1=0}">. Tale equazione ha per soluzioni <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dfrac {1\pm {\sqrt {5}}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mn>1</mn> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dfrac {1\pm {\sqrt {5}}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dbef19e0052217293113271b933b69759ba125b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.937ex; height:5.843ex;" alt="{\displaystyle {\dfrac {1\pm {\sqrt {5}}}{2}}}">, ma <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\rightarrow \infty }{\dfrac {F_{n}}{F_{n-1}}}>1}"> <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"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <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> </mfrac> </mstyle> </mrow> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\rightarrow \infty }{\dfrac {F_{n}}{F_{n-1}}}>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aef9088b9f7eba81fa5389211386c840e862bc0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.57ex; height:5.676ex;" alt="{\displaystyle \lim _{n\rightarrow \infty }{\dfrac {F_{n}}{F_{n-1}}}>1}"> perché la successione di Fibonacci è definitivamente crescente: perciò <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }{F_{n} \over F_{n-1}}={\dfrac {1+{\sqrt {5}}}{2}}.}"> <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> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }{F_{n} \over F_{n-1}}={\dfrac {1+{\sqrt {5}}}{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11ac6603aa08d89c896371b4cf413774aa5e2f15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.992ex; height:6.343ex;" alt="{\displaystyle \lim _{n\to \infty }{F_{n} \over F_{n-1}}={\dfrac {1+{\sqrt {5}}}{2}}.}"></dd></dl> Il rapporto tra un numero di Fibonacci e il suo successivo tende al reciproco della sezione aurea <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\varphi }}=0,6180339887498948\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>φ</mi> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>6180339887498948</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\varphi }}=0,6180339887498948\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e226ec7949038483a5bc8d42b4240ef7e8c94fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.361ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{\varphi }}=0,6180339887498948\ldots }"> Per <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><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 }"> valgono le seguenti relazioni: <dl><dd>a) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi -1={\frac {1}{\varphi }}={-1+{\sqrt {5}} \over 2},}"> <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> <mn>1</mn> <mi>φ</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</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 \varphi -1={\frac {1}{\varphi }}={-1+{\sqrt {5}} \over 2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/731178f8b27ff2ed248edca5de68be86e61d0129" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.468ex; height:6.343ex;" alt="{\displaystyle \varphi -1={\frac {1}{\varphi }}={-1+{\sqrt {5}} \over 2},}"></dd> <dd>b) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-\varphi =-{\frac {1}{\varphi }}={1-{\sqrt {5}} \over 2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-\varphi =-{\frac {1}{\varphi }}={1-{\sqrt {5}} \over 2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a289666062073628a0932c8be8f06588752dd777" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.468ex; height:6.343ex;" alt="{\displaystyle 1-\varphi =-{\frac {1}{\varphi }}={1-{\sqrt {5}} \over 2}.}"></dd></dl> L'<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><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}">-esimo numero di Fibonacci si può esprimere con la formula:<a href="#cite_note-4">[4]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}={\frac {\varphi ^{n}}{\sqrt {5}}}-{\frac {(1-\varphi )^{n}}{\sqrt {5}}}={\frac {\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> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msqrt> <mn>5</mn> </msqrt> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>φ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <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> <msqrt> <mn>5</mn> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}={\frac {\varphi ^{n}}{\sqrt {5}}}-{\frac {(1-\varphi )^{n}}{\sqrt {5}}}={\frac {\varphi ^{n}-(-\varphi )^{-n}}{\sqrt {5}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b44f363139004619f9cdd0b5b4b2994e0b5573f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:39.768ex; height:6.676ex;" alt="{\displaystyle F_{n}={\frac {\varphi ^{n}}{\sqrt {5}}}-{\frac {(1-\varphi )^{n}}{\sqrt {5}}}={\frac {\varphi ^{n}-(-\varphi )^{-n}}{\sqrt {5}}}.}"></dd></dl> Questa elegante formula è nota come <a href="/wiki/Jacques_Philippe_Marie_Binet#Formula_di_Binet_per_la_successione_di_Fibonacci" title="Jacques Philippe Marie Binet">formula di Binet</a>. <a href="/wiki/Jacques_Binet" class="mw-redirect" title="Jacques Binet">Jacques Binet</a> la <a href="/wiki/Jacques_Philippe_Marie_Binet#Dimostrazione" title="Jacques Philippe Marie Binet">dimostrò</a> nel 1843, tuttavia essa era già nota nel XVIII secolo a <a href="/wiki/Leonhard_Euler" class="mw-redirect" title="Leonhard Euler">Eulero</a>, <a href="/wiki/Abraham_de_Moivre" title="Abraham de Moivre">Abraham de Moivre</a> e <a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a>. Questa espressione per <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><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}}"> può essere calcolata per mezzo della <a href="/wiki/Trasformata_zeta" title="Trasformata zeta">trasformata zeta</a><style data-mw-deduplicate="TemplateStyles:r140554517">.mw-parser-output .chiarimento{background:#ffeaea;color:#444444}.mw-parser-output .chiarimento-apice{color:#EE0700}@media screen{html.skin-theme-clientpref-night .mw-parser-output .chiarimento{background:rgba(179,36,36,0.21);color:inherit}html.skin-theme-clientpref-night .mw-parser-output .chiarimento-apice{color:#b32424}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .chiarimento{background:rgba(179,36,36,0.21);color:inherit}html.skin-theme-clientpref-os .mw-parser-output .chiarimento-apice{color:#b32424}}</style>[<a href="/wiki/Wikipedia:Uso_delle_fonti" title="Wikipedia:Uso delle fonti">senza fonte</a>]. Talora risulta comodo servirsi della successione bilatera, cioè una successione definita sugli interi invece che sui naturali, costituita da numeri interi aggiungendo ai precedenti i termini <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{-n}:=(-1)^{n+1}F_{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <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> <mo>+</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{-n}:=(-1)^{n+1}F_{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30b3e478b0dbd4735f17cdc2759c488ed76736f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.196ex; height:3.176ex;" alt="{\displaystyle F_{-n}:=(-1)^{n+1}F_{n},}"> per <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=0,1,2,\ldots ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=0,1,2,\ldots ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/014e63e55d9280586bfbb9be65d8816a67c105c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.84ex; height:2.509ex;" alt="{\displaystyle n=0,1,2,\ldots ,}"> di cui alcuni termini sono: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dots \ 233,-144,89,-55,34,-21,13,-8,5,-3,2,-1,1,0,1,1,2,3,5,8,13,21,34,55,89,144,233,\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>…</mo> <mtext> </mtext> <mn>233</mn> <mo>,</mo> <mo>−</mo> <mn>144</mn> <mo>,</mo> <mn>89</mn> <mo>,</mo> <mo>−</mo> <mn>55</mn> <mo>,</mo> <mn>34</mn> <mo>,</mo> <mo>−</mo> <mn>21</mn> <mo>,</mo> <mn>13</mn> <mo>,</mo> <mo>−</mo> <mn>8</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mn>13</mn> <mo>,</mo> <mn>21</mn> <mo>,</mo> <mn>34</mn> <mo>,</mo> <mn>55</mn> <mo>,</mo> <mn>89</mn> <mo>,</mo> <mn>144</mn> <mo>,</mo> <mn>233</mn> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dots \ 233,-144,89,-55,34,-21,13,-8,5,-3,2,-1,1,0,1,1,2,3,5,8,13,21,34,55,89,144,233,\dots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb743a5f37644860b4f93842e06475b32f2dbdc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:97.49ex; height:2.509ex;" alt="{\displaystyle \dots \ 233,-144,89,-55,34,-21,13,-8,5,-3,2,-1,1,0,1,1,2,3,5,8,13,21,34,55,89,144,233,\dots }"></dd></dl> A partire dai numeri di Fibonacci e dalla sezione aurea si possono definire alcune funzioni speciali: <a href="/w/index.php?title=Coseno_iperbolico_di_Fibonacci&action=edit&redlink=1" class="new" title="Coseno iperbolico di Fibonacci (la pagina non esiste)">coseno iperbolico di Fibonacci</a>, <a href="/w/index.php?title=Cotangente_iperbolica_di_Fibonacci&action=edit&redlink=1" class="new" title="Cotangente iperbolica di Fibonacci (la pagina non esiste)">cotangente iperbolica di Fibonacci</a>, <a href="/w/index.php?title=Seno_iperbolico_di_Fibonacci&action=edit&redlink=1" class="new" title="Seno iperbolico di Fibonacci (la pagina non esiste)">seno iperbolico di Fibonacci</a>, <a href="/w/index.php?title=Tangente_iperbolica_di_Fibonacci&action=edit&redlink=1" class="new" title="Tangente iperbolica di Fibonacci (la pagina non esiste)">tangente iperbolica di Fibonacci</a>. <div class="mw-heading mw-heading2"><h2 id="Proprietà_di_una_qualunque_successione_definita_come_quella_di_Fibonacci">Proprietà di una qualunque successione definita come quella di Fibonacci</h2>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=3" title="Modifica la sezione Proprietà di una qualunque successione definita come quella di Fibonacci" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=3" title="Edit section's source code: Proprietà di una qualunque successione definita come quella di Fibonacci">modifica wikitesto</a>]</div> Dati due numeri Naturali qualunque <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbf42ecda092975c9c69dae84e16182ba5fe2e07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{1}}">e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/270580da7333505d9b73697417d0543c43c98b9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{2}}"> con <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}<a_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}<a_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/444d1d8e0860f5e383541a869aa4dbfc131d17ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.667ex; height:2.176ex;" alt="{\displaystyle a_{1}<a_{2}}"> e costruita, analogamente a quella di Fibonacci, la successione: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbf42ecda092975c9c69dae84e16182ba5fe2e07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{1}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/270580da7333505d9b73697417d0543c43c98b9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{2}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{3}=a_{2}+a_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{3}=a_{2}+a_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da53f8ff4fc6fe51af135fd65528fac9c0e39972" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.791ex; height:2.343ex;" alt="{\displaystyle a_{3}=a_{2}+a_{1}}">, ..., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n+1}=a_{n}+a_{n-1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>a</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 a_{n+1}=a_{n}+a_{n-1},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e38ba916ca83b31b3cbea36715816ff38be9a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.132ex; height:2.343ex;" alt="{\displaystyle a_{n+1}=a_{n}+a_{n-1},}"></dd></dl> allora <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }{a_{n+1} \over a_{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>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>a</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 }{a_{n+1} \over a_{n}}=\varphi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49e48e649b13d8138b810955ef0dfff5d909cf12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.31ex; height:5.176ex;" alt="{\displaystyle \lim _{n\to \infty }{a_{n+1} \over a_{n}}=\varphi .}"></dd></dl> Dimostrazione Ogni numero <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"> della successione è la somma dei due precedenti che a sua volta sono la somma dei due precedenti, di fatto (un po’ come per i coniglietti di Fibonacci) ogni <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"> è esprimibie come somma di un certo numero di volte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbf42ecda092975c9c69dae84e16182ba5fe2e07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{1}}"> più un certo numero di volte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/270580da7333505d9b73697417d0543c43c98b9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{2}}">. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/602d08dd865689204f563ce6f0de095c8ca67410" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{3}}">= 1 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/270580da7333505d9b73697417d0543c43c98b9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{2}}">+1 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbf42ecda092975c9c69dae84e16182ba5fe2e07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{1}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9fde542c1a0ee6390f05d9c0a58e9de213e4415" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{4}}">= <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/602d08dd865689204f563ce6f0de095c8ca67410" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{3}}">+ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {a_{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {a_{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2906ce2f81becffa85ff358466662b9e3a861139" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle {a_{2}}}">= <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1a_{2}+1a_{1})+a_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1a_{2}+1a_{1})+a_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0694cec3bb6aeeef6a7715ff591560761143fdf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.667ex; height:2.843ex;" alt="{\displaystyle (1a_{2}+1a_{1})+a_{2}}"> = 2 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/270580da7333505d9b73697417d0543c43c98b9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{2}}">+ 1 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbf42ecda092975c9c69dae84e16182ba5fe2e07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{1}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{5}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c8009bf62d3e32fef185d63ee7039384e54be05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{5}}">= <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9fde542c1a0ee6390f05d9c0a58e9de213e4415" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{4}}">+ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {a_{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {a_{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e7e009d63738682750d1047c0cf9dd0b7d49cb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle {a_{3}}}">= <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2a_{2}+1a_{1})+(1a_{2}+1a_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2a_{2}+1a_{1})+(1a_{2}+1a_{1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03f0cdfbe13a02499d55b08446c019b3be25fe33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.926ex; height:2.843ex;" alt="{\displaystyle (2a_{2}+1a_{1})+(1a_{2}+1a_{1})}"> = 3 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/270580da7333505d9b73697417d0543c43c98b9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{2}}">+ 2 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbf42ecda092975c9c69dae84e16182ba5fe2e07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{1}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{6}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e982a909d1777b59abc6fb749f670de898e8c1d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{6}}"> = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{5}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c8009bf62d3e32fef185d63ee7039384e54be05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{5}}">+ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {a_{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {a_{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/794d89d9c370162dbf2b94cb4c3a22c5436b2c9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle {a_{4}}}">= <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (3a_{2}+2a_{1})+(2a_{2}+1a_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>3</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (3a_{2}+2a_{1})+(2a_{2}+1a_{1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21fa8421fc5b17c4dec0848bcd64d35b5badcad9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.926ex; height:2.843ex;" alt="{\displaystyle (3a_{2}+2a_{1})+(2a_{2}+1a_{1})}"> = 5 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/270580da7333505d9b73697417d0543c43c98b9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{2}}">+ 3 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbf42ecda092975c9c69dae84e16182ba5fe2e07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{1}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \;\vdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mspace width="thickmathspace" /> <mo>⋮</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \;\vdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41980af70562984aa7fb38489325c31387466af2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.615ex; height:3.676ex;" alt="{\displaystyle \quad \;\vdots }"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}={F_{n-1}}{a_{2}}+{F_{n-2}}a_{1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <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"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}={F_{n-1}}{a_{2}}+{F_{n-2}}a_{1},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69d9462a9734c82248d1194377088b5a0c3c68ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.229ex; height:2.509ex;" alt="{\displaystyle a_{n}={F_{n-1}}{a_{2}}+{F_{n-2}}a_{1},}"></dd></dl> dove <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><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}}"> è l'ennesimo numero di Fibonacci con <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}=F_{2}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da58e6f2110984c6b6f3179d983877eb1d519ddb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.457ex; height:2.509ex;" alt="{\displaystyle F_{1}=F_{2}=1}">, <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><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}">, <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><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}">, ... Quindi: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }{a_{n+1} \over a_{n}}=\lim _{n\to \infty }{F_{n}a_{2}+F_{n-1}a_{1} \over F_{n-1}a_{2}+F_{n-2}a_{1}}=\lim _{n\to \infty }\left({F_{n}a_{2} \over F_{n-1}a_{2}+F_{n-2}a_{1}}+{F_{n-1}a_{1} \over F_{n-1}a_{2}+F_{n-2}a_{1}}\right)=}"> <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>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <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> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <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> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <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> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }{a_{n+1} \over a_{n}}=\lim _{n\to \infty }{F_{n}a_{2}+F_{n-1}a_{1} \over F_{n-1}a_{2}+F_{n-2}a_{1}}=\lim _{n\to \infty }\left({F_{n}a_{2} \over F_{n-1}a_{2}+F_{n-2}a_{1}}+{F_{n-1}a_{1} \over F_{n-1}a_{2}+F_{n-2}a_{1}}\right)=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d40057ab534c01de7ba65461ed71dccebab19a7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:87.891ex; height:6.176ex;" alt="{\displaystyle \lim _{n\to \infty }{a_{n+1} \over a_{n}}=\lim _{n\to \infty }{F_{n}a_{2}+F_{n-1}a_{1} \over F_{n-1}a_{2}+F_{n-2}a_{1}}=\lim _{n\to \infty }\left({F_{n}a_{2} \over F_{n-1}a_{2}+F_{n-2}a_{1}}+{F_{n-1}a_{1} \over F_{n-1}a_{2}+F_{n-2}a_{1}}\right)=}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\lim _{n\to \infty }\left({F_{n} \over F_{n-1}+F_{n-2}{a_{1} \over a_{2}}}+{F_{n-1} \over {F_{n-1}{a_{2} \over a_{1}}+F_{n-2}}}\right)=\lim _{n\to \infty }\left({F_{n} \over {F_{n-1}+F_{n-2}{a_{1} \over a_{2}}}}+{1 \over {a_{2} \over a_{1}}+{F_{n-2} \over F_{n-1}}}\right)=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <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> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <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>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>+</mo> <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> <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 class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <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> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\lim _{n\to \infty }\left({F_{n} \over F_{n-1}+F_{n-2}{a_{1} \over a_{2}}}+{F_{n-1} \over {F_{n-1}{a_{2} \over a_{1}}+F_{n-2}}}\right)=\lim _{n\to \infty }\left({F_{n} \over {F_{n-1}+F_{n-2}{a_{1} \over a_{2}}}}+{1 \over {a_{2} \over a_{1}}+{F_{n-2} \over F_{n-1}}}\right)=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d0569ee1937936aa01335fd9aa72c0cc5851afc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:89.053ex; height:9.843ex;" alt="{\displaystyle =\lim _{n\to \infty }\left({F_{n} \over F_{n-1}+F_{n-2}{a_{1} \over a_{2}}}+{F_{n-1} \over {F_{n-1}{a_{2} \over a_{1}}+F_{n-2}}}\right)=\lim _{n\to \infty }\left({F_{n} \over {F_{n-1}+F_{n-2}{a_{1} \over a_{2}}}}+{1 \over {a_{2} \over a_{1}}+{F_{n-2} \over F_{n-1}}}\right)=}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\lim _{n\to \infty }\left({F_{n} \over {F_{n-1}+F_{n-2}{a_{1} \over a_{2}}}}+{1 \over {a_{2} \over a_{1}}+{1 \over \varphi }}\right)=\lim _{n\to \infty }\left({F_{n} \over {F_{n-1}+F_{n-2}{a_{1} \over a_{2}}}}\right)+{\varphi a_{1} \over \varphi a_{2}+a_{1}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <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> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>φ</mi> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <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> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi>φ</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\lim _{n\to \infty }\left({F_{n} \over {F_{n-1}+F_{n-2}{a_{1} \over a_{2}}}}+{1 \over {a_{2} \over a_{1}}+{1 \over \varphi }}\right)=\lim _{n\to \infty }\left({F_{n} \over {F_{n-1}+F_{n-2}{a_{1} \over a_{2}}}}\right)+{\varphi a_{1} \over \varphi a_{2}+a_{1}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c790858f43b634d7de4590551ff259f52533afe6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:78.518ex; height:8.343ex;" alt="{\displaystyle =\lim _{n\to \infty }\left({F_{n} \over {F_{n-1}+F_{n-2}{a_{1} \over a_{2}}}}+{1 \over {a_{2} \over a_{1}}+{1 \over \varphi }}\right)=\lim _{n\to \infty }\left({F_{n} \over {F_{n-1}+F_{n-2}{a_{1} \over a_{2}}}}\right)+{\varphi a_{1} \over \varphi a_{2}+a_{1}}.}"></dd></dl> Mediante la seguente formula di Binet: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}={\frac {\sqrt {5}}{5}}\left[\left({\frac {1+{\sqrt {5}}}{2}}\right)^{n}-\left({\frac {1-{\sqrt {5}}}{2}}\right)^{n}\right]}"> <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> <msqrt> <mn>5</mn> </msqrt> <mn>5</mn> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <msup> <mrow> <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> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <msup> <mrow> <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> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}={\frac {\sqrt {5}}{5}}\left[\left({\frac {1+{\sqrt {5}}}{2}}\right)^{n}-\left({\frac {1-{\sqrt {5}}}{2}}\right)^{n}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2d1b6068cfcc09f6558c05c16bbd9a25dc8a50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:40.838ex; height:7.509ex;" alt="{\displaystyle F_{n}={\frac {\sqrt {5}}{5}}\left[\left({\frac {1+{\sqrt {5}}}{2}}\right)^{n}-\left({\frac {1-{\sqrt {5}}}{2}}\right)^{n}\right]}"></dd></dl> e facendo una serie di passaggi algebrici, si ottiene <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }\left({F_{n} \over {F_{n-1}+F_{n-2}{a_{1} \over a_{2}}}}\right)+{\varphi a_{1} \over \varphi a_{2}+a_{1}}={{{1 \over 4}(1+{\sqrt {5}})^{2}} \over {{1 \over 2}(1+{\sqrt {5}})+{a_{1} \over a_{2}}}}+{\varphi a_{1} \over \varphi a_{2}+a_{1}}={\varphi ^{2} \over \varphi +{a_{1} \over a_{2}}}+{\varphi a_{1} \over \varphi a_{2}+a_{1}}=\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> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi>φ</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi>φ</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>φ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi>φ</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }\left({F_{n} \over {F_{n-1}+F_{n-2}{a_{1} \over a_{2}}}}\right)+{\varphi a_{1} \over \varphi a_{2}+a_{1}}={{{1 \over 4}(1+{\sqrt {5}})^{2}} \over {{1 \over 2}(1+{\sqrt {5}})+{a_{1} \over a_{2}}}}+{\varphi a_{1} \over \varphi a_{2}+a_{1}}={\varphi ^{2} \over \varphi +{a_{1} \over a_{2}}}+{\varphi a_{1} \over \varphi a_{2}+a_{1}}=\varphi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/547d002cde3cf2706daab443ad3883cffc72e4d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:97.976ex; height:8.009ex;" alt="{\displaystyle \lim _{n\to \infty }\left({F_{n} \over {F_{n-1}+F_{n-2}{a_{1} \over a_{2}}}}\right)+{\varphi a_{1} \over \varphi a_{2}+a_{1}}={{{1 \over 4}(1+{\sqrt {5}})^{2}} \over {{1 \over 2}(1+{\sqrt {5}})+{a_{1} \over a_{2}}}}+{\varphi a_{1} \over \varphi a_{2}+a_{1}}={\varphi ^{2} \over \varphi +{a_{1} \over a_{2}}}+{\varphi a_{1} \over \varphi a_{2}+a_{1}}=\varphi .}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Relazioni_con_il_triangolo_di_Tartaglia_e_i_coefficienti_binomiali">Relazioni con il triangolo di Tartaglia e i coefficienti binomiali</h2>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=4" title="Modifica la sezione Relazioni con il triangolo di Tartaglia e i coefficienti binomiali" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=4" title="Edit section's source code: Relazioni con il triangolo di Tartaglia e i coefficienti binomiali">modifica wikitesto</a>]</div> Il <a href="/wiki/Triangolo_di_Tartaglia" title="Triangolo di Tartaglia">triangolo di Tartaglia</a> è una famosa rappresentazione dei <a href="/wiki/Coefficienti_binomiali" class="mw-redirect" title="Coefficienti binomiali">coefficienti binomiali</a> che si ottengono dallo sviluppo del <a href="/wiki/Binomio_di_Newton" class="mw-redirect" title="Binomio di Newton">binomio di Newton</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+b)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+b)^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bdfac89abdc81ad9084425d7401403b48d41e7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.096ex; height:2.843ex;" alt="{\displaystyle (a+b)^{n}}">, dove <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><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}"> è una riga del triangolo: <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Pascal_triangle.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Pascal_triangle.svg/671px-Pascal_triangle.svg.png" decoding="async" width="671" height="286" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Pascal_triangle.svg/1007px-Pascal_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Pascal_triangle.svg/1342px-Pascal_triangle.svg.png 2x" data-file-width="588" data-file-height="251" /></a><figcaption>Le prime righe del triangolo di Tartaglia</figcaption></figure> Per mostrare che esiste una relazione tra il triangolo e i numeri di Fibonacci, riscriviamo i numeri del triangolo nel seguente modo: <figure class="mw-default-size mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:PascalFibonacci.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/PascalFibonacci.svg/730px-PascalFibonacci.svg.png" decoding="async" width="730" height="840" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/PascalFibonacci.svg/1095px-PascalFibonacci.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6c/PascalFibonacci.svg/1460px-PascalFibonacci.svg.png 2x" data-file-width="1000" data-file-height="1150" /></a><figcaption>Serie di Fibonacci ricavata dal triangolo di Tartaglia</figcaption></figure> A partire dalla prima linea rossa in alto, se si sommano i numeri attraversati da ogni linea, si ottiene la successione di Fibonacci. La relazione con i coefficienti binomiali è: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=\sum _{k=0}^{n-1}{n-k-1 \choose k}=\sum _{k=1}^{n}{n-k \choose k-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> <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> <mo>−</mo> <mn>1</mn> </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> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</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"> <mrow> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </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}^{n-1}{n-k-1 \choose k}=\sum _{k=1}^{n}{n-k \choose k-1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da2f293778edd325bf64983004289d5604db9a0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:38.779ex; height:7.509ex;" alt="{\displaystyle F_{n}=\sum _{k=0}^{n-1}{n-k-1 \choose k}=\sum _{k=1}^{n}{n-k \choose k-1}.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Numeri_di_Fibonacci_e_fattori_comuni">Numeri di Fibonacci e fattori comuni</h3>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=5" title="Modifica la sezione Numeri di Fibonacci e fattori comuni" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=5" title="Edit section's source code: Numeri di Fibonacci e fattori comuni">modifica wikitesto</a>]</div> Se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\mid m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∣</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\mid m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5ddf4a363079b14cbafe893d81d16f3a15f20a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.372ex; height:2.843ex;" alt="{\displaystyle n\mid m}">, allora <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}\mid F_{m}}"> <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>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}\mid F_{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c3e79e6ec15cac501ef8c0a557bbfb63e7073d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.82ex; height:2.843ex;" alt="{\displaystyle F_{n}\mid F_{m}}">, cioè ogni multiplo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle nk}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle nk}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89aeba53b00a81ae0496ccdc34536a6a1dcf1457" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.606ex; height:2.176ex;" alt="{\displaystyle nk}"> di <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><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}"> individua un numero di Fibonacci <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{nk}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{nk}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b379e159834050499a9bcc33443ed7ea2eb6fd71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.57ex; height:2.509ex;" alt="{\displaystyle F_{nk}}"> multiplo di <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><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}}">. Visivamente, si può costruire una tabella mettendo "x" se <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><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}}"> non è un divisore di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18f9a5ed62f131c8ad9acdf7f8539f103f1e0ec9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.294ex; height:2.509ex;" alt="{\displaystyle F_{i}}">: <pre> i 3 4 5 6 7 8 9 10 11 12 F(i) 2 3 5 8 13 21 34 55 89 144 F(3)=2 x x x x x x F(4)=3 x x x x x x x F(5)=5 x x x x x x x x </pre> Da cui si vede che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"> è un fattore di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{3n}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{3n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8afc7d9f338f61368c1814243bdf5938fbbc4f51" 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_{3n}}"> per ogni <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><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}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991e33c6e207b12546f15bdfee8b5726eafbbb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}"> è un fattore di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{4n}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{4n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9dc79db979380485bb3bc3bc17fd0d5d0e9415f" 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_{4n}}"> per ogni <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><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}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29483407999b8763f0ea335cf715a6a5e809f44b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 5}"> è un fattore di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{5n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{5n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1800aa77f95f1e4d47065b392aeede07a4fedd19" 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_{5n}}"> per ogni <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><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}">, e così via. La dimostrazione segue dai coefficienti binomiali <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{k}=\sum _{i=1}^{k-1}{k-i \choose i-1}=m,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </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>k</mi> <mo>−</mo> <mi>i</mi> </mrow> <mrow> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mi>m</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{k}=\sum _{i=1}^{k-1}{k-i \choose i-1}=m,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbe787fed7997a1d8dabfa29c5d8efb85debbe29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.485ex; height:7.343ex;" alt="{\displaystyle F_{k}=\sum _{i=1}^{k-1}{k-i \choose i-1}=m,}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{nk}=\sum _{i=1}^{nk-1}{nk-i \choose i-1}=t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>k</mi> </mrow> </msub> <mo>=</mo> <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> <mi>k</mi> <mo>−</mo> <mn>1</mn> </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> <mi>k</mi> <mo>−</mo> <mi>i</mi> </mrow> <mrow> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{nk}=\sum _{i=1}^{nk-1}{nk-i \choose i-1}=t.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/350b597c29dbef4cb35ddb978a9b239c641258b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.253ex; height:7.343ex;" alt="{\displaystyle F_{nk}=\sum _{i=1}^{nk-1}{nk-i \choose i-1}=t.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Numeri_di_Fibonacci_vicini">Numeri di Fibonacci vicini</h3>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=6" title="Modifica la sezione Numeri di Fibonacci vicini" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=6" title="Edit section's source code: Numeri di Fibonacci vicini">modifica wikitesto</a>]</div> Due numeri di Fibonacci consecutivi <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n},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},F_{n+1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68b84510393683343d02334216e8bca04db924eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.561ex; height:2.509ex;" alt="{\displaystyle F_{n},F_{n+1}}"> non hanno fattori comuni, cioè sono <a href="/wiki/Coprimi" class="mw-redirect" title="Coprimi">coprimi</a>. Infatti, sia <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=ma}"> <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> <mi>m</mi> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=ma}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c77368d583a19aa43d3d270173b4bc525586cfd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.082ex; height:2.509ex;" alt="{\displaystyle F_{n}=ma}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n+1}=mb}"> <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> <mi>m</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n+1}=mb}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05d4a96bf397e81dca191996c23f09382023d68c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.95ex; height:2.509ex;" alt="{\displaystyle F_{n+1}=mb}"> per qualche <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><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}">, in cui <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> è un divisore comune. Si ha <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n-1}=F_{n+1}-F_{n}=m(b-a)}"> <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>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>m</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n-1}=F_{n+1}-F_{n}=m(b-a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf1ce8cf57016da1b01c59b69e63c055d8dd797" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.295ex; height:2.843ex;" alt="{\displaystyle F_{n-1}=F_{n+1}-F_{n}=m(b-a)}">, cioè anche <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><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}}"> ha <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> come divisore e, proseguendo il ragionamento per i termini precedenti <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n-2},F_{n-3},\dots }"> <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> <mo>−</mo> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n-2},F_{n-3},\dots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81ded3f947e665102abca8568747cae6dd9dd51e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.418ex; height:2.509ex;" alt="{\displaystyle F_{n-2},F_{n-3},\dots }">, si arriva che anche <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><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}"> ha <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> come divisore, quindi <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6100c5ebd48c6fd848709f2be624465203eb173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=1}"> <div class="mw-heading mw-heading3"><h3 id="Numeri_di_Fibonacci_primi">Numeri di Fibonacci primi</h3>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=7" title="Modifica la sezione Numeri di Fibonacci primi" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=7" title="Edit section's source code: Numeri di Fibonacci primi">modifica wikitesto</a>]</div> Dato che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{nm}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{nm}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/739121f821cf200c4155fdf9047a9b158d19b085" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.156ex; height:2.509ex;" alt="{\displaystyle F_{nm}}"> è divisibile per <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><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}}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{m}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afc15d41d3176d0fb9b4474762c53d49add76fbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.17ex; height:2.509ex;" alt="{\displaystyle F_{m}}">, se un numero <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4eb94e88d1b1887f734606180a126abfdca3a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.583ex; height:2.509ex;" alt="{\displaystyle F_{k}}"> è primo, anche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> è primo, fatta eccezione per <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><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}">. Non è vero il contrario. Infatti ad esempio <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 19}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>19</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 19}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d12e89044b8eb4a6034d16a39e9c9d0b5c2518ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 19}"> è primo, mentre <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{19}=113\cdot 37=4181}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>19</mn> </mrow> </msub> <mo>=</mo> <mn>113</mn> <mo>⋅</mo> <mn>37</mn> <mo>=</mo> <mn>4181</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{19}=113\cdot 37=4181}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a86264665c4b71abd6966a0cfecb9e5264766cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.709ex; height:2.509ex;" alt="{\displaystyle F_{19}=113\cdot 37=4181}"> non è primo. Il più grande numero di Fibonacci primo noto <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{81839}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>81839</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{81839}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaeb889468af157d5235e61f89519f16eebcf40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.837ex; height:2.509ex;" alt="{\displaystyle F_{81839}}"> è stato segnalato in aprile 2001 da David Broadbent e Bouk de Water. La serie di numeri indice dei numeri primi di Fibonacci è la sequenza <a href="//oeis.org/A001605" class="extiw" title="oeis:A001605">A001605</a>. <div class="mw-heading mw-heading3"><h3 id="Teorema_di_Carmichael_e_fattori_primi_caratteristici">Teorema di Carmichael e fattori primi caratteristici</h3>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=8" title="Modifica la sezione Teorema di Carmichael e fattori primi caratteristici" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=8" title="Edit section's source code: Teorema di Carmichael e fattori primi caratteristici">modifica wikitesto</a>]</div> Per ogni <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>12}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>12</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>12}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e825724bb3853baccbf64cb7959c9784745a8933" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.818ex; height:2.176ex;" alt="{\displaystyle n>12}">, esiste un fattore <a href="/wiki/Numero_primo" title="Numero primo">primo</a> del numero di Fibonacci <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><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}}"> che non è mai apparso come fattore dei numeri di Fibonacci <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4eb94e88d1b1887f734606180a126abfdca3a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.583ex; height:2.509ex;" alt="{\displaystyle F_{k}}">, con <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k<n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo><</mo> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k<n.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3683c640b701bba9563f0497ddae90153a393d98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.351ex; height:2.176ex;" alt="{\displaystyle k<n.}"> Questo teorema è noto come <a href="/wiki/Teorema_di_Carmichael" title="Teorema di Carmichael">teorema di Carmichael</a>. Per <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\leq 12}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≤</mo> <mn>12</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\leq 12}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ced0c8199572399fda990e12bbbbc5192f3c0cae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.818ex; height:2.343ex;" alt="{\displaystyle n\leq 12}"> si hanno i seguenti casi particolari: <dl><dd><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><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}"> (non ha fattori primi);</dd></dl> <dl><dd><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><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}"> (non ha fattori primi);</dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{6}=8}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{6}=8}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ba2ddf4c8e0c05b3b7a2f91bd0907b9f1a2e1b2" 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_{6}=8}">, che ha solo il fattore primo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}">, che è anche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afd489b4f91358d1094a63dec2abf9699478e6c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.549ex; height:2.509ex;" alt="{\displaystyle F_{3}}">;</dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{12}=144}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>=</mo> <mn>144</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{12}=144}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f06ce288f20910858924d892b2a281eddbbc0ee1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.957ex; height:2.509ex;" alt="{\displaystyle F_{12}=144}">, che ha solo i fattori <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991e33c6e207b12546f15bdfee8b5726eafbbb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}">, come i suoi fattori primi e questi sono apparsi in precedenza come <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><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}"> e <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><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}">.</dd></dl> Si noti che questo non significa che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2662434791194da8930c0a1c6d194e189a4470b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.554ex; height:2.843ex;" alt="{\displaystyle F_{p}}"> deve essere un numero primo per ogni <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> primo. Ad esempio <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{19}=4181=37\times 113}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>19</mn> </mrow> </msub> <mo>=</mo> <mn>4181</mn> <mo>=</mo> <mn>37</mn> <mo>×</mo> <mn>113</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{19}=4181=37\times 113}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd8db8a0302473be05e0a2cb4717c4a1a05d564" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.87ex; height:2.509ex;" alt="{\displaystyle F_{19}=4181=37\times 113}">, dove <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 19}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>19</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 19}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d12e89044b8eb4a6034d16a39e9c9d0b5c2518ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 19}"> è un numero primo, ma <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{19}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>19</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{19}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b96f778de41f57d1215e7f2b7b0d5c1f042feb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle F_{19}}"> no. I fattori primi di un numero di Fibonacci <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><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}}"> che non dividono nessun numero di Fibonacci precedente sono detti fattori caratteristici o divisori primi primitivi. Un fattore primitivo di <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><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}}"> è congruente a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1\mod n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mspace width="1em" /> <mi>mod</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1\mod n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fce2dc1e5f3ace64007adc57a617dcd8e16c8a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.627ex; height:2.343ex;" alt="{\displaystyle -1\mod n}">, con l'eccezione <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdb41e9a10a8fd7179b9170149a8d70949ba5d03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=5}">. Se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\equiv 3\mod 10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≡</mo> <mn>3</mn> <mspace width="1em" /> <mi>mod</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\equiv 3\mod 10}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f55c29f0dab5ab6ff0db7af7eb1cb44786680dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.242ex; height:2.176ex;" alt="{\displaystyle n\equiv 3\mod 10}"> e <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><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}"> è un divisore primitivo di <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><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}}">, allora <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><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}"> è primo. Se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\equiv 1\,{\bmod {1}}0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≡</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mrow> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\equiv 1\,{\bmod {1}}0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c92421f0f3701a3aa3e977674ba8db7720688bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.049ex; height:2.176ex;" alt="{\displaystyle n\equiv 1\,{\bmod {1}}0}"> e <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><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}"> è un divisore primitivo di <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><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}}">, allora <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><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}"> è primo (questo teorema è stato citato per la prima volta da <a href="/wiki/%C3%89douard_Lucas" title="Édouard Lucas">Édouard Lucas</a>, ma non dimostrato). <div class="mw-heading mw-heading3"><h3 id="Proprietà_di_divisibilità">Proprietà di divisibilità</h3>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=9" title="Modifica la sezione Proprietà di divisibilità" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=9" title="Edit section's source code: Proprietà di divisibilità">modifica wikitesto</a>]</div> I numeri di Fibonacci godono in generale delle seguenti proprietà di divisibilità: <ul><li>se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\mid k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>∣</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\mid k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17339d7c62447e8d8ad96dd132b1e192225020ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.189ex; height:2.843ex;" alt="{\displaystyle m\mid k}"> allora <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{m}\mid F_{k},}"> <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> <mo>∣</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{m}\mid F_{k},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f4ef3dce09771ce5903a98ff4e08ddcae7b2184" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.337ex; height:2.843ex;" alt="{\displaystyle F_{m}\mid F_{k},}"></li></ul> dove il simbolo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mid y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∣</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mid y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b3e9724af70db6d7d99f205a0c739faaa12cc88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.422ex; height:2.843ex;" alt="{\displaystyle x\mid y}"> significa che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> è un divisore di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83f72471aff7c6fbb27df0f971283a068efe091f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.802ex; height:2.009ex;" alt="{\displaystyle y.}"> Un altro risultato è il seguente: scelti <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><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}"> numeri di Fibonacci da un insieme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1},F_{2},F_{3},\ldots ,F_{2n}}"> <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> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <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 F_{1},F_{2},F_{3},\ldots ,F_{2n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90c4df4d59dfa42eeb9069ba2d4279ef922fc24e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.428ex; height:2.509ex;" alt="{\displaystyle F_{1},F_{2},F_{3},\ldots ,F_{2n}}">, allora uno dei numeri scelti divide un altro esattamente (Weinstein 1966). Mihàly Bencze trovò una nuova proprietà di divisibilità con una nuova sequenza. La sequenza ha i primi quattro valori fissati e la regola <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(n+4)=B(n+1)+B(n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B(n+4)=B(n+1)+B(n).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30d87bf73b9b3b7d96f493b91812d35c1b19c1eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.495ex; height:2.843ex;" alt="{\displaystyle B(n+4)=B(n+1)+B(n).}"> <table class="wikitable"> <tbody><tr> <th>n </th> <td width="20">0 </td> <td width="20">1 </td> <td width="20">2 </td> <td width="20">3 </td> <td width="20">4 </td> <td width="20">5 </td> <td width="20">6 </td> <td width="20">7 </td> <td width="20">8 </td> <td width="20">9 </td> <td width="20">10 </td> <td width="20">11 </td> <td width="20">12 </td> <td width="20">13 </td></tr> <tr> <th>B(n) </th> <td>4 </td> <td>0 </td> <td>0 </td> <td>3 </td> <td>4 </td> <td>0 </td> <td>3 </td> <td>7 </td> <td>4 </td> <td>3 </td> <td>10 </td> <td>11 </td> <td>7 </td> <td>13 </td></tr></tbody></table> Ora si osserva che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c9e64e5bffc2309489a7635add24c595d060382" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.968ex; height:2.843ex;" alt="{\displaystyle B(n)}"> è sempre divisibile per <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><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}">, quando <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><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}"> è un numero primo (Bencze 1998). <div class="mw-heading mw-heading3"><h3 id="Primalità">Primalità</h3>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=10" title="Modifica la sezione Primalità" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=10" title="Edit section's source code: Primalità">modifica wikitesto</a>]</div> Se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> è un numero primo maggiore di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 7}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>7</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 7}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee716ec61382a6b795092c0edd859d12e64cbba8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 7}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\equiv 2\,{\bmod {5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>≡</mo> <mn>2</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\equiv 2\,{\bmod {5}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e1ab22342fd31bcbca40232913c3ee02888db03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:12.75ex; height:2.509ex;" alt="{\displaystyle p\equiv 2\,{\bmod {5}}}"> oppure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\equiv 4\,{\bmod {5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>≡</mo> <mn>4</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\equiv 4\,{\bmod {5}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de9d1bdbafd942bd59a9e6953784b1f5a8a8c1ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:12.75ex; height:2.509ex;" alt="{\displaystyle p\equiv 4\,{\bmod {5}}}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2p-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2p-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f90341d703bdb83f4fdc89bffdfa135e8cea5d6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.335ex; height:2.509ex;" alt="{\displaystyle 2p-1}"> è un numero primo (una condizione che ricorda quella sulla primalità di Sophie Germain), allora <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2p-1)\mid F_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>∣</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2p-1)\mid F_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2609ac2df51d5edaec9c3a50484b9e808055bc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.635ex; height:3.009ex;" alt="{\displaystyle (2p-1)\mid F_{p}}">, quindi <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2662434791194da8930c0a1c6d194e189a4470b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.554ex; height:2.843ex;" alt="{\displaystyle F_{p}}"> è composto. Se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> è primo, allora <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{p}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{p}^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e411619de21fb92e54b9dfe5711d71cfa6e13802" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.033ex; height:2.843ex;" alt="{\displaystyle F_{p}^{n}}"> non è un quadrato perfetto ad eccezione di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261ae92684f7423f53230594c6aca12906781b79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=5}">, nel qual caso però è <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{5}^{n}=5^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo>=</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{5}^{n}=5^{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b50492cfd7a2604b966a1bc04ef436e0503929" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.969ex; height:3.009ex;" alt="{\displaystyle F_{5}^{n}=5^{m}}">, con <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> non <a href="/wiki/Quadrato_perfetto" title="Quadrato perfetto">quadrato perfetto</a>. <div class="mw-heading mw-heading3"><h3 id="Relazioni_con_il_massimo_comun_divisore_e_la_divisibilità">Relazioni con il massimo comun divisore e la divisibilità</h3>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=11" title="Modifica la sezione Relazioni con il massimo comun divisore e la divisibilità" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=11" title="Edit section's source code: Relazioni con il massimo comun divisore e la divisibilità">modifica wikitesto</a>]</div> Un'importante proprietà dei numeri di Fibonacci riguarda il loro <a href="/wiki/Massimo_comun_divisore" title="Massimo comun divisore">massimo comun divisore</a>. Infatti è soddisfatta l'identità <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {MCD} (F_{n},F_{m})=F_{\mathrm {MCD} (n,m)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">D</mi> </mrow> <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>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">D</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {MCD} (F_{n},F_{m})=F_{\mathrm {MCD} (n,m)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8419aef4a5842cb27333de6727d5547e454573e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:27.251ex; height:3.176ex;" alt="{\displaystyle \mathrm {MCD} (F_{n},F_{m})=F_{\mathrm {MCD} (n,m)}}"> (teorema di Vorob'ev).</dd></dl> Da questo segue che <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><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}}"> è divisibile per <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{m}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afc15d41d3176d0fb9b4474762c53d49add76fbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.17ex; height:2.509ex;" alt="{\displaystyle F_{m}}"> se e solo se <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><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}"> è divisibile per <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}">. Questa proprietà è importante perché ne segue che un numero di Fibonacci <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><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}}"> può essere un <a href="/wiki/Numero_primo" title="Numero primo">numero primo</a> solamente se <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><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}"> stesso è un numero primo, con l'unica eccezione di <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><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}"> (l'unico numero di Fibonacci per cui potrebbe essere divisibile è <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><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}">).<a href="#cite_note-5">[5]</a> Il viceversa tuttavia non è vero: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{19}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>19</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{19}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b96f778de41f57d1215e7f2b7b0d5c1f042feb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle F_{19}}">, ad esempio, è uguale a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4181=37\cdot 113}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4181</mn> <mo>=</mo> <mn>37</mn> <mo>⋅</mo> <mn>113</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4181=37\cdot 113}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/123f41c7779d983caf4cc4104c763ca1580c1f3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.24ex; height:2.176ex;" alt="{\displaystyle 4181=37\cdot 113}">. Non è noto se i numeri primi che sono anche numeri di Fibonacci siano o meno infiniti. Inoltre si può dimostrare che ogni numero primo divide almeno uno, e di conseguenza infiniti, numeri di Fibonacci. <div class="mw-heading mw-heading2"><h2 id="Altre_proprietà">Altre proprietà</h2>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=12" title="Modifica la sezione Altre proprietà" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=12" title="Edit section's source code: Altre proprietà">modifica wikitesto</a>]</div> Tra le altre proprietà minori della sequenza di Fibonacci vi sono le seguenti. <ul><li>Charles Raine trovò quanto segue. Si considerino 4 numeri di Fibonacci consecutivi <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{k},F_{k+1},F_{k+2},F_{k+3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>,</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{k},F_{k+1},F_{k+2},F_{k+3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee259b298b79664e23f03c16da35acea5ba4583a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.737ex; height:2.509ex;" alt="{\displaystyle F_{k},F_{k+1},F_{k+2},F_{k+3}}"> e un <a href="/wiki/Triangolo_rettangolo" title="Triangolo rettangolo">triangolo rettangolo</a> con cateti <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> e ipotenusa <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}">. Allora, se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> è uguale al prodotto dei termini esterni e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> è uguale al doppio del prodotto dei termini interni (ovvero se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=F_{k}F_{k+3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=F_{k}F_{k+3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd3629d1bcabfa1c9ce7a6e14b481cadb73c412f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.595ex; height:2.509ex;" alt="{\displaystyle a=F_{k}F_{k+3}}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=2F_{k+1}F_{k+2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>2</mn> <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> <mo>+</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=2F_{k+1}F_{k+2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3e3d03879b8cce5a1e88913d47e13ba5db558aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.626ex; height:2.509ex;" alt="{\displaystyle b=2F_{k+1}F_{k+2}}">), anche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"> è un numero di Fibonacci. Inoltre, l'area del triangolo è uguale al prodotto dei quattro numeri.</li></ul> Prendendo ad esempio i numeri <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3,5,8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3,5,8}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/149b3cbebc589e73e53c2659d10e0f37b16afb0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.555ex; height:2.509ex;" alt="{\displaystyle 3,5,8}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 13,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>13</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 13,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eec95299911f39f2e54999eb8ef3b45373ed69f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.972ex; height:2.509ex;" alt="{\displaystyle 13,}"> allora è <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=3\cdot 13=39,b=2\cdot (5\cdot 8)=80}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>3</mn> <mo>⋅</mo> <mn>13</mn> <mo>=</mo> <mn>39</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>5</mn> <mo>⋅</mo> <mn>8</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>80</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=3\cdot 13=39,b=2\cdot (5\cdot 8)=80}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a40d566fe316418035f6908f0d023d32dc81ad99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.126ex; height:2.843ex;" alt="{\displaystyle a=3\cdot 13=39,b=2\cdot (5\cdot 8)=80}">. Sommando i quadrati ed estraendo la <a href="/wiki/Radice_quadrata" title="Radice quadrata">radice quadrata</a> otteniamo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c={\sqrt {39^{2}+80^{2}}}={\sqrt {1521+6400}}={\sqrt {7921}}=89}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mn>39</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>80</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1521</mn> <mo>+</mo> <mn>6400</mn> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7921</mn> </msqrt> </mrow> <mo>=</mo> <mn>89</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c={\sqrt {39^{2}+80^{2}}}={\sqrt {1521+6400}}={\sqrt {7921}}=89}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2058c6bd1fbb4040e225a8f942077ea61534a7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.31ex; height:3.509ex;" alt="{\displaystyle c={\sqrt {39^{2}+80^{2}}}={\sqrt {1521+6400}}={\sqrt {7921}}=89}">, che è l'undicesimo numero di Fibonacci. L'area del triangolo sarà <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\cdot 5\cdot 8\cdot 13=1560}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>⋅</mo> <mn>5</mn> <mo>⋅</mo> <mn>8</mn> <mo>⋅</mo> <mn>13</mn> <mo>=</mo> <mn>1560</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\cdot 5\cdot 8\cdot 13=1560}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ccc1152e2644690e2b553b3e0b5fc4ed27743bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.598ex; height:2.176ex;" alt="{\displaystyle 3\cdot 5\cdot 8\cdot 13=1560}">. <ul><li>Dati quattro numeri di Fibonacci consecutivi, il prodotto del primo col quarto è sempre pari al prodotto del secondo col terzo aumentato o diminuito di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}">.</li> <li>Se si prende la sequenza dei quadrati dei numeri di Fibonacci e si costruisce una sequenza sommando a due a due i numeri della prima sequenza, la sequenza risultante è costituita da tutti e soli i numeri di Fibonacci di posto dispari.</li> <li>Data la sequenza dei numeri di Fibonacci di posto dispari, se si costruisce la sequenza ottenuta sottraendo a due a due i numeri adiacenti della prima sequenza, si ottiene la sequenza dei numeri di Fibonacci di posto pari.</li> <li>Ogni numero di Fibonacci corrisponde alla somma dei numeri che lo precedono eccetto l'ultimo, aumentata di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}">.</li> <li>Gli unici numeri di Fibonacci che sono anche quadrati sono <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0,1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>,</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0,1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6d7ca945b5a3fb32ceb9513350b49ae0a19a4dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.359ex; height:2.509ex;" alt="{\displaystyle 0,1}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 144,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>144</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 144,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/252e921b7172d45deab44b7f0e3cd05cbe2c4c38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.134ex; height:2.509ex;" alt="{\displaystyle 144,}"> come dimostrato nel 1963 da <a href="/w/index.php?title=John_H._E._Cohn&action=edit&redlink=1" class="new" title="John H. E. Cohn (la pagina non esiste)">John H. E. Cohn</a><a href="#cite_note-6">[6]</a>.</li> <li>L'<a href="/wiki/Identit%C3%A0_di_Cassini" title="Identità di Cassini">identità di Cassini</a>, scoperta nel 1680 da <a href="/wiki/Jean-Dominique_Cassini" class="mw-redirect" title="Jean-Dominique Cassini">Jean-Dominique Cassini</a>, afferma che per ogni intero <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><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}">,</li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n-1}F_{n+1}-F_{n}^{2}=(-1)^{n}.}"> <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> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n-1}F_{n+1}-F_{n}^{2}=(-1)^{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8285bf334efb332814378cdfdc20e0a99092b6d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.08ex; height:3.009ex;" alt="{\displaystyle F_{n-1}F_{n+1}-F_{n}^{2}=(-1)^{n}.}"></dd></dl></dd> <dd>Tale identità è stata generalizzata nel 1879 da <a href="/wiki/Eug%C3%A8ne_Charles_Catalan" title="Eugène Charles Catalan">Eugène Charles Catalan</a>: <dl><dd><math 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"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</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> <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> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>.</mo> </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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd42f69e8869acaf514ba420b4ecc02d1080ea03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.808ex; height:3.009ex;" alt="{\displaystyle F_{n}^{2}-F_{n-r}F_{n+r}=(-1)^{n+r}F_{r}^{2}.}"></dd></dl></dd></dl> <ul><li>La somma dei <a href="/wiki/Reciproco" title="Reciproco">reciproci</a> dei numeri di Fibonacci <a href="/wiki/Serie_convergente" title="Serie convergente">converge</a>, come si può vedere applicando il <a href="/wiki/Criteri_di_convergenza" title="Criteri di convergenza">criterio del rapporto</a>, ricordando che il rapporto tra due numeri di Fibonacci consecutivi tende a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \approx 1,618>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>≈</mo> <mn>1</mn> <mo>,</mo> <mn>618</mn> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi \approx 1,618>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8bad1b14f0f15c6b1ac3113745686946e3a3fb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.429ex; height:2.509ex;" alt="{\displaystyle \phi \approx 1,618>1}">. La somma di questa serie è circa <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3,35988566624;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>,</mo> <mn>35988566624</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3,35988566624;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec3b65ed77efff618c6c3e1b1472e36f7237e474" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.63ex; height:2.509ex;" alt="{\displaystyle 3,35988566624;}"> è stato dimostrato che questo numero è <a href="/wiki/Numero_irrazionale" title="Numero irrazionale">irrazionale</a>. Si può ricavare già da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 100}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>100</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 100}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0572cd017c6d7936a12737c9d614a2f801f94a36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.487ex; height:2.176ex;" alt="{\displaystyle 100}"> termini con PARI/GP: sum(i=1,100,1.0/fibonacci(i))</li></ul> <div class="mw-heading mw-heading2"><h2 id="Algoritmo_di_Euclide_con_ciclo_più_lungo">Algoritmo di Euclide con ciclo più lungo</h2>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=13" title="Modifica la sezione Algoritmo di Euclide con ciclo più lungo" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=13" title="Edit section's source code: Algoritmo di Euclide con ciclo più lungo">modifica wikitesto</a>]</div> Lamé dimostrò nel 1844 che l'<a href="/wiki/Algoritmo_di_Euclide" title="Algoritmo di Euclide">algoritmo di Euclide</a> ha un ciclo più lungo se in input ci sono numeri di Fibonacci. <div class="mw-heading mw-heading2"><h2 id="Frazioni_continue">Frazioni continue</h2>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=14" title="Modifica la sezione Frazioni continue" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=14" title="Edit section's source code: Frazioni continue">modifica wikitesto</a>]</div> Ci sono legami con le frazioni continue da parte dei numeri di Fibonacci e anche con le frazioni di Farey e la sezione aurea. Una particolare frazione continua infinita è la <a href="/wiki/Sezione_aurea" title="Sezione aurea">sezione aurea</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi =[1;1,1,1,1,1,1,1,1,\ldots ].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>;</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi =[1;1,1,1,1,1,1,1,1,\ldots ].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe157d6aed481cf9e27b59d89cb87b24b887b6fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.915ex; height:2.843ex;" alt="{\displaystyle \phi =[1;1,1,1,1,1,1,1,1,\ldots ].}"> La frazione continua precedente si può anche considerare come vari pezzetti di termini convergenti; ad esempio: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0)=0;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0)=0;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48d67005d205994213a6f9441543e3e7bb65012a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.88ex; height:2.843ex;" alt="{\displaystyle (0)=0;}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1)=1;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1)=1;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c727e028d02279dbc9789449e2fc68706c486faa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.076ex; height:2.843ex;" alt="{\displaystyle (0,1)=1;}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1,1)=1/2;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1,1)=1/2;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05c4ae3ce82b45a63620fa14185279070cc845df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.597ex; height:2.843ex;" alt="{\displaystyle (0,1,1)=1/2;}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1,1,1)=2/3;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1,1,1)=2/3;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7662568a75fa0304dd70e82f477db8eccdba70b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.794ex; height:2.843ex;" alt="{\displaystyle (0,1,1,1)=2/3;}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1,1,1,1)=3/5;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</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> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>5</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1,1,1,1)=3/5;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1dd6c47af9ec1d46dd436cef17e8a55f170ba15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.99ex; height:2.843ex;" alt="{\displaystyle (0,1,1,1,1)=3/5;}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1,1,1,1,1)=5/8;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</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> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>8</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1,1,1,1,1)=5/8;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec83720fce0a01586d4ba13b975f11e652b4a60a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.186ex; height:2.843ex;" alt="{\displaystyle (0,1,1,1,1,1)=5/8;}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1,1,1,1,1,1)=8/13.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</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> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>13.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1,1,1,1,1,1)=8/13.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6f60d231e46a6a0c7ba326bc6838ffb2a7923e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.545ex; height:2.843ex;" alt="{\displaystyle (0,1,1,1,1,1,1)=8/13.}"></dd></dl> I vari pezzetti visti prima danno due legami inattesi della sezione Aurea: uno con la successione di Fibonacci, l'altro con la successione di Farey. Difatti tra i pezzetti si ripete la sequenza <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1,2,3,5,8,13,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mn>13</mn> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1,2,3,5,8,13,\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a65971693987807896a13cbf12a1398b8c0a02d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.064ex; height:2.509ex;" alt="{\displaystyle 1,2,3,5,8,13,\ldots }"> come nei numeri di Fibonacci. Escludendo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf6d00d0d700cc9a652fb7e4b7f858b6eb2e9925" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle (0)}">, per ottenere il terzo elemento si devono sommare i primi due, per ottenere poi il successivo termine si devono sommare i precedenti due, ecc. Sempre dai pezzetti si osserva che due successivi convergenti della sezione aurea soddisfano la relazione <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (ps-qr)=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mi>s</mi> <mo>−</mo> <mi>q</mi> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (ps-qr)=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a63b7be0e83a668b064cc9898ed009231b0e47cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.936ex; height:2.843ex;" alt="{\displaystyle (ps-qr)=1.}"> Ad esempio con <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5/8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5/8}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dc6aa52fb652d2fb4f9107b290537a2a55cb8da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle 5/8}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8/13}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>13</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8/13}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcc1f4e4c620ed6bab49e4892815e036bd08d10b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.65ex; height:2.843ex;" alt="{\displaystyle 8/13}"> si ha che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5\cdot 13-8\cdot 8=65-64=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>⋅</mo> <mn>13</mn> <mo>−</mo> <mn>8</mn> <mo>⋅</mo> <mn>8</mn> <mo>=</mo> <mn>65</mn> <mo>−</mo> <mn>64</mn> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5\cdot 13-8\cdot 8=65-64=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6826beb2c200383af82de0751a0d6e096234bae4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:26.86ex; height:2.343ex;" alt="{\displaystyle 5\cdot 13-8\cdot 8=65-64=1}">, come nella serie di Farey. <div class="mw-heading mw-heading2"><h2 id="Generalizzazioni">Generalizzazioni</h2>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=15" title="Modifica la sezione Generalizzazioni" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=15" title="Edit section's source code: Generalizzazioni">modifica wikitesto</a>]</div> Una generalizzazione si può ottenere ponendo: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{0}(a,b,h,k)=a,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>h</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{0}(a,b,h,k)=a,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d850595c1090116db1ec28e143fe28c4282c4b97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.912ex; height:2.843ex;" alt="{\displaystyle W_{0}(a,b,h,k)=a,}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{1}(a,b,h,k)=b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>h</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{1}(a,b,h,k)=b,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72fde9e6f31e27c97210536574928f06be915b1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.679ex; height:2.843ex;" alt="{\displaystyle W_{1}(a,b,h,k)=b,}"></dd></dl> e per ogni <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee74e1cc07e7041edf0fcbd4481f5cd32ad17b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>1}"> sia <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{n}(a,b,h,k)=hW_{n-1}(a,b,h,k)-kW_{n-2}(a,b,h,k).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>h</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>h</mi> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>h</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>k</mi> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>h</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{n}(a,b,h,k)=hW_{n-1}(a,b,h,k)-kW_{n-2}(a,b,h,k).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d4cb22aa2d1648e3302eac5024bfbbe67ea726e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.639ex; height:2.843ex;" alt="{\displaystyle W_{n}(a,b,h,k)=hW_{n-1}(a,b,h,k)-kW_{n-2}(a,b,h,k).}"></dd></dl> Le <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{n}(a,b,h,k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>h</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{n}(a,b,h,k)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c288de20d86e641c9c773648a8038a20c1b39a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.101ex; height:2.843ex;" alt="{\displaystyle W_{n}(a,b,h,k)}"> sono successioni ricorrenti lineari, dove ogni elemento è <a href="/wiki/Combinazione_lineare" title="Combinazione lineare">combinazione lineare</a> dei due precedenti. Si dice successione generalizzata di Fibonacci la sequenza <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{n}(a,b,h,k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>h</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{n}(a,b,h,k)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c288de20d86e641c9c773648a8038a20c1b39a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.101ex; height:2.843ex;" alt="{\displaystyle W_{n}(a,b,h,k)}"> con valori iniziali <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}">: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}(h,k)=W_{n}(0,1,h,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 stretchy="false">(</mo> <mi>h</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mi>h</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}(h,k)=W_{n}(0,1,h,k).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba051a7472424a1e5886ff65848cec85c4c115d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.05ex; height:2.843ex;" alt="{\displaystyle F_{n}(h,k)=W_{n}(0,1,h,k).}"> La classica successione di Fibonacci è: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{F_{n}(1,-1)\}=\{0,1,1,2,3,5,8,13,21,34,55,89,144,\dots \}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mn>13</mn> <mo>,</mo> <mn>21</mn> <mo>,</mo> <mn>34</mn> <mo>,</mo> <mn>55</mn> <mo>,</mo> <mn>89</mn> <mo>,</mo> <mn>144</mn> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{F_{n}(1,-1)\}=\{0,1,1,2,3,5,8,13,21,34,55,89,144,\dots \}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0141bd419d392c7bf4167f5b7fa6abb55660ba6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.498ex; height:2.843ex;" alt="{\displaystyle \{F_{n}(1,-1)\}=\{0,1,1,2,3,5,8,13,21,34,55,89,144,\dots \}.}"></dd></dl> Si dice successione generalizzata di Lucas la successione: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{n}(h,k)=W_{n}(2,h,h,k).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>h</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mi>h</mi> <mo>,</mo> <mi>h</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{n}(h,k)=W_{n}(2,h,h,k).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d09a9b1a64af320ee3013f86b8b3fe1b8046da5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.315ex; height:2.843ex;" alt="{\displaystyle L_{n}(h,k)=W_{n}(2,h,h,k).}"></dd></dl> La classica successione dei numeri di Lucas è: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{L_{n}(1,-1)\}=\{2,1,3,4,7,11,18,29,47,76,123,199,\dots \}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>7</mn> <mo>,</mo> <mn>11</mn> <mo>,</mo> <mn>18</mn> <mo>,</mo> <mn>29</mn> <mo>,</mo> <mn>47</mn> <mo>,</mo> <mn>76</mn> <mo>,</mo> <mn>123</mn> <mo>,</mo> <mn>199</mn> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{L_{n}(1,-1)\}=\{2,1,3,4,7,11,18,29,47,76,123,199,\dots \}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0754320d85f3a47e9706d0be035c91e4f23552a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.715ex; height:2.843ex;" alt="{\displaystyle \{L_{n}(1,-1)\}=\{2,1,3,4,7,11,18,29,47,76,123,199,\dots \}.}"></dd></dl> I numeri di Lucas e quelli di Fibonacci sono collegati da moltissime relazioni. Si noti per esempio che: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+2=3,1+3=4,2+5=7,3+8=11,\dots ,F_{k}+F_{k+2}=L_{k+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>+</mo> <mn>3</mn> <mo>=</mo> <mn>4</mn> <mo>,</mo> <mn>2</mn> <mo>+</mo> <mn>5</mn> <mo>=</mo> <mn>7</mn> <mo>,</mo> <mn>3</mn> <mo>+</mo> <mn>8</mn> <mo>=</mo> <mn>11</mn> <mo>,</mo> <mo>…</mo> <mo>,</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>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+2=3,1+3=4,2+5=7,3+8=11,\dots ,F_{k}+F_{k+2}=L_{k+1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/348e7762d03680cef5c8cfefc05bbf90cf04a41a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:65.126ex; height:2.509ex;" alt="{\displaystyle 1+2=3,1+3=4,2+5=7,3+8=11,\dots ,F_{k}+F_{k+2}=L_{k+1}}">. Quindi si deduce che una successione di Fibonacci può anche non cominciare necessariamente con due <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}">. Questa successione è detta successione di Fibonacci generica o generalizzata. Ogni successione generica di Fibonacci ha una singolare caratteristica, la somma dei primi dieci elementi è sempre uguale a 11 volte il settimo elemento. La dimostrazione è molto semplice: si elenchino i primi dieci elementi in questo modo: <dl><dd>1º elemento: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8ed05f47d4ca397c6c51bfff7fbfa0f7e136344" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.687ex; height:2.009ex;" alt="{\displaystyle m;}"></dd> <dd>2º elemento: <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> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9baf48b143df618b7fe8d7fa3edabd59914a833" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.042ex; height:2.009ex;" alt="{\displaystyle n;}"></dd> <dd>3º elemento: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m+n;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>+</mo> <mi>n</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m+n;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07d1f4ee9c5ed295958f75c150a1d332e6fe3e1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.922ex; height:2.343ex;" alt="{\displaystyle m+n;}"></dd> <dd>4º elemento: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m+2n;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>+</mo> <mn>2</mn> <mi>n</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m+2n;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2c2e12deef004a99ea2846118b387d91f6a115c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.085ex; height:2.509ex;" alt="{\displaystyle m+2n;}"></dd> <dd>5º elemento: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2m+3n;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>m</mi> <mo>+</mo> <mn>3</mn> <mi>n</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2m+3n;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13ee4b746d30eafaccf89c0a825badb0d9230ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.247ex; height:2.509ex;" alt="{\displaystyle 2m+3n;}"></dd> <dd>6º elemento: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3m+5n;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>m</mi> <mo>+</mo> <mn>5</mn> <mi>n</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3m+5n;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38ca6306473f70819d5f86dcdd17b6000a5500e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.247ex; height:2.509ex;" alt="{\displaystyle 3m+5n;}"></dd> <dd>7º elemento: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5m+8n;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mi>m</mi> <mo>+</mo> <mn>8</mn> <mi>n</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5m+8n;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7395fd6f2b332fe79059b7882b8b04934bfd32e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.247ex; height:2.509ex;" alt="{\displaystyle 5m+8n;}"></dd> <dd>8º elemento: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8m+13n;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> <mi>m</mi> <mo>+</mo> <mn>13</mn> <mi>n</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8m+13n;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5581b97cd3d267393023d80c80933c575648dcf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.41ex; height:2.509ex;" alt="{\displaystyle 8m+13n;}"></dd> <dd>9º elemento: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 13m+21n;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>13</mn> <mi>m</mi> <mo>+</mo> <mn>21</mn> <mi>n</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 13m+21n;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b9702e70369cc16b3d06fd00e97edad093fcdf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.572ex; height:2.509ex;" alt="{\displaystyle 13m+21n;}"></dd> <dd>10º elemento: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 21m+34n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>21</mn> <mi>m</mi> <mo>+</mo> <mn>34</mn> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 21m+34n.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edd0889cc5b0039606feef779c90e4bbfd6a96e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.572ex; height:2.343ex;" alt="{\displaystyle 21m+34n.}"></dd></dl> Sommando tutti i dieci elementi, si otterrà <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 55m+88n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>55</mn> <mi>m</mi> <mo>+</mo> <mn>88</mn> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 55m+88n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcadd6bf8e7bf852a54ca5ffd7d9abecd9725bfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.925ex; height:2.343ex;" alt="{\displaystyle 55m+88n}"> che è proprio uguale a 11 volte il settimo elemento. Ogni successione generalizzata conserva la proprietà che il rapporto tra due numeri consecutivi tende alla sezione aurea. Una particolare successione di Fibonacci generalizzata, quella ottenuta ponendo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b32de1b0dc05f6e525ad6a3e8ddeeb4321fd79e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=2}"> e <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}">, è detta <a href="/wiki/Successione_di_Lucas" title="Successione di Lucas">successione di Lucas</a>. <div class="mw-heading mw-heading3"><h3 id="Calcolo_con_le_matrici">Calcolo con le matrici</h3>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=16" title="Modifica la sezione Calcolo con le matrici" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=16" title="Edit section's source code: Calcolo con le matrici">modifica wikitesto</a>]</div> Un metodo efficace per calcolare numeri di Fibonacci generalizzati con indice grande è fare ricorso alle matrici. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M={\begin{bmatrix}0&1\\-k&h\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>k</mi> </mtd> <mtd> <mi>h</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M={\begin{bmatrix}0&1\\-k&h\end{bmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d26e7b29bfb3a65ffa982f0d2b810d978d542e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.075ex; height:6.176ex;" alt="{\displaystyle M={\begin{bmatrix}0&1\\-k&h\end{bmatrix}},}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M^{n}={\begin{bmatrix}-kF_{n-1}(h,k)&F_{n}(h,k)\\-kF_{n}(h,k)&F_{n+1}(h,k)\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>M</mi> <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> <mo>−</mo> <mi>k</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>h</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>h</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>k</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>h</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>h</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M^{n}={\begin{bmatrix}-kF_{n-1}(h,k)&F_{n}(h,k)\\-kF_{n}(h,k)&F_{n+1}(h,k)\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1473c919232792180f33d6f043401e6639eb5799" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.425ex; height:6.176ex;" alt="{\displaystyle M^{n}={\begin{bmatrix}-kF_{n-1}(h,k)&F_{n}(h,k)\\-kF_{n}(h,k)&F_{n+1}(h,k)\end{bmatrix}}.}"></dd></dl> Se <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{r}(M^{n})=L_{r}(h,k),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>h</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{r}(M^{n})=L_{r}(h,k),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ad7909b54ea6300535e10b8dc249828549b8052" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.553ex; height:2.843ex;" alt="{\displaystyle T_{r}(M^{n})=L_{r}(h,k),}"></dd></dl> allora <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M^{n}=T_{n}(h,k)I+F_{n}(h,k)M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>h</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mi>I</mi> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>h</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M^{n}=T_{n}(h,k)I+F_{n}(h,k)M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef33790f6d337c3107db61f8993966210a42a4e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.346ex; height:2.843ex;" alt="{\displaystyle M^{n}=T_{n}(h,k)I+F_{n}(h,k)M}"></dd></dl> dove <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{n}(h,k)=W_{n}(1,0,h,k).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>h</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mi>h</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{n}(h,k)=W_{n}(1,0,h,k).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eedc20b8a5bb71a5ebbe033a0a0a4b8fba2a142" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.913ex; height:2.843ex;" alt="{\displaystyle T_{n}(h,k)=W_{n}(1,0,h,k).}"> <div class="mw-heading mw-heading3"><h3 id="Successioni_Tribonacci_e_Tetranacci">Successioni Tribonacci e Tetranacci</h3>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=17" title="Modifica la sezione Successioni Tribonacci e Tetranacci" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=17" title="Edit section's source code: Successioni Tribonacci e Tetranacci">modifica wikitesto</a>]</div> La successione di Fibonacci può essere anche generalizzata richiedendo che ogni numero sia la somma degli ultimi <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><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}">, dove <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><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}"> è un qualsiasi numero intero. Se <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}"> si ottiene una successione degenere i cui termini sono tutti <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}">, se <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a02c8bd752d2cc859747ca1f3a508281bdbc3b34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=2}"> si ottiene la successione di Fibonacci, mentre per <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c5a5a42ced00df920fad4ab2d4acdb960a4105b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=3}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/295b4bf1de7cd3500e740e0f4f0635db22d87b42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 4}"> si ottengono rispettivamente le cosiddette <a href="/wiki/Successione_Tribonacci" title="Successione Tribonacci">successione Tribonacci</a> e <a href="/wiki/Successione_Tetranacci" title="Successione Tetranacci">Tetranacci</a>. Caratteristica comune di queste successioni è che il rapporto tra due termini consecutivi tende alla radice reale compresa tra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"> del polinomio <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{n}-x^{n-1}-x^{n-2}-\dots -x-1.}"> <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> <mo>⋯</mo> <mo>−</mo> <mi>x</mi> <mo>−</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{n}-x^{n-1}-x^{n-2}-\dots -x-1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/917e2a7a968eb803eddff399cec6a0dd0acd143e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:31.91ex; height:2.843ex;" alt="{\displaystyle x^{n}-x^{n-1}-x^{n-2}-\dots -x-1.}"></dd></dl> Anche la somma dei reciproci degli elementi di questa successione converge (se <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee74e1cc07e7041edf0fcbd4481f5cd32ad17b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>1}">), come si può vedere facilmente considerando che ogni <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">-esimo elemento di una successione è maggiore o uguale del corrispondente elemento <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4eb94e88d1b1887f734606180a126abfdca3a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.583ex; height:2.509ex;" alt="{\displaystyle F_{k}}"> della successione di Fibonacci, e quindi il reciproco è minore. <div class="mw-heading mw-heading2"><h2 id="Numeri_complessi_di_Fibonacci">Numeri complessi di Fibonacci</h2>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=18" title="Modifica la sezione Numeri complessi di Fibonacci" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=18" title="Edit section's source code: Numeri complessi di Fibonacci">modifica wikitesto</a>]</div> Un numero complesso di Fibonacci è un <a href="/wiki/Numero_complesso" title="Numero complesso">numero complesso</a> la cui parte reale è un numero di Fibonacci. Ad esempio <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=8-i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>8</mn> <mo>−</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=8-i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca03e91f13fe858dc73309ad9cb937f37add8ada" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.992ex; height:2.343ex;" alt="{\displaystyle z=8-i}"> è un numero complesso di Fibonacci perché <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Re} (z)=8=F_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>8</mn> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Re} (z)=8=F_{6}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67434340a928e7c85a4b815a280830cf8bfd7fce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.549ex; height:2.843ex;" alt="{\displaystyle \mathrm {Re} (z)=8=F_{6}}">. Il rapporto di numeri complessi di Fibonacci con <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> dispari e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27a6a5d982d54202a14f111cb8a49210501b2c96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>0}"> è tale che: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {F_{k}-ni}{F_{k-1}-(n-1)i}}={\frac {F_{k+n}+i(-1)^{n-1}}{F_{k+(n-1)}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <mi>n</mi> <mi>i</mi> </mrow> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>i</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mi>i</mi> <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> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {F_{k}-ni}{F_{k-1}-(n-1)i}}={\frac {F_{k+n}+i(-1)^{n-1}}{F_{k+(n-1)}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bac8f3ce8e6a4eb119d5da9d33e81153eb8bdb40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.541ex; height:6.676ex;" alt="{\displaystyle {\frac {F_{k}-ni}{F_{k-1}-(n-1)i}}={\frac {F_{k+n}+i(-1)^{n-1}}{F_{k+(n-1)}}},}"></dd></dl> dove <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{k+n}=\sum _{i=k+1}^{n-1}F_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>k</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{k+n}=\sum _{i=k+1}^{n-1}F_{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f73f455a2af8c09591747f120e41ae7f6918e9b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:16.078ex; height:7.509ex;" alt="{\displaystyle F_{k+n}=\sum _{i=k+1}^{n-1}F_{i}.}"> Ad esempio: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {5-i}{3-i}}={\frac {8+i}{5}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>5</mn> <mo>−</mo> <mi>i</mi> </mrow> <mrow> <mn>3</mn> <mo>−</mo> <mi>i</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>8</mn> <mo>+</mo> <mi>i</mi> </mrow> <mn>5</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5-i}{3-i}}={\frac {8+i}{5}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b853afb7beeaa0447bbf161264f98b70335af0de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.028ex; height:5.343ex;" alt="{\displaystyle {\frac {5-i}{3-i}}={\frac {8+i}{5}},}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {13-i}{8-i}}={\frac {21+i}{13}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>13</mn> <mo>−</mo> <mi>i</mi> </mrow> <mrow> <mn>8</mn> <mo>−</mo> <mi>i</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>21</mn> <mo>+</mo> <mi>i</mi> </mrow> <mn>13</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {13-i}{8-i}}={\frac {21+i}{13}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10b3b4a5e5a7562c33c44fed8a5442429361eef8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.353ex; height:5.343ex;" alt="{\displaystyle {\frac {13-i}{8-i}}={\frac {21+i}{13}},}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {8-2i}{5-i}}={\frac {21-i}{13}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>8</mn> <mo>−</mo> <mn>2</mn> <mi>i</mi> </mrow> <mrow> <mn>5</mn> <mo>−</mo> <mi>i</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>21</mn> <mo>−</mo> <mi>i</mi> </mrow> <mn>13</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {8-2i}{5-i}}={\frac {21-i}{13}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1841db29f57943ef4499b7f9455609ead400a5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.353ex; height:5.343ex;" alt="{\displaystyle {\frac {8-2i}{5-i}}={\frac {21-i}{13}},}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {13-3i}{8-2i}}={\frac {55+i}{34}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>13</mn> <mo>−</mo> <mn>3</mn> <mi>i</mi> </mrow> <mrow> <mn>8</mn> <mo>−</mo> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>55</mn> <mo>+</mo> <mi>i</mi> </mrow> <mn>34</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {13-3i}{8-2i}}={\frac {55+i}{34}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b32b6bd65ecc0bc906a747e3d0a63fb8d1142d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.516ex; height:5.343ex;" alt="{\displaystyle {\frac {13-3i}{8-2i}}={\frac {55+i}{34}}.}"></dd></dl> Per <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> pari e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27a6a5d982d54202a14f111cb8a49210501b2c96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>0}"> la formula non vale per i numeri complessi ma solo per i numeri interi sostituendo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"> a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}">, ovvero <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {F_{k}-n}{F_{k-1}-(n-1)}}={\frac {F_{k+n}+(-1)^{n-1}}{F_{k+(n-1)}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <mi>n</mi> </mrow> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <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> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {F_{k}-n}{F_{k-1}-(n-1)}}={\frac {F_{k+n}+(-1)^{n-1}}{F_{k+(n-1)}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3c25b2b06a21b2f878c8fc8e24603298d90162d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:35.936ex; height:6.676ex;" alt="{\displaystyle {\frac {F_{k}-n}{F_{k-1}-(n-1)}}={\frac {F_{k+n}+(-1)^{n-1}}{F_{k+(n-1)}}},}"></dd></dl> dove <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{k+n}=\sum _{i=k+1}^{n-1}F_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>k</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{k+n}=\sum _{i=k+1}^{n-1}F_{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f73f455a2af8c09591747f120e41ae7f6918e9b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:16.078ex; height:7.509ex;" alt="{\displaystyle F_{k+n}=\sum _{i=k+1}^{n-1}F_{i}.}"> Ad esempio: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {8-1}{5-1}}={\frac {13+1}{8}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>8</mn> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mn>5</mn> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>13</mn> <mo>+</mo> <mn>1</mn> </mrow> <mn>8</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {8-1}{5-1}}={\frac {13+1}{8}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c274291e169306788c7b855509de49d39c559e60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.911ex; height:5.343ex;" alt="{\displaystyle {\frac {8-1}{5-1}}={\frac {13+1}{8}},}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {13-2}{8-1}}={\frac {34-1}{21}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>13</mn> <mo>−</mo> <mn>2</mn> </mrow> <mrow> <mn>8</mn> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>34</mn> <mo>−</mo> <mn>1</mn> </mrow> <mn>21</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {13-2}{8-1}}={\frac {34-1}{21}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b30679e72321c32e3bb76bc2ecbbffca0f644388" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.073ex; height:5.343ex;" alt="{\displaystyle {\frac {13-2}{8-1}}={\frac {34-1}{21}},}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {8-3}{5-2}}={\frac {34+1}{21}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>8</mn> <mo>−</mo> <mn>3</mn> </mrow> <mrow> <mn>5</mn> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>34</mn> <mo>+</mo> <mn>1</mn> </mrow> <mn>21</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {8-3}{5-2}}={\frac {34+1}{21}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cfe02c4192547d2fd4d88dc840710c5ef4e3754" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.911ex; height:5.343ex;" alt="{\displaystyle {\frac {8-3}{5-2}}={\frac {34+1}{21}},}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {8-3}{5-2}}={\frac {34+1}{21}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>8</mn> <mo>−</mo> <mn>3</mn> </mrow> <mrow> <mn>5</mn> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>34</mn> <mo>+</mo> <mn>1</mn> </mrow> <mn>21</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {8-3}{5-2}}={\frac {34+1}{21}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a38ece8cb17a6185898dd77ed017550fecb232d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.911ex; height:5.343ex;" alt="{\displaystyle {\frac {8-3}{5-2}}={\frac {34+1}{21}}.}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Sequenza_casuale_di_Fibonacci">Sequenza casuale di Fibonacci</h2>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=19" title="Modifica la sezione Sequenza casuale di Fibonacci" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=19" title="Edit section's source code: Sequenza casuale di Fibonacci">modifica wikitesto</a>]</div> Nel 1999, <a href="/w/index.php?title=Divikar_Viswanath&action=edit&redlink=1" class="new" title="Divikar Viswanath (la pagina non esiste)">Divikar Viswanath</a> considerò una sequenza casuale di Fibonacci, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ebc5a637019ce3415183f06995aeeca93547767" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.574ex; height:2.509ex;" alt="{\displaystyle V_{n}}">, in cui <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ebc5a637019ce3415183f06995aeeca93547767" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.574ex; height:2.509ex;" alt="{\displaystyle V_{n}}"> è definito come <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{n-1}\pm V_{n-2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>±</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{n-1}\pm V_{n-2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68c070368067b58fae13ccb8a81560a3a71127aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.189ex; height:2.509ex;" alt="{\displaystyle V_{n-1}\pm V_{n-2}}">, dove <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/869e366caf596564de4de06cb0ba124056d4064b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \pm }"> è + o - con uguale probabilità. Questa sequenza fu detta sequenza di Vibonacci oppure sequenza casuale di Viswanath. Viswanath scoprì una costante simile al rapporto aureo nella sua successione. Dal momento che la sequenza non è sempre crescente, Viswanath sapeva che la costante sarebbe stata inferiore al rapporto aureo. Tale costante è nota come <a href="/wiki/Costante_di_Viswanath" title="Costante di Viswanath">costante di Viswanath</a>. <div class="mw-heading mw-heading2"><h2 id="Sequenze_Repfigit">Sequenze Repfigit</h2>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=20" title="Modifica la sezione Sequenze Repfigit" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=20" title="Edit section's source code: Sequenze Repfigit">modifica wikitesto</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Numeri_Repfigit">Numeri Repfigit</h3>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=21" title="Modifica la sezione Numeri Repfigit" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=21" title="Edit section's source code: Numeri Repfigit">modifica wikitesto</a>]</div> Il nome deriva da "replicating Fibonacci digit" ed indica i "numeri riproduttori di Fibonacci". Si definisce numero Repfigit o numero di Keith un numero <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><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}"> intero, costituito da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> cifre <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=\sum _{i=0}^{m-1}10^{i}{d_{i}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <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>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=\sum _{i=0}^{m-1}10^{i}{d_{i}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08cf0644ed934e03756b91d2e112fd997ff6a571" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.204ex; height:7.343ex;" alt="{\displaystyle n=\sum _{i=0}^{m-1}10^{i}{d_{i}},}"></dd></dl> che si rigenera all'interno di una sequenza del tipo <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{1},d_{2},\ldots ,d_{m},s_{1},s_{2},s_{3},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{1},d_{2},\ldots ,d_{m},s_{1},s_{2},s_{3},\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/570f97df23050d4785f236e84f440b9573d3c96b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.915ex; height:2.509ex;" alt="{\displaystyle d_{1},d_{2},\ldots ,d_{m},s_{1},s_{2},s_{3},\ldots }"></dd></dl> con <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{1}=d_{1}+d_{2}+d_{3}+\ldots +d_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mo>…</mo> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{1}=d_{1}+d_{2}+d_{3}+\ldots +d_{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/012530e653756985522b766f6300fc24785cfd9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.001ex; height:2.509ex;" alt="{\displaystyle s_{1}=d_{1}+d_{2}+d_{3}+\ldots +d_{m}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{2}=s_{1}+d_{2}+d_{3}+\ldots +d_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mo>…</mo> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{2}=s_{1}+d_{2}+d_{3}+\ldots +d_{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d691f550ebea0a89eb29e65a0ba0e7d024dca3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.883ex; height:2.509ex;" alt="{\displaystyle s_{2}=s_{1}+d_{2}+d_{3}+\ldots +d_{m}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{3}=s_{1}+s_{2}+d_{3}+\ldots +d_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mo>…</mo> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{3}=s_{1}+s_{2}+d_{3}+\ldots +d_{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4676237ab27b4265780e5316741bdbf59352b95e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.764ex; height:2.509ex;" alt="{\displaystyle s_{3}=s_{1}+s_{2}+d_{3}+\ldots +d_{m}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \;\vdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mspace width="thickmathspace" /> <mo>⋮</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \;\vdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41980af70562984aa7fb38489325c31387466af2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.615ex; height:3.676ex;" alt="{\displaystyle \quad \;\vdots }"></dd></dl> Generalizzando si consideri la sequenza definita in maniera ricorsiva da <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{1}=\sum _{k=1}^{m}d_{k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <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>m</mi> </mrow> </munderover> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{1}=\sum _{k=1}^{m}d_{k},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5a4443a0aec5fb2d6304f9c785f2dfdc0546b5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:11.93ex; height:6.843ex;" alt="{\displaystyle s_{1}=\sum _{k=1}^{m}d_{k},}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{k}=\sum _{i=1}^{k-1}s_{i}+\sum _{j=k}^{m}d_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{k}=\sum _{i=1}^{k-1}s_{i}+\sum _{j=k}^{m}d_{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3896533a2a1c95838c7a3999b34d82e60b974a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:19.611ex; height:7.676ex;" alt="{\displaystyle s_{k}=\sum _{i=1}^{k-1}s_{i}+\sum _{j=k}^{m}d_{j}}"> per <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>></mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k>1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfd00bc5bddda614f914a4944f185ebd29e813f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.119ex; height:2.176ex;" alt="{\displaystyle k>1.}"></dd></dl> Se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{k}=n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{k}=n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a19c85238b12ac0afa720254f17f95e0af369479" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.672ex; height:2.009ex;" alt="{\displaystyle s_{k}=n}"> per qualche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">, <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><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}"> è un numero riproduttore di Fibonacci o <a href="/wiki/Numero_di_Keith" title="Numero di Keith">numero di Keith</a>. Esempi di repfigit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=47,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>47</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=47,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2c3ad7376d8529e8866e7ca2305e30dac1c2d32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.465ex; height:2.509ex;" alt="{\displaystyle n=47,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b32de1b0dc05f6e525ad6a3e8ddeeb4321fd79e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=2}"> cifre <dl><dd>4, 7, 11, 18, 29, 47, 76 , ...</dd></dl> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=197,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>197</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=197,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2519be003c0fa2660a8aca200e3c78710e5d0de7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.627ex; height:2.509ex;" alt="{\displaystyle n=197,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/918bb386a7ca6891255b62ef91ccc022883f3809" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=3}"> cifre <dl><dd>1, 9, 7, 17, 33, 57, 107, 197, 361, ...</dd></dl> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1537,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>1537</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=1537,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0524c8384e85afd7615a1202964e4a35014a7066" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.79ex; height:2.509ex;" alt="{\displaystyle n=1537,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0002ab187a5f0920f4c5eff6741f9964cbe2abfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=4}"> cifre <dl><dd>1, 5, 3, 7, 16, 31, 57, 111, 215, 414, 797, 1537, 2963 , ...</dd></dl> Nel 1987 Michael Keith ha introdotto il concetto dei numeri riproduttori di Fibonacci. Nel 1987 il numero repfigit più grande conosciuto era un numero di 7 cifre, 7.913.837. Nel novembre 1989, fu scoperto 44.121.607 e nello stesso anno il dottor Googol trovò che i numeri 129.572.008 e 251.133.297 sono repfigit nell'intervallo definito tra 100 e 1.000 milioni. Oggi sono stati scoperti numeri di questo tipo molto più grandi. Numeri riproduttori di Fibonacci fino a 5 cifre <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b32de1b0dc05f6e525ad6a3e8ddeeb4321fd79e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=2}"></dd> <dd>14 , 19 , 28 , 47 , 61 , 75</dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/918bb386a7ca6891255b62ef91ccc022883f3809" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=3}"></dd> <dd>197 , 742</dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0002ab187a5f0920f4c5eff6741f9964cbe2abfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=4}"></dd> <dd>1104 , 1537 , 2208 , 2580 , 3684 , 4788 , 7385 , 7647 , 7909</dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed436becdbbbc62c94b057f6922d53e6df39d67b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=5}"></dd> <dd>31331 , 34285 , 34348 , 55604 , 62662 , 86935 , 93993</dd></dl> Vedi <a href="#cite_note-7">[7]</a> A007629 in Sloane's OEIS per una lista completa. <div class="mw-heading mw-heading3"><h3 id="Numeri_Repfigit_inversi">Numeri Repfigit inversi</h3>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=22" title="Modifica la sezione Numeri Repfigit inversi" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=22" title="Edit section's source code: Numeri Repfigit inversi">modifica wikitesto</a>]</div> Esistono anche i numeri di Keith inversi, detti sinteticamente revRepfigit. Ad esempio 12 è un numero revRepfigit perché con la tecnica vista prima si può ottenere una sequenza che mi dà il numero rovesciato ovvero 21: 1,2,3,5,8,13,21 Sono revRepfigit anche 12, 24, 36, 48, 52, 71, 341, 682, 1285, 5532, 8166, 17593, 28421, 74733, 90711, 759664, 901921, 1593583, 4808691, ecc. <div class="mw-heading mw-heading3"><h3 id="Congetture">Congetture</h3>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=23" title="Modifica la sezione Congetture" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=23" title="Edit section's source code: Congetture">modifica wikitesto</a>]</div> Ci sono almeno due congetture da verificare, in particolare (1) se i numeri repfigit sono infiniti e (2) se esistono repfigit con <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m>34.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>></mo> <mn>34.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m>34.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/882ce88729ae0fe826adbd59635054623eaaa04f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.111ex; height:2.176ex;" alt="{\displaystyle m>34.}"> <div class="mw-heading mw-heading2"><h2 id="Numeri_di_Fibonacci_e_legami_con_altri_settori">Numeri di Fibonacci e legami con altri settori</h2>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=24" title="Modifica la sezione Numeri di Fibonacci e legami con altri settori" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=24" title="Edit section's source code: Numeri di Fibonacci e legami con altri settori">modifica wikitesto</a>]</div> In matematica i numeri di Fibonacci sono legati in qualche modo alla <a href="/wiki/Sezione_aurea" title="Sezione aurea">sezione aurea</a>, alla <a href="/wiki/Sequenza_di_Farey" title="Sequenza di Farey">sequenza di Farey</a>, alle <a href="/wiki/Frazioni_continue" class="mw-redirect" title="Frazioni continue">frazioni continue</a>, alla zeta di Fibonacci, alla <a href="/wiki/Zeta_di_Riemann" class="mw-redirect" title="Zeta di Riemann">zeta di Riemann</a>, ai <a href="/wiki/Gruppi_di_Lie" class="mw-redirect" title="Gruppi di Lie">gruppi di Lie</a>, ai <a href="/wiki/Frattali" class="mw-redirect" title="Frattali">frattali</a>. In fisica sussiste il legame con la <a href="/wiki/Teoria_delle_stringhe" title="Teoria delle stringhe">teoria delle stringhe</a>. Molti altri legami sono evidenti con la biologia, la cristallografia, la musica, l'economia, l'arte, l'elettrotecnica, l'informatica, ecc. Tuttavia non mancano esempi di "avvistamenti" della successione di Fibonacci un po' forzati: lo rivelano Gael Mariani e Martin Scott dell'Università di Warwick, con un articolo su New Scientist del settembre <a href="/wiki/2005" title="2005">2005</a>.<a href="#cite_note-8">[8]</a> <div class="mw-heading mw-heading3"><h3 id="In_chimica">In chimica</h3>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=25" title="Modifica la sezione In chimica" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=25" title="Edit section's source code: In chimica">modifica wikitesto</a>]</div> Nel 2010 un gruppo di scienziati capeggiato da R. Coldea dell'università di Oxford ha osservato come in un <a href="/wiki/Composto_chimico" title="Composto chimico">composto chimico</a> (niobato di cobalto), portato artificialmente in uno stato quantistico critico, appare una simmetria riconducibile al gruppo di Lie E8, con due picchi alle basse energie in un rapporto simile a quello aureo.<a href="#cite_note-9">[9]</a><a href="#cite_note-10">[10]</a> Tramite il principio geometrico delle teorie di stringa si può trovare che i numeri di Fibonacci conservano la simmetria e sono abbastanza vicini ai "numeri di Lie", sui quali, invece, si basano i cinque gruppi eccezionali di simmetria <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2},F_{4},E_{6},E_{7},E_{8}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{2},F_{4},E_{6},E_{7},E_{8}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0322ec870caab48c662be2db67637abef0775c1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.521ex; height:2.509ex;" alt="{\displaystyle G_{2},F_{4},E_{6},E_{7},E_{8}.}"> Il gruppo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{8}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{8}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48479e96d90b4cfabc7784106cc3cfff907dda34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{8}}"> ha dimensione 57, che è un numero di Lie per <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=7,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>7</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=7,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e7f2bf74832ce645fa3be5abd2f28dd7e139f8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.302ex; height:2.509ex;" alt="{\displaystyle n=7,}"> infatti <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 7^{2}+7+1=57,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>7</mn> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>57</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 7^{2}+7+1=57,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37619e533456ff3cdbcf1bbfba387fd0123f5700" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.293ex; height:3.009ex;" alt="{\displaystyle 7^{2}+7+1=57,}"> vicinissimo al numero di Fibonacci <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 55=7^{2}+7-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>55</mn> <mo>=</mo> <msup> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>7</mn> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 55=7^{2}+7-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/824ada69524c18df20c48341c783c7ac397ad30a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.646ex; height:2.843ex;" alt="{\displaystyle 55=7^{2}+7-1}"> (i numeri di Lie e i numeri di Fibonacci hanno quindi lo stesso DNA geometrico (simmetria) e numerico corrispondente (parabola <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{2}+n+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{2}+n+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66c7cf47e221662db436cb3da1f298f6615c928d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.687ex; height:2.843ex;" alt="{\displaystyle n^{2}+n+1}"> per i numeri di Lie, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{2}+n\pm c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>n</mi> <mo>±</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{2}+n\pm c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9217a576f71b8a04bb0542799cde8d6cf18b4240" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.531ex; height:2.843ex;" alt="{\displaystyle n^{2}+n\pm c}"> con <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><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}"> primo e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"> molto piccolo). Ma il numero 248, collegato a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{8},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{8},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd66083328f421d32e95e350ba1e3045ce9f3fe9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.416ex; height:2.509ex;" alt="{\displaystyle E_{8},}"> è anche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 248=15^{2}+15+8=225+15+8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>248</mn> <mo>=</mo> <msup> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>15</mn> <mo>+</mo> <mn>8</mn> <mo>=</mo> <mn>225</mn> <mo>+</mo> <mn>15</mn> <mo>+</mo> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 248=15^{2}+15+8=225+15+8}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83239009f01fe4b1cda3298841ef3321b72567e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:34.887ex; height:2.843ex;" alt="{\displaystyle 248=15^{2}+15+8=225+15+8}"> con numero vicino di Fibonacci <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 233=15^{2}+15-7.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>233</mn> <mo>=</mo> <msup> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>15</mn> <mo>−</mo> <mn>7.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 233=15^{2}+15-7.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3c9b83507f873948b16fa5cf5b9bd7382cf9fc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.78ex; height:2.843ex;" alt="{\displaystyle 233=15^{2}+15-7.}"> <div class="mw-heading mw-heading3"><h3 id="Nella_musica">Nella musica</h3>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=26" title="Modifica la sezione Nella musica" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=26" title="Edit section's source code: Nella musica">modifica wikitesto</a>]</div> La musica ha numerosi <a href="/wiki/Rapporto_tra_musica_e_matematica" title="Rapporto tra musica e matematica">legami con la matematica</a>, e molti ritengono<a href="#cite_note-11">[11]</a> che importante sia in essa il ruolo della <a href="/wiki/Sezione_aurea" title="Sezione aurea">sezione aurea</a> e dei numeri di Fibonacci.<a href="#cite_note-12">[12]</a> Sul piano <a href="/wiki/Composizione_musicale" class="mw-redirect" title="Composizione musicale">compositivo</a>, attraverso la successione di Fibonacci la sezione aurea può essere rapportata a qualsiasi unità di misura concernente la musica, cioè durata temporale di un brano, numero di note o di battute, etc. Anche se vi sono stati fraintendimenti numerici: nel <a href="/wiki/1978" title="1978">1978</a>, per esempio, nei <a href="/wiki/Kyrie_eleison" title="Kyrie eleison">Kyrie</a> contenuti nel <a href="/wiki/Liber_Usualis" title="Liber Usualis">Liber Usualis</a> <a href="/w/index.php?title=Paul_Larson&action=edit&redlink=1" class="new" title="Paul Larson (la pagina non esiste)">Paul Larson</a> riscontrò il rapporto aureo a livello delle proporzioni melodiche, ma in mancanza di una documentazione che ne attesti un'effettiva volontà di inserimento, la non casualità della ricorrenza rimane tutta a livello puramente congetturale. Simili illazioni sono più volte state espresse circa le opere di <a href="/wiki/Mozart" class="mw-redirect" title="Mozart">Mozart</a>, anche se recentemente John Putz, matematico all'Alma College, convinto anche lui di tale teoria (specialmente per quanto riguarda le sue <a href="/wiki/Wolfgang_Amadeus_Mozart#Sonate" title="Wolfgang Amadeus Mozart">sonate per pianoforte</a>), dovette ricredersi riscontrando un risultato decente soltanto per la <a href="/wiki/Sonata_per_pianoforte_n._1_(Mozart)" title="Sonata per pianoforte n. 1 (Mozart)">Sonata n. 1 in Do maggiore</a>. I musicologi hanno trovato altre applicazioni nei rapporti fra le durate (in <a href="/wiki/Misura_(musica)" title="Misura (musica)">misure</a>) delle varie parti dei brani musicali, in particolare si trovano questi rapporti nelle opere di <a href="/wiki/Claude_Debussy" title="Claude Debussy">Claude Debussy</a><a href="#cite_note-13">[13]</a><a href="#cite_note-14">[14]</a> e di <a href="/wiki/B%C3%A9la_Bart%C3%B3k" title="Béla Bartók">Béla Bartók</a><a href="#cite_note-15">[15]</a><a href="#cite_note-16">[16]</a>. Tra i compositori del <a href="/wiki/XX_secolo" title="XX secolo">XX secolo</a> si evidenziano in proposito <a href="/wiki/Igor%27_F%C3%ABdorovi%C4%8D_Stravinskij" title="Igor' Fëdorovič Stravinskij">Stravinsky</a>, <a href="/wiki/Iannis_Xenakis" title="Iannis Xenakis">Xenakis</a>, <a href="/wiki/Karlheinz_Stockhausen" title="Karlheinz Stockhausen">Stockhausen</a> (nel cui brano Klavierstücke IX si hanno frequenti rimandi alle successioni fibonacciane nelle segnature di tempo), <a href="/wiki/Luigi_Nono_(compositore)" title="Luigi Nono (compositore)">Luigi Nono</a>, <a href="/wiki/Gy%C3%B6rgy_Ligeti" title="György Ligeti">Ligeti</a>, <a href="/wiki/Giacomo_Manzoni" title="Giacomo Manzoni">Giacomo Manzoni</a> e <a href="/wiki/Sofija_Asgatovna_Gubajdulina" title="Sofija Asgatovna Gubajdulina">Sofija Asgatovna Gubajdulina</a> che disse a proposito di Bartok: <style data-mw-deduplicate="TemplateStyles:r139517313">.mw-parser-output .itwiki-template-citazione{margin-bottom:.5em;font-size:95%;padding-left:2.4em;padding-right:1.2em}.mw-parser-output .itwiki-template-citazione-doppia{display:flex;gap:1.2em}.mw-parser-output .itwiki-template-citazione-doppia>div{width:0;flex:1 1 0}.mw-parser-output .itwiki-template-citazione-footer{padding:0 1.2em 0 0;margin:0}</style><div class="itwiki-template-citazione"> <div class="itwiki-template-citazione-singola"> «[...] L'aspetto ritmico della musica di Bartók mi interessa moltissimo, al punto che vorrei studiare a fondo la sua applicazione della Sezione Aurea.» </div></div> Tuttavia è molto difficile stabilire se l'artista abbia voluto consciamente strutturare l'opera con la sezione aurea o se questa non sia piuttosto frutto della sua sensibilità artistica<a href="#cite_note-17">[17]</a>, dato che la sezione aurea si riscontra spesso in natura<a href="#cite_note-18">[18]</a> (come ad esempio nelle stelle marine, in ammoniti, conchiglie, <a href="/wiki/Ananas" title="Ananas">ananas</a>, pigne e nella forma dell'<a href="/wiki/Ovale" title="Ovale">uovo</a><a href="#cite_note-Pappas-19">[19]</a>). Infatti, mentre alcuni ritengono che i sopra citati Debussy e Bartok abbiano deliberatamente impiegato la sezione aurea, per altri questo è meno scontato. D'altronde Debussy stesso<a href="#cite_note-20">[20]</a> scrisse esplicitamente al suo editore Durand (nell'agosto 1903): <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139517313"><div class="itwiki-template-citazione"> <div class="itwiki-template-citazione-doppia"> <div>(<abbr title="francese">FR</abbr>) «Vous verrez, à la page 8 de "Jardins sous la Pluie", qu'il manque une mesure; c'est d'ailleurs un oubli de ma part, car elle n'est pas dans le manuscrit. Pourtant, elle est nécessaire, quant au nombre; le divine nombre [...].» </div> <div>(<abbr title="italiano">IT</abbr>) «Lei vedrà, alla pagina 8 di "<a href="/wiki/Estampes_(Debussy)#Jardins_Sous_La_Pluie" title="Estampes (Debussy)">Jardins sous la Pluie</a>" che manca una battuta; è del resto una mia dimenticanza, perché non è nel manoscritto. Eppure, è necessaria, per il numero; il divino numero [...].» </div></div></div> Nel <a href="/wiki/Novecento" class="mw-redirect" title="Novecento">Novecento</a> le <a href="/wiki/Avanguardie" class="mw-redirect" title="Avanguardie">avanguardie</a> della musica colta e molti tra gli eredi del <a href="/wiki/Serialismo" title="Serialismo">serialismo</a>, come i già citati <a href="/wiki/Karlheinz_Stockhausen" title="Karlheinz Stockhausen">Karlheinz Stockhausen</a>, <a href="/wiki/Gy%C3%B6rgy_Ligeti" title="György Ligeti">György Ligeti</a> e <a href="/wiki/Iannis_Xenakis" title="Iannis Xenakis">Iannis Xenakis</a>, applicarono invece sistematicamente e intenzionalmente - a differenza della maggioranza dei loro predecessori - i numeri di Fibonacci alla musica, approfondendone lo studio e la conoscenza; facendo evolvere i precedenti utilizzi della matematica in musica, hanno introdotto un utilizzo più strutturato della matematica (soprattutto il <a href="/wiki/Teoria_della_probabilit%C3%A0" title="Teoria della probabilità">calcolo delle probabilità</a> e del <a href="/wiki/Computer" title="Computer">computer</a> per la composizione musicale). Xenakis in particolare ha fondato a tale fine, a <a href="/wiki/Parigi" title="Parigi">Parigi</a> nel <a href="/wiki/1972" title="1972">1972</a>, un gruppo di ricerca universitario chiamato CEMAMU, che ha appunto come obiettivo l'applicazione delle conoscenze scientifiche moderne e del computer alla composizione musicale e alla creazione di nuovi suoni tramite <a href="/wiki/Sintetizzatore" title="Sintetizzatore">sintetizzatori</a>. Anche la musica <a href="/wiki/Rock" title="Rock">rock</a>, specialmente nel cosiddetto <a href="/wiki/Rock_progressivo" title="Rock progressivo">rock progressivo</a>, si è confrontata con gli aspetti <a href="/wiki/Mistica" title="Mistica">mistico</a>-<a href="/wiki/Esoterismo" title="Esoterismo">esoterici</a> della sezione aurea, e più precisamente dalla successione di Fibonacci. L'esempio più emblematico è la musica dei <a href="/wiki/Genesis" title="Genesis">Genesis</a>, che hanno usato assiduamente questa successione nella costruzione armonico-temporale dei loro brani; <a href="/wiki/Firth_of_Fifth" title="Firth of Fifth">Firth of Fifth</a> è tutto basato su numeri aurei: ad esempio ci sono <a href="/wiki/Assolo" title="Assolo">assoli</a> di 55, 34, 13 battute, di questi alcuni sono formati da 144 note, etc. Oltre ai Genesis, altre rock band hanno usato, seppure più sporadicamente, i numeri aurei nelle loro composizioni. Fra questi i <a href="/wiki/Deep_Purple" title="Deep Purple">Deep Purple</a> nel brano <a href="/wiki/Child_in_Time" title="Child in Time">Child in Time</a> e i <a href="/wiki/Dream_Theater" title="Dream Theater">Dream Theater</a> nell'album <a href="/wiki/Octavarium" title="Octavarium">Octavarium</a>, interamente concepito secondo il rapporto tra i numeri 8 e 5 e termini consecutivi della sequenza di Fibonacci. Risale invece al 2001 <a href="/wiki/Lateralus" title="Lateralus">Lateralus</a> album della band statunitense <a href="/wiki/Tool_(gruppo_musicale)" title="Tool (gruppo musicale)">Tool</a> che contiene il singolo omonimo "Lateralus" costruito fedelmente sulla successione di Fibonacci: i Tool fanno un sapiente uso dei primi elementi della successione di Fibonacci: contando infatti le sillabe della prima strofa si ottiene 1,1,2,3,5,8,5,3,2,1,1,2,3,5,8,13,8,5,3. Inoltre la ritmica della canzone alterna battute da 9/8, 8/8 e 7/8, il numero ottenuto è 987 che è il sedicesimo numero della sequenza. Da notare che la canzone fa un continuo riferimento alla figura della spirale ([...] To swing on the spiral [...] Spiral out. Keep going [...]). <div class="mw-heading mw-heading3"><h3 id="In_botanica">In botanica</h3>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=27" title="Modifica la sezione In botanica" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=27" title="Edit section's source code: In botanica">modifica wikitesto</a>]</div> Quasi tutti i <a href="/wiki/Fiori" class="mw-redirect" title="Fiori">fiori</a> hanno tre, cinque, otto, tredici, ventuno, trentaquattro, cinquantacinque o ottantanove petali: ad esempio i <a href="/wiki/Lilium" title="Lilium">gigli</a> ne hanno tre, i ranuncoli cinque, il <a href="/wiki/Delphinium" title="Delphinium">delphinium</a> spesso ne ha otto, la <a href="/wiki/Calendula" title="Calendula">calendula</a> tredici, l'<a href="/wiki/Astro_(botanica)" class="mw-redirect" title="Astro (botanica)">astro</a> ventuno, e le <a href="/wiki/Margherite" class="mw-redirect" title="Margherite">margherite</a> di solito ne hanno trentaquattro o cinquantacinque o ottantanove. <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Pflanze-Sonnenblume1-Asio.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Pflanze-Sonnenblume1-Asio.JPG/220px-Pflanze-Sonnenblume1-Asio.JPG" decoding="async" width="220" height="282" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Pflanze-Sonnenblume1-Asio.JPG/330px-Pflanze-Sonnenblume1-Asio.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Pflanze-Sonnenblume1-Asio.JPG/440px-Pflanze-Sonnenblume1-Asio.JPG 2x" data-file-width="1944" data-file-height="2491" /></a><figcaption>La disposizione dei fiori nel <a href="/wiki/Capolino" title="Capolino">capolino</a> del <a href="/wiki/Helianthus_annuus" title="Helianthus annuus">girasole</a></figcaption></figure> I numeri di Fibonacci sono presenti anche in altre piante come il <a href="/wiki/Helianthus_annuus" title="Helianthus annuus">girasole</a>; difatti i piccoli fiori al centro del girasole (che è in effetti una infiorescenza) sono disposti lungo due insiemi di spirali che girano rispettivamente in senso orario e antiorario. I <a href="/wiki/Pistilli" class="mw-redirect" title="Pistilli">pistilli</a> sulle <a href="/wiki/Corolle" class="mw-redirect" title="Corolle">corolle</a> dei fiori spesso si dispongono secondo uno schema preciso formato da spirali il cui numero corrisponde ad uno della successione di Fibonacci. Di solito le <a href="/wiki/Spirale" title="Spirale">spirali</a> orientate in senso orario sono trentaquattro mentre quelle orientate in senso antiorario cinquantacinque (due numeri di Fibonacci); altre volte sono rispettivamente cinquantacinque e ottantanove, o ottantanove e centoquarantaquattro. Si tratta sempre di numeri di Fibonacci consecutivi. I numeri di Fibonacci sono presenti anche nel numero di infiorescenze di ortaggi come il <a href="/wiki/Broccolo_romanesco" title="Broccolo romanesco">broccolo romanesco</a>. Le <a href="/wiki/Foglie" class="mw-redirect" title="Foglie">foglie</a> sono disposte sui rami in modo tale da non coprirsi l'una con l'altra per permettere a ciascuna di esse di ricevere la luce del sole. Se prendiamo come punto di partenza la prima foglia di un ramo e contiamo quante foglie ci sono fino a quella perfettamente allineata, spesso questo numero è un numero di Fibonacci, e anche il numero di giri in senso orario o antiorario che si compiono per raggiungere tale foglia allineata dovrebbe essere un numero di Fibonacci. Il rapporto tra il numero di foglie e il numero di giri si chiama “rapporto fillotattico” (vedi <a href="/wiki/Fillotassi" title="Fillotassi">Fillotassi</a>). <div class="mw-heading mw-heading3"><h3 id="Nel_corpo_umano">Nel corpo umano</h3>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=28" title="Modifica la sezione Nel corpo umano" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=28" title="Edit section's source code: Nel corpo umano">modifica wikitesto</a>]</div> Il rapporto fra le lunghezze delle <a href="/wiki/Falange_(anatomia)" title="Falange (anatomia)">falangi</a> del dito medio e anulare di un uomo adulto è aureo, come anche il rapporto tra la lunghezza del braccio e l'avambraccio, e tra la lunghezza della gamba e la sua parte inferiore.<a href="#cite_note-21">[21]</a><a href="#cite_note-22">[22]</a> Se si misura l'altezza dividendola per la distanza da terra del nostro <a href="/wiki/Ombelico" title="Ombelico">ombelico</a>, si ottiene Phi. Lo stesso accade se si misura la distanza dalla spalla alla punta delle dita e poi la dividiamo per la distanza dal gomito alla punta delle dita.<a href="#cite_note-La_Nazione-23">[23]</a> <div class="mw-heading mw-heading3"><h3 id="In_geometria_e_in_natura">In geometria e in natura</h3>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=29" title="Modifica la sezione In geometria e in natura" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=29" title="Edit section's source code: In geometria e in natura">modifica wikitesto</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Fibonacci_spiral_2019.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Fibonacci_spiral_2019.svg/180px-Fibonacci_spiral_2019.svg.png" decoding="async" width="180" height="114" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Fibonacci_spiral_2019.svg/270px-Fibonacci_spiral_2019.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Fibonacci_spiral_2019.svg/360px-Fibonacci_spiral_2019.svg.png 2x" data-file-width="512" data-file-height="324" /></a><figcaption>La spirale di Fibonacci, creata mediante l'unione di quadrati con i lati equivalenti ai numeri della successione di Fibonacci.</figcaption></figure> Se si disegna un rettangolo con i lati in rapporto aureo fra di loro, lo si può dividere in un quadrato e un altro rettangolo, simile a quello grande nel senso che anche i suoi lati stanno fra loro nel rapporto aureo. A questo punto il rettangolo minore può essere diviso in un quadrato e un rettangolo che ha pure i lati in rapporto aureo, e così via. <style data-mw-deduplicate="TemplateStyles:r140555421">.mw-parser-output .itwiki-template-approfondimento{border:1px solid var(--border-color-subtle,#c8ccd1);background-color:var(--background-color-interactive-subtle,#f8f9fa);padding:2px;box-sizing:border-box}@media all and (max-width:720px){.mw-parser-output .itwiki-template-approfondimento{width:100%!important}}@media all and (min-width:720px){.mw-parser-output .itwiki-template-approfondimento-sinistra{clear:left;float:left;margin-right:10px;margin-left:0}.mw-parser-output .itwiki-template-approfondimento-centro{margin-left:auto;margin-right:auto}.mw-parser-output .itwiki-template-approfondimento-destra{clear:right;float:right;margin-left:10px}}.mw-parser-output .itwiki-template-approfondimento-intestazione{background:#C3D0DF;padding:2px;text-align:center}@media screen{html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-approfondimento-intestazione{background-color:var(--background-color-neutral,#27292d)}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-approfondimento-intestazione{background-color:var(--background-color-neutral,#27292d)}}</style><div role="complementary" class="itwiki-template-approfondimento itwiki-template-approfondimento-sinistra" style="width:300px;" id="In_apicoltura"> <div class="itwiki-template-approfondimento-intestazione">In <a href="/wiki/Apicoltura" title="Apicoltura">apicoltura</a></div> <div style="margin: 0.4em 0; font-size:95%"> Leonardo da Pisa o Fibonacci visse vicino a <a href="/wiki/B%C3%A9ja%C3%AFa" title="Béjaïa">Béjaïa</a>, a quell'epoca importante città esportatrice di cera (da ciò deriva la versione francese del nome della città, "bougie", che significa "candela" in francese). Una recente analisi matematico-storica del periodo e della regione in cui visse Fibonacci suggerisce che, in realtà, furono gli apicoltori di Bejaia e le loro conoscenze sulla riproduzione delle api la fonte di ispirazione per la Successione di Fibonacci e non il più noto modello della riproduzione dei conigli<a href="#cite_note-24">[24]</a>. </div> </div> La curva che passa per vertici consecutivi di questa successione di rettangoli è una <a href="/wiki/Spirale" title="Spirale">spirale</a> che troviamo spesso nelle <a href="/wiki/Conchiglie" class="mw-redirect" title="Conchiglie">conchiglie</a> e nella disposizione dei semi del <a href="/wiki/Helianthus_annuus" title="Helianthus annuus">girasole</a> sopra descritta e delle foglie su un ramo, oltre che negli alveari delle <a href="/wiki/Apis" title="Apis">api</a> nel rapporto numerico tra femmine e maschi.<a href="#cite_note-La_Nazione-23">[23]</a> <div class="mw-heading mw-heading3"><h3 id="Nell'arte">Nell'arte</h3>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=30" title="Modifica la sezione Nell'arte" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=30" title="Edit section's source code: Nell'arte">modifica wikitesto</a>]</div> I numeri di Fibonacci sono stati usati in alcune opere d'arte: nella <a href="/wiki/Piramide_di_Cheope" title="Piramide di Cheope">piramide di Cheope</a>, come nel <a href="/w/index.php?title=Partenone_di_Atene&action=edit&redlink=1" class="new" title="Partenone di Atene (la pagina non esiste)">Partenone di Atene</a>.<a href="#cite_note-La_Nazione-23">[23]</a> Secondo Pietro Armienti, docente all'Università di Pisa ed esperto di <a href="/wiki/Petrologia" title="Petrologia">petrologia</a>, le geometrie presenti sulla facciata della <a href="/wiki/Chiesa_di_San_Nicola_(Pisa)" title="Chiesa di San Nicola (Pisa)">chiesa pisana di San Nicola</a> sarebbero un chiaro riferimento alla successione del matematico.<a href="#cite_note-25">[25]</a> <a href="/wiki/Mario_Merz" title="Mario Merz">Mario Merz</a> li ha usati nell'installazione luminosa denominata Il volo dei numeri, su una delle fiancate della <a href="/wiki/Mole_Antonelliana" title="Mole Antonelliana">Mole Antonelliana</a> di <a href="/wiki/Torino" title="Torino">Torino</a>. Sulle mura di <a href="/wiki/San_Casciano_in_Val_di_Pesa" title="San Casciano in Val di Pesa">San Casciano in Val di Pesa</a>, inoltre, accanto ad un <a href="/wiki/Cervo" class="mw-redirect" title="Cervo">cervo</a> imbalsamato, sono permanentemente installati i numeri al <a href="/wiki/Neon" title="Neon">neon</a> riportanti le cifre 55, 89, 144, 233, 377 e 610. Si tratta di una creazione di Merz realizzata in occasione della mostra Tuscia Electa del <a href="/wiki/1997" title="1997">1997</a><a href="#cite_note-26">[26]</a>. Lo stesso autore ha inoltre realizzato nel <a href="/wiki/1994" title="1994">1994</a> un'installazione permanente sulla ciminiera della compagnia elettrica <a href="/w/index.php?title=Turku_Energia&action=edit&redlink=1" class="new" title="Turku Energia (la pagina non esiste)">Turku Energia</a> a <a href="/wiki/Turku" title="Turku">Turku</a>, in <a href="/wiki/Finlandia" title="Finlandia">Finlandia</a>. Tutta l'opera di <a href="/wiki/Tobia_Rav%C3%A0" title="Tobia Ravà">Tobia Ravà</a> fa riferimento alla successione di Fibonacci, scoprendone anche una specifica proprietà. Anche il pittore <a href="/wiki/Austria" title="Austria">austriaco</a> <a href="/w/index.php?title=Helmutt_Bruck&action=edit&redlink=1" class="new" title="Helmutt Bruck (la pagina non esiste)">Helmutt Bruck</a> ha dipinto quadri omaggianti Fibonacci e prodotto opere in serie di 21. A <a href="/wiki/Barcellona" title="Barcellona">Barcellona</a> e a <a href="/wiki/Napoli" title="Napoli">Napoli</a> è stata creata un'installazione luminosa: nella città spagnola si trova nell'area della <a href="/wiki/La_Barceloneta" title="La Barceloneta">Barceloneta</a>, all'interno dell'area pedonale, dove i numeri sono posti a distanze proporzionali alla loro differenza, mentre a Napoli sono disposti a spirale all'interno della <a href="/wiki/Vanvitelli_(metropolitana_di_Napoli)" title="Vanvitelli (metropolitana di Napoli)">stazione Vanvitelli</a> della <a href="/wiki/Linea_1_(metropolitana_di_Napoli)" title="Linea 1 (metropolitana di Napoli)">linea 1</a> della <a href="/wiki/Metropolitana_di_Napoli" title="Metropolitana di Napoli">metropolitana</a>, e più precisamente sul soffitto che sovrasta le scale mobili quando, superate le obliteratrici, si scende all'interno della stazione vera e propria. Nel 2017, ad <a href="/wiki/Albissola_Marina" title="Albissola Marina">Albissola Marina</a>, nella Piazzetta Poggi del centro storico, è stato installato un mosaico pavimentale dal titolo Fiore di Fibonacci, dovuto all'artista Gabriele Gelatti. <div class="mw-heading mw-heading3"><h3 id="Nell'economia">Nell'economia</h3>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=31" title="Modifica la sezione Nell'economia" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=31" title="Edit section's source code: Nell'economia">modifica wikitesto</a>]</div> I numeri di Fibonacci sono utilizzati anche in economia nell'<a href="/wiki/Analisi_tecnica" title="Analisi tecnica">Analisi tecnica</a> per le previsioni dell'andamento dei titoli in borsa, secondo la <a href="/w/index.php?title=Teoria_delle_onde_di_Elliott&action=edit&redlink=1" class="new" title="Teoria delle onde di Elliott (la pagina non esiste)">teoria delle onde di Elliott</a>.<a href="#cite_note-27">[27]</a> Studiando i grafici storici dei titoli, <a href="/w/index.php?title=Ralph_Nelson_Elliott&action=edit&redlink=1" class="new" title="Ralph Nelson Elliott (la pagina non esiste)">Ralph Nelson Elliott</a> sviluppò un metodo basato su tredici conformazioni grafiche dette onde, simili per forma ma non necessariamente per dimensione. A differenza di altre applicazioni grafiche come medie mobili, trendline, macd, rsi ecc. che si limitano ad indicare il livello di resistenza e di supporto e le angolature del trend "Il principio delle onde di Elliott" è l'unico metodo in grado di individuare un movimento del mercato dall'inizio alla fine e quindi di presumere i futuri andamenti dei prezzi. <div class="mw-heading mw-heading3"><h3 id="In_informatica">In informatica</h3>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=32" title="Modifica la sezione In informatica" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=32" title="Edit section's source code: In informatica">modifica wikitesto</a>]</div> I numeri di Fibonacci sono utilizzati anche nel sistema informatico di molti computer. In particolare vi è un complesso meccanismo basato su tali numeri, detto "<a href="/w/index.php?title=Fibonacci_heap&action=edit&redlink=1" class="new" title="Fibonacci heap (la pagina non esiste)">Fibonacci heap</a>" che viene utilizzato nel processore Pentium della Intel per la risoluzione di particolari algoritmi.<a href="#cite_note-28">[28]</a> Il seguente algoritmo in Python permette di trovare l'n-esimo numero della serie di Fibonacci. <div class="mw-highlight mw-highlight-lang-python3 mw-content-ltr mw-highlight-lines" dir="ltr"><pre>def fibonacci(n): if n < 2: return 1 return fibonacci(n-2) + fibonacci(n-1) </pre></div> <div class="mw-heading mw-heading3"><h3 id="Nei_frattali">Nei frattali</h3>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=33" title="Modifica la sezione Nei frattali" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=33" title="Edit section's source code: Nei frattali">modifica wikitesto</a>]</div> Nei frattali di <a href="/wiki/Beno%C3%AEt_Mandelbrot" title="Benoît Mandelbrot">Mandelbrot</a>, governati dalla proprietà dell'autosomiglianza, si ritrovano i numeri di Fibonacci. L'autosomiglianza difatti è governata da una regola o formula ripetibile, così come la successione di Fibonacci. <div class="mw-heading mw-heading3"><h3 id="In_elettrotecnica">In elettrotecnica</h3>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=34" title="Modifica la sezione In elettrotecnica" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=34" title="Edit section's source code: In elettrotecnica">modifica wikitesto</a>]</div> Una rete di resistori, per esempio un Ladder Network (Rete a scala), ha una resistenza equivalente ai morsetti A e B esprimibile sia come frazione continua che tramite la <a href="/wiki/Sezione_aurea" title="Sezione aurea">sezione aurea</a> o i numeri di Fibonacci (infatti si ha Req/R = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }">).<a href="#cite_note-29">[29]</a> <div class="mw-heading mw-heading3"><h3 id="Nei_giochi_sistemici">Nei giochi sistemici</h3>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=35" title="Modifica la sezione Nei giochi sistemici" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=35" title="Edit section's source code: Nei giochi sistemici">modifica wikitesto</a>]</div> In qualunque gioco sistemico come totocalcio, superenalotto o roulette i numeri di Fibonacci possono essere utilizzati come montanti per le puntate. <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=36" title="Modifica la sezione Note" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=36" title="Edit section's source code: Note">modifica wikitesto</a>]</div> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://oeis.org/A000045">A000045 - OEIS</a>, su oeis.org. URL consultato il 6 marzo 2019.</cite> </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite class="citation pubblicazione" style="font-style:normal"> T.C. Scott e P. Marketos, <a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/Publications/fibonacci.pdf">On the Origin of the Fibonacci Sequence</a> (<abbr title="documento in formato PDF">PDF</abbr>), <a href="/wiki/MacTutor_History_of_Mathematics_archive" class="mw-redirect" title="MacTutor History of Mathematics archive">MacTutor History of Mathematics archive</a>, University of St Andrews, marzo 2014.</cite> </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite class="citation libro" style="font-style:normal"> Leonardo Fibonacci, <a rel="nofollow" class="external text" href="https://bibdig.museogalileo.it/tecanew/opera?bid=1072400&seq=253">Liber Abaci</a>, Biblioteca nazionale centrale, Firenze.</cite> </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite class="citation libro" style="font-style:normal"> Samuele Maschio, Principi di induzione, in Tecniche dimostrative, Trieste, Scienza Express, 2019, pp. 66-67, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-88-969-7375-2" title="Speciale:RicercaISBN/978-88-969-7375-2">978-88-969-7375-2</a>.</cite> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> La sequenza <cite class="citation testo" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://oeis.org/A005478">A005478</a>.</cite> dell'<a href="/wiki/OEIS" class="mw-redirect" title="OEIS">OEIS</a> elenca i primi numeri primi presenti nella successione di Fibonacci; la sequenza <cite class="citation testo" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://oeis.org/A001605">A001605</a>.</cite> ne elenca invece gli indici </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite class="citation news" style="font-style:normal"> J H E Cohn, <a rel="nofollow" class="external text" href="https://archive.org/details/sim_fibonacci-quarterly_1964-04_2_2/page/109">Square Fibonacci Numbers Etc</a>, in Fibonacci Quarterly, vol. 2, 1964, pp. 109-113.</cite> </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://oeis.org/A007629">A007629 - OEIS</a>, su oeis.org. URL consultato il 29 novembre 2024.</cite> </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.newscientist.com/letter/mg18725180-400-not-so-fibonacci/">Not so Fibonacci</a>, su newscientist.com.</cite> </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite class="citation news" style="font-style:normal"> <a rel="nofollow" class="external text" href="http://www.lescienze.it/news/2010/01/08/news/il_rapporto_aureo_governa_la_musica_quantistica-576543/">Il rapporto aureo governa la "musica" quantistica - Le Scienze</a>, in Le Scienze. URL consultato il 15 novembre 2016.</cite> </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite class="citation news" style="font-style:normal"> <a rel="nofollow" class="external text" href="http://science.sciencemag.org/content/327/5962/177">Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry</a>, in Science, 8 gennaio 2010. URL consultato il 18 dicembre 2016.</cite> </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> Ad esempio, fra gli studi più recenti, Michele Emmer, Matematica e cultura, Springer, 2001 - <a href="/wiki/Speciale:RicercaISBN/8847001412" class="internal mw-magiclink-isbn">ISBN 8847001412</a>, oppure Ian Bent, William Drabkin, Analisi musicale, EDT srl Editore, 1990 - <a href="/wiki/Speciale:RicercaISBN/8870630730" class="internal mw-magiclink-isbn">ISBN 8870630730</a>. </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <cite class="citation pubblicazione" style="font-style:normal"> Canal 104 Plus, <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=JLTq_KpjnrQ">Sequenza in musica Fibonacci, una teoria originale</a>, 29 gennaio 2018. URL consultato il 29 novembre 2024.</cite> </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> <a href="/wiki/Mario_Livio" title="Mario Livio">Mario Livio</a>, La Sezione Aurea, Storia di un numero e di un mistero che dura da tremila anni - Bur, 2003, p. 280. <a href="/wiki/Speciale:RicercaISBN/9788817872010" class="internal mw-magiclink-isbn">ISBN 978-88-17-87201-0</a> </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> (<abbr title="inglese">EN</abbr>) Roy Howat, Debussy in proportion: a musical analysis, Cambridge University Press, 1986. <a href="/wiki/Speciale:RicercaISBN/9780521311458" class="internal mw-magiclink-isbn">ISBN 978-0-521-31145-8</a> </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <a href="/wiki/Mario_Livio" title="Mario Livio">Mario Livio</a>, La Sezione Aurea, Storia di un numero e di un mistero che dura da tremila anni - Bur, 2003, p. 276-279. <a href="/wiki/Speciale:RicercaISBN/9788817872010" class="internal mw-magiclink-isbn">ISBN 978-88-17-87201-0</a> </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> (<abbr title="inglese">EN</abbr>) Ernö Lendvai, Béla Bartók: an analysis of his music, Kahn & Averill, 1971. <a href="/wiki/Speciale:RicercaISBN/9780900707049" class="internal mw-magiclink-isbn">ISBN 9780900707049</a> </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> <cite class="citation testo" style="font-style:normal"> <a rel="nofollow" class="external text" href="http://www.sectioaurea.com/sectioaurea/S.A.&Musica.htm">Sectio Aurea</a>.</cite>: Sezione Aurea e Musica: Breve storia del "Numero d'Oro" da Dufay al «progressive-rock» dei Genesis., di Gaudenzio Temporelli </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> <cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="http://www.liceoberchet.it/ricerche/sezioneaurea/sez3.htm">Sezione Aurea in natura</a>, su liceoberchet.it. URL consultato il 1º maggio 2014.</cite> </li> <li id="cite_note-Pappas-19"><a href="#cite_ref-Pappas_19-0">^</a> Le gioie della matematica di Theoni Pappas, Franco Muzzio Editore. (<a href="/wiki/Speciale:RicercaISBN/8874131127" class="internal mw-magiclink-isbn">ISBN 88-7413-112-7</a>) </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> Di cui si cita la composizione Reflets dans l'eau, in L 110, Images, Set 1 per piano (1905): in questo brano la sequenza degli accordi è segnata dagli intervalli 34, 21, 13 e 8. Si veda in proposito Peter F. Smith, The Dynamics of Delight: Architecture and Aesthetics, Routledge, New York, 2003 - p. 83, <a href="/wiki/Speciale:RicercaISBN/041530010X" class="internal mw-magiclink-isbn">ISBN 0-415-30010-X</a> </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> <cite class="citation libro" style="font-style:normal"> Dan Brown, Il codice da Vinci.</cite> </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> <cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="http://www.atuttoportale.it/didattica/botanica/OltreBotanica/OltreBotanica-01-CorpoUmano.pdf">La sezione aurea nel corpo umano</a> (<abbr title="documento in formato PDF">PDF</abbr>), su atuttoportale.it.</cite> </li> <li id="cite_note-La_Nazione-23">^ <a href="#cite_ref-La_Nazione_23-0">a</a> <a href="#cite_ref-La_Nazione_23-1">b</a> <a href="#cite_ref-La_Nazione_23-2">c</a> <cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.lanazione.it/firenze/fibonacci-numero-dio-2ed45a7c">Fibonacci e la scoperta del Numero di Dio. Sapete che è in tutti noi?</a>, su La Nazione, 23 novembre 2024. URL consultato il 30 novembre 2024.</cite> </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> <cite class="citation web" style="font-style:normal">(<abbr title="inglese">EN</abbr>) T.C. Scott e P. Marketos, <a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/Publications/fibonacci.pdf">On the Origin of the Fibonacci Sequence</a> (<abbr title="documento in formato PDF">PDF</abbr>), su www-history.mcs.st-andrews.ac.uk, <a href="/wiki/MacTutor_History_of_Mathematics_archive" class="mw-redirect" title="MacTutor History of Mathematics archive">MacTutor History of Mathematics archive</a>, University of St Andrews, marzo 2014.</cite> </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> <cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://firenze.repubblica.it/cronaca/2015/09/18/news/fibonacci-123140907/">Un messaggio rimasto segreto per 800 anni: la scoperta sulla facciata di una chiesa</a>, su la Repubblica, 18 settembre 2015. URL consultato il 29 novembre 2024.</cite> </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> <cite class="citation testo" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080615144229/http://www.tusciaelecta.it/itamerz.htm">Tuscia Electa</a> (archiviato dall'<abbr title="http://www.tusciaelecta.it/itamerz.htm">url originale</abbr> il 15 giugno 2008).</cite> </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> <cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="http://www.fib30online.it/it/teoria-di-elliott.htm">Teoria di Elliott</a>, su Fib30Online, 2009.</cite> </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> <cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180128132447/https://www.atuttoportale.it/didattica/botanica/OltreBotanica/OltreBotanica-06-Informatica.pdf">La sezione aurea in informatica</a> (<abbr title="documento in formato PDF">PDF</abbr>), su atuttoportale.it. URL consultato il 15 novembre 2016 (archiviato dall'<abbr title="http://www.atuttoportale.it/didattica/botanica/OltreBotanica/OltreBotanica-06-Informatica.pdf">url originale</abbr> il 28 gennaio 2018).</cite> </li> <li id="cite_note-29"><a href="#cite_ref-29">^</a> <cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="http://www.electroportal.net/renzodf/wiki/articolo6">Resistenze e simmetrie nei Ladder Network</a>, su ElectroPortal, 2009.</cite> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=37" title="Modifica la sezione Bibliografia" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=37" title="Edit section's source code: Bibliografia">modifica wikitesto</a>]</div> <ul><li><cite class="citation libro" style="font-style:normal"> Marcel Danesi, Labirinti, quadrati magici e paradossi logici. I dieci più grandi, <a href="/wiki/Bari" title="Bari">Bari</a>, <a href="/wiki/Edizioni_Dedalo" title="Edizioni Dedalo">Dedalo</a>, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/88-220-6293-0" title="Speciale:RicercaISBN/88-220-6293-0">88-220-6293-0</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal"> Paolo Camagni, Algoritmi e basi della programmazione, <a href="/wiki/Milano" title="Milano">Milano</a>, <a href="/wiki/Hoepli_(casa_editrice)" title="Hoepli (casa editrice)">Hoepli</a>, 2003, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/88-203-3601-4" title="Speciale:RicercaISBN/88-203-3601-4">88-203-3601-4</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal"> Rob Eastaway, Probabilità, numeri e code. La matematica nascosta nella vita, Dedalo, 2003, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/88-220-6263-9" title="Speciale:RicercaISBN/88-220-6263-9">88-220-6263-9</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal"> Rita Laganà, Marco Righi e Francesco Romani, Informatica. Concetti e sperimentazioni, Milano, <a href="/wiki/Apogeo_(casa_editrice)" title="Apogeo (casa editrice)">Apogeo</a>, 2007, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/88-503-2493-6" title="Speciale:RicercaISBN/88-503-2493-6">88-503-2493-6</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal"> Adam Drozdek, Algoritmi e strutture dati in Java, Milano, Apogeo, 2001, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/88-7303-895-6" title="Speciale:RicercaISBN/88-7303-895-6">88-7303-895-6</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal"> Gianclaudio Floria e Andrea Terzaghi, Giocare e vincere con Excel, FAG, 2006, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/88-8233-529-1" title="Speciale:RicercaISBN/88-8233-529-1">88-8233-529-1</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal"> Daniele Marsero, Of game, UNI, 2006, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/88-88859-40-3" title="Speciale:RicercaISBN/88-88859-40-3">88-88859-40-3</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal"> Peter Higgins, Divertirsi con la matematica. Curiosità e stranezze del mondo dei numeri, Dedalo, 2001, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/88-220-6216-7" title="Speciale:RicercaISBN/88-220-6216-7">88-220-6216-7</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal"> Michael Schneider e Judith Gersting, Informatica, Apogeo, 2007, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/88-503-2383-2" title="Speciale:RicercaISBN/88-503-2383-2">88-503-2383-2</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal"> Joseph Mayo, C#, Apogeo, 2002, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/88-503-2011-6" title="Speciale:RicercaISBN/88-503-2011-6">88-503-2011-6</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal">(<abbr title="inglese">EN</abbr>) Thomas Koshy, <a rel="nofollow" class="external text" href="https://archive.org/details/fibonaccilucasnu0000kosh">Fibonacci and Lucas Numbers with Applications</a>, <a href="/wiki/New_York" title="New York">New York</a>, <a href="/wiki/John_Wiley_%26_Sons" title="John Wiley & Sons">Wiley</a>, 2001, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/0-471-39969-8" title="Speciale:RicercaISBN/0-471-39969-8">0-471-39969-8</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal"> Nikolay Vorobyov, I numeri di Fibonacci, Milano, <a href="/wiki/Progresso_tecnico_editoriale" title="Progresso tecnico editoriale">Progresso tecnico editoriale</a>, 1964.</cite></li> <li><cite class="citation libro" style="font-style:normal">(<abbr title="inglese">EN</abbr>) Keith Devlin, The Man of Numbers: Fibonacci's Arithmetic Revolution, Walker Publishing Co, 2011, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-0-8027-7812-3" title="Speciale:RicercaISBN/978-0-8027-7812-3">978-0-8027-7812-3</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal">(<abbr title="inglese">EN</abbr>) Leland Wilkinson, The Grammar of Graphics, <a href="/wiki/Berlino" title="Berlino">Berlino</a>, <a href="/wiki/Springer_(azienda)" title="Springer (azienda)">Springer</a>, 2005, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/0-387-24544-8" title="Speciale:RicercaISBN/0-387-24544-8">0-387-24544-8</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal"> <a href="/wiki/Mario_Livio" title="Mario Livio">Mario Livio</a>, La Sezione Aurea, Storia di un numero e di un mistero che dura da tremila anni, Milano, <a href="/wiki/Biblioteca_Universale_Rizzoli" title="Biblioteca Universale Rizzoli">Bur</a>, 2003, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-88-17-87201-0" title="Speciale:RicercaISBN/978-88-17-87201-0">978-88-17-87201-0</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal"> <a href="/wiki/Clifford_A._Pickover" title="Clifford A. Pickover">Clifford A. Pickover</a>, La magia dei numeri - Sfide Matematiche, <a href="/wiki/Barcellona" title="Barcellona">Barcellona</a>, RBA.</cite></li> <li><cite class="citation libro" style="font-style:normal"> Alfred Posamentier e Ingmar Lehmann, I (favolosi) numeri di Fibonacci, <a href="/wiki/Padova" title="Padova">Padova</a>, <a href="/wiki/Franco_Muzzio_%26_C._Editore" title="Franco Muzzio & C. Editore">Gruppo editoriale Muzzio</a>, 2010, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-88-96159-24-8" title="Speciale:RicercaISBN/978-88-96159-24-8">978-88-96159-24-8</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal">(<abbr title="inglese">EN</abbr>) Hrant Babkeni Arakelyan, Mathematics and History of the Golden Section, Logos, 2014, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-5-98704-663-0" title="Speciale:RicercaISBN/978-5-98704-663-0">978-5-98704-663-0</a>.</cite> p. 404</li> <li><cite class="citation web" style="font-style:normal">(<abbr title="inglese">EN</abbr>) <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110715225419/http://www.newscientist.com/article/mg18725180.400.html">Not so Fibonacci</a>, in New Scientist, n. 251, 24 settembre 2005, p. 24 (archiviato dall'<abbr title="https://www.newscientist.com/article/mg18725180.400.html">url originale</abbr> il 15 luglio 2011).</cite></li></ul> <div class="mw-heading mw-heading2"><h2 id="Voci_correlate">Voci correlate</h2>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=38" title="Modifica la sezione Voci correlate" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=38" title="Edit section's source code: Voci correlate">modifica wikitesto</a>]</div> <ul><li><a href="/wiki/Leonardo_Fibonacci" title="Leonardo Fibonacci">Leonardo Fibonacci</a></li> <li><a href="/wiki/Successione_Tribonacci" title="Successione Tribonacci">Successione Tribonacci</a></li> <li><a href="/wiki/Successione_Tetranacci" title="Successione Tetranacci">Successione Tetranacci</a></li> <li><a href="/wiki/Triangolo_di_Tartaglia" title="Triangolo di Tartaglia">Triangolo di Tartaglia</a></li> <li><a href="/wiki/Sezione_aurea" title="Sezione aurea">Sezione aurea</a></li> <li><a href="/wiki/Polinomi_di_Fibonacci" title="Polinomi di Fibonacci">Polinomi di Fibonacci</a></li> <li><a href="/wiki/Teorema_di_Zeckendorf" title="Teorema di Zeckendorf">Teorema di Zeckendorf</a></li> <li><a href="/wiki/Generatore_di_Fibonacci_ritardato" title="Generatore di Fibonacci ritardato">Generatore di Fibonacci ritardato</a></li> <li><a href="/wiki/Costante_di_Viswanath" title="Costante di Viswanath">Costante di Viswanath</a></li> <li><a href="/wiki/Fillotassi" title="Fillotassi">Fillotassi</a></li> <li><a href="/wiki/Teorema_di_Matijasevi%C4%8D" title="Teorema di Matijasevič">Teorema di Matijasevič</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Altri_progetti">Altri progetti</h2>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=39" title="Modifica la sezione Altri progetti" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=39" title="Edit section's source code: Altri progetti">modifica wikitesto</a>]</div> <div id="interProject" class="toccolours" style="display: none; clear: both; margin-top: 2em">Altri progetti<ul title="Collegamenti verso gli altri progetti Wikimedia"> <li class="" title=""><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Fibonacci_numbers?uselang=it">Wikimedia Commons</a></li></ul></div> <ul><li><a href="https://commons.wikimedia.org/wiki/?uselang=it" title="Collabora a Wikimedia Commons"><img alt="Collabora a Wikimedia Commons" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png" decoding="async" width="18" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/27px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/36px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a> <a class="external text" href="https://commons.wikimedia.org/wiki/?uselang=it">Wikimedia Commons</a> contiene immagini o altri file sulla <a class="external text" href="https://commons.wikimedia.org/wiki/Category:Fibonacci_numbers?uselang=it">successione di Fibonacci</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Collegamenti_esterni">Collegamenti esterni</h2>[<a href="/w/index.php?title=Successione_di_Fibonacci&veaction=edit&section=40" title="Modifica la sezione Collegamenti esterni" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Successione_di_Fibonacci&action=edit&section=40" title="Edit section's source code: Collegamenti esterni">modifica wikitesto</a>]</div> <ul><li class="mw-empty-elt"></li> <li>(<abbr title="inglese">EN</abbr>) "<cite class="citation testo" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100109045556/http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm">Fibonacci Flim-Flam</a>. URL consultato il 27 dicembre 2022 (archiviato dall'<abbr title="http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm">url originale</abbr> il 9 gennaio 2010).</cite>", by Donald E. Simanek</li> <li><cite class="citation web" style="font-style:normal">(<abbr title="inglese">EN</abbr>) <a rel="nofollow" class="external text" href="https://web.archive.org/web/20061205091146/http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html">The Golden Section: Phi</a>, su mcs.surrey.ac.uk. URL consultato il 4 novembre 2004 (archiviato dall'<abbr title="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html">url originale</abbr> il 5 dicembre 2006).</cite></li> <li><cite class="citation web" style="font-style:normal">(<abbr title="inglese">EN</abbr>) <a rel="nofollow" class="external text" href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/Fibonomials.html">Fibonomial and Factorial</a>, su maths.surrey.ac.uk.</cite></li> <li><cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20081024104016/http://www.marianospadaccini.it/progetti.php">Progetto che permette il calcolo numerico di valori arbitrariamente grandi della successione (implementazione ed esempi)</a>, su marianospadaccini.it. URL consultato il 10 ottobre 2008 (archiviato dall'<abbr title="http://www.marianospadaccini.it/progetti.php">url originale</abbr> il 24 ottobre 2008).</cite></li> <li><cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.sosmatematica.it/contenuti/la-formula-di-binet-per-i-numeri-di-fibonacci-una-dimostrazione-con-lalgebra-lineare/">Dimostrazione della formula di Binet</a>, su sosmatematica.it.</cite></li> <li><cite class="citation web" style="font-style:normal">(<abbr title="inglese">EN</abbr>) <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100621044909/http://planetmath.org/encyclopedia/KeithNumber.html">Keith's site</a>, su planetmath.org. URL consultato l'11 aprile 2010 (archiviato dall'<abbr title="http://planetmath.org/encyclopedia/KeithNumber.html">url originale</abbr> il 21 giugno 2010).</cite></li></ul> <style data-mw-deduplicate="TemplateStyles:r141815314">.mw-parser-output .navbox{border:1px solid #aaa;clear:both;margin:auto;padding:2px;width:100%}.mw-parser-output .navbox th{padding-left:1em;padding-right:1em;text-align:center}.mw-parser-output .navbox>tbody>tr:first-child>th{background:#ccf;font-size:90%;width:100%;color:var(--color-base,black)}.mw-parser-output .navbox_navbar{float:left;margin:0;padding:0 10px 0 0;text-align:left;width:6em}.mw-parser-output .navbox_title{font-size:110%}.mw-parser-output .navbox_abovebelow{background:#ddf;font-size:90%;font-weight:normal}.mw-parser-output .navbox_group{background:#ddf;font-size:90%;padding:0 10px;white-space:nowrap}.mw-parser-output .navbox_list{font-size:90%;width:100%}.mw-parser-output .navbox_list a{white-space:nowrap}html:not(.vector-feature-night-mode-enabled) .mw-parser-output .navbox_odd{background:#fdfdfd;color:var(--color-base,black)}html:not(.vector-feature-night-mode-enabled) .mw-parser-output .navbox_even{background:#f7f7f7;color:var(--color-base,black)}.mw-parser-output .navbox a.mw-selflink{color:var(--color-base,black)}.mw-parser-output .navbox_center{text-align:center}.mw-parser-output .navbox .navbox_image{padding-left:7px;vertical-align:middle;width:0}.mw-parser-output .navbox+.navbox{margin-top:-1px}.mw-parser-output .navbox .mw-collapsible-toggle{font-weight:normal;text-align:right;width:7em}body.skin--responsive .mw-parser-output .navbox_image img{max-width:none!important}.mw-parser-output .subnavbox{margin:-3px;width:100%}.mw-parser-output .subnavbox_group{background:#e6e6ff;padding:0 10px}@media screen{html.skin-theme-clientpref-night .mw-parser-output .navbox>tbody>tr:first-child>th{background:var(--background-color-interactive)!important}html.skin-theme-clientpref-night .mw-parser-output .navbox th{color:var(--color-base)!important}html.skin-theme-clientpref-night .mw-parser-output .navbox_abovebelow,html.skin-theme-clientpref-night .mw-parser-output .navbox_group{background:var(--background-color-interactive-subtle)!important}html.skin-theme-clientpref-night .mw-parser-output .subnavbox_group{background:var(--background-color-neutral-subtle)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbox>tbody>tr:first-child>th{background:var(--background-color-interactive)!important}html.skin-theme-clientpref-os .mw-parser-output .navbox th{color:var(--color-base)!important}html.skin-theme-clientpref-os .mw-parser-output .navbox_abovebelow,html.skin-theme-clientpref-os .mw-parser-output .navbox_group{background:var(--background-color-interactive-subtle)!important}html.skin-theme-clientpref-os .mw-parser-output .subnavbox_group{background:var(--background-color-neutral-subtle)!important}}</style><table class="navbox mw-collapsible mw-collapsed noprint metadata" id="navbox-Teoria_dei_numeri"><tbody><tr><th colspan="3" style="background:#ffc0cb;"><div class="navbox_navbar"><div class="noprint plainlinks" style="background-color:transparent; padding:0; font-size:xx-small; color:var(--color-base, #000000); white-space:nowrap;"><a href="/wiki/Template:Teoria_dei_numeri" title="Template:Teoria dei numeri">V</a> · <a href="/w/index.php?title=Discussioni_template:Teoria_dei_numeri&action=edit&redlink=1" class="new" title="Discussioni template:Teoria dei numeri (la pagina non esiste)">D</a> · <a class="external text" href="https://it.wikipedia.org/w/index.php?title=Template:Teoria_dei_numeri&action=edit">M</a></div></div><a href="/wiki/Teoria_dei_numeri" title="Teoria dei numeri">Teoria dei numeri</a></th></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;"><a href="/wiki/Numero" title="Numero">Numeri</a> più usati</th><td colspan="1" class="navbox_list navbox_odd" style="text-align:left;"><a href="/wiki/Numero_naturale" title="Numero naturale">Naturali</a> · <a href="/wiki/Numero_intero" title="Numero intero">Interi</a> · <a href="/wiki/Numeri_pari_e_dispari" title="Numeri pari e dispari">Pari e dispari</a></td><td rowspan="10" class="navbox_image"><figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics-p.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/58px-Nuvola_apps_edu_mathematics-p.svg.png" decoding="async" width="58" height="58" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/87px-Nuvola_apps_edu_mathematics-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/116px-Nuvola_apps_edu_mathematics-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a><figcaption></figcaption></figure></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;">Principi generali</th><td colspan="1" class="navbox_list navbox_even" style="text-align:left;"><a href="/wiki/Principio_d%27induzione" title="Principio d'induzione">Principio d'induzione</a> · <a href="/wiki/Principio_del_buon_ordinamento" title="Principio del buon ordinamento">Principio del buon ordinamento</a> · <a href="/wiki/Relazione_di_equivalenza" title="Relazione di equivalenza">Relazione di equivalenza</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;"><a href="/wiki/Successione_di_interi" title="Successione di interi">Successioni di interi</a></th><td colspan="1" class="navbox_list navbox_odd" style="text-align:left;"><a href="/wiki/Fattoriale" title="Fattoriale">Fattoriale</a> · <a class="mw-selflink selflink">Successione di Fibonacci</a> · <a href="/wiki/Numero_di_Catalan" title="Numero di Catalan">Numero di Catalan</a> · <a href="/wiki/Numero_di_Perrin" title="Numero di Perrin">Numero di Perrin</a> · <a href="/wiki/Numero_di_Eulero_(teoria_dei_numeri)" class="mw-redirect" title="Numero di Eulero (teoria dei numeri)">Numero di Eulero</a> · <a href="/wiki/Successione_di_Mian-Chowla" title="Successione di Mian-Chowla">Successione di Mian-Chowla</a> · <a href="/wiki/Successione_di_Thue-Morse" title="Successione di Thue-Morse">Successione di Thue-Morse</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;">Caratteristiche dei numeri primi</th><td colspan="1" class="navbox_list navbox_even" style="text-align:left;"><a href="/wiki/Numero_primo" title="Numero primo">Numero primo</a> · <a href="/wiki/Lemma_di_Euclide" title="Lemma di Euclide">Lemma di Euclide</a> · <a href="/wiki/Teorema_dell%27infinit%C3%A0_dei_numeri_primi" title="Teorema dell'infinità dei numeri primi">Teorema dell'infinità dei numeri primi</a> · <a href="/wiki/Crivello_di_Eratostene" title="Crivello di Eratostene">Crivello di Eratostene</a> · <a href="/wiki/Test_di_primalit%C3%A0" title="Test di primalità">Test di primalità</a> · <a href="/wiki/Teorema_fondamentale_dell%27aritmetica" title="Teorema fondamentale dell'aritmetica">Teorema fondamentale dell'aritmetica</a> · <a href="/wiki/Interi_coprimi" title="Interi coprimi">Interi coprimi</a> · <a href="/wiki/Identit%C3%A0_di_B%C3%A9zout" title="Identità di Bézout">Identità di Bézout</a> · <a href="/wiki/Massimo_comun_divisore" title="Massimo comun divisore">MCD</a> · <a href="/wiki/Minimo_comune_multiplo" title="Minimo comune multiplo">mcm</a> · <a href="/wiki/Algoritmo_di_Euclide" title="Algoritmo di Euclide">Algoritmo di Euclide</a> · <a href="/wiki/Algoritmo_esteso_di_Euclide" title="Algoritmo esteso di Euclide">Algoritmo esteso di Euclide</a> · <a href="/wiki/Teorema_dei_numeri_primi" title="Teorema dei numeri primi">Teorema dei numeri primi</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;"><a href="/wiki/Funzione_aritmetica" title="Funzione aritmetica">Funzioni aritmetiche</a></th><td colspan="1" class="navbox_list navbox_odd" style="text-align:left;"><a href="/wiki/Funzione_moltiplicativa" title="Funzione moltiplicativa">Funzione moltiplicativa</a> · <a href="/wiki/Funzione_additiva" title="Funzione additiva">Funzione additiva</a> · <a href="/wiki/Convoluzione_di_Dirichlet" title="Convoluzione di Dirichlet">Convoluzione di Dirichlet</a> · <a href="/wiki/Funzione_%CF%86_di_Eulero" title="Funzione φ di Eulero">Funzione φ di Eulero</a> · <a href="/wiki/Funzione_di_M%C3%B6bius" title="Funzione di Möbius">Funzione di Möbius</a> · <a href="/wiki/Funzione_tau_sui_positivi" title="Funzione tau sui positivi">Funzione tau sui positivi</a> · <a href="/wiki/Funzione_sigma" title="Funzione sigma">Funzione sigma</a> · <a href="/wiki/Funzione_di_Liouville" title="Funzione di Liouville">Funzione di Liouville</a> · <a href="/wiki/Funzione_di_Mertens" title="Funzione di Mertens">Funzione di Mertens</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;"><a href="/wiki/Aritmetica_modulare" title="Aritmetica modulare">Aritmetica modulare</a></th><td colspan="1" class="navbox_list navbox_even" style="text-align:left;"><a href="/wiki/Teorema_cinese_del_resto" title="Teorema cinese del resto">Teorema cinese del resto</a> · <a href="/wiki/Piccolo_teorema_di_Fermat" title="Piccolo teorema di Fermat">Piccolo teorema di Fermat</a> · <a href="/wiki/Teorema_di_Eulero_(aritmetica_modulare)" title="Teorema di Eulero (aritmetica modulare)">Teorema di Eulero</a> · <a href="/wiki/Criteri_di_divisibilit%C3%A0" title="Criteri di divisibilità">Criteri di divisibilità</a> · <a href="/wiki/Teorema_di_Fermat_sulle_somme_di_due_quadrati" title="Teorema di Fermat sulle somme di due quadrati">Teorema di Fermat sulle somme di due quadrati</a> · <a href="/wiki/Teorema_di_Wilson" title="Teorema di Wilson">Teorema di Wilson</a> · <a href="/wiki/Reciprocit%C3%A0_quadratica" title="Reciprocità quadratica">Legge di reciprocità quadratica</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;"><a href="/wiki/Congettura" title="Congettura">Congetture</a></th><td colspan="1" class="navbox_list navbox_odd" style="text-align:left;"><a href="/wiki/Congettura_di_Goldbach" title="Congettura di Goldbach">Congettura di Goldbach</a> · <a href="/wiki/Congettura_di_Polignac" title="Congettura di Polignac">Congettura di Polignac</a> · <a href="/wiki/Congettura_abc" title="Congettura abc">Congettura abc</a> · <a href="/wiki/Congettura_dei_numeri_primi_gemelli" title="Congettura dei numeri primi gemelli">Congettura dei numeri primi gemelli</a> · <a href="/wiki/Congettura_di_Legendre" title="Congettura di Legendre">Congettura di Legendre</a> · <a href="/wiki/Nuova_congettura_di_Mersenne" title="Nuova congettura di Mersenne">Nuova congettura di Mersenne</a> · <a href="/wiki/Congettura_di_Collatz" title="Congettura di Collatz">Congettura di Collatz</a> · <a href="/wiki/Ipotesi_di_Riemann" title="Ipotesi di Riemann">Ipotesi di Riemann</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;">Altro</th><td colspan="1" class="navbox_list navbox_even" style="text-align:left;"><a href="/wiki/Problema_di_Waring" title="Problema di Waring">Problema di Waring</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;">Principali teorici</th><td colspan="1" class="navbox_list navbox_odd" style="text-align:left;"><a href="/wiki/Leonardo_Fibonacci" title="Leonardo Fibonacci">Fibonacci</a> · <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Fermat</a> · <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a> · <a href="/wiki/Eulero" title="Eulero">Eulero</a> · <a href="/wiki/Adrien-Marie_Legendre" title="Adrien-Marie Legendre">Legendre</a> · <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Riemann</a> · <a href="/wiki/Peter_Gustav_Lejeune_Dirichlet" title="Peter Gustav Lejeune Dirichlet">Dirichlet</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;">Discipline connesse</th><td colspan="1" class="navbox_list navbox_even" style="text-align:left;"><a href="/wiki/Teoria_algebrica_dei_numeri" title="Teoria algebrica dei numeri">Teoria algebrica dei numeri</a> · <a href="/wiki/Teoria_analitica_dei_numeri" title="Teoria analitica dei numeri">Teoria analitica dei numeri</a> · <a href="/wiki/Crittografia" title="Crittografia">Crittografia</a> · <a href="/wiki/Teoria_computazionale_dei_numeri" title="Teoria computazionale dei numeri">Teoria computazionale dei numeri</a></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r141815314"><table class="navbox mw-collapsible autocollapse noprint metadata" id="navbox-Serie_(matematica)"><tbody><tr><th colspan="2"><div class="navbox_navbar"><div class="noprint plainlinks" style="background-color:transparent; padding:0; font-size:xx-small; color:var(--color-base, #000000); white-space:nowrap;"><a href="/wiki/Template:Serie_(matematica)" title="Template:Serie (matematica)">V</a> · <a href="/w/index.php?title=Discussioni_template:Serie_(matematica)&action=edit&redlink=1" class="new" title="Discussioni template:Serie (matematica) (la pagina non esiste)">D</a> · <a class="external text" href="https://it.wikipedia.org/w/index.php?title=Template:Serie_(matematica)&action=edit">M</a></div></div><a href="/wiki/Successione_(matematica)" title="Successione (matematica)">Successioni</a> e <a href="/wiki/Serie_(matematica)" title="Serie (matematica)">Serie</a></th></tr><tr><th colspan="1" class="navbox_group"><a href="/wiki/Successione_di_interi" title="Successione di interi">Successioni di interi</a></th><td colspan="1" class="navbox_list navbox_odd"><table class="subnavbox" style="font-size:12px;"><tbody><tr><th class="subnavbox_group" style="text-align:left; width:6em;">Base</th><td colspan="1"><a href="/wiki/Progressione_aritmetica" title="Progressione aritmetica">Progressione aritmetica</a> · <a href="/wiki/Progressione_geometrica" title="Progressione geometrica">Progressione geometrica</a> · <a href="/w/index.php?title=Progressione_armonica_(matematica)&action=edit&redlink=1" class="new" title="Progressione armonica (matematica) (la pagina non esiste)">Progressione armonica</a> · <a href="/wiki/Progressione_aritmetica" title="Progressione aritmetica">Progressione aritmetica</a> · <a href="/wiki/Quadrato_perfetto" title="Quadrato perfetto">Numeri quadrati</a> · <a href="/wiki/Cubo_perfetto" title="Cubo perfetto">Numeri cubici</a> · <a href="/wiki/Fattoriale" title="Fattoriale">Fattoriale</a> · <a href="/wiki/Potenza_di_due" title="Potenza di due">Potenze di 2</a> · <a href="/w/index.php?title=Potenza_di_tre&action=edit&redlink=1" class="new" title="Potenza di tre (la pagina non esiste)">Potenze di 3</a> · <a href="/w/index.php?title=Potenza_di_dieci&action=edit&redlink=1" class="new" title="Potenza di dieci (la pagina non esiste)">Potenze di 10</a></td></tr><tr><th class="subnavbox_group" style="text-align:left; width:6em;">Avanzate</th><td colspan="1"><a href="/wiki/Successione_completa" title="Successione completa">Successione completa</a> · <a class="mw-selflink selflink">Numeri di Fibonacci</a> · <a href="/wiki/Numero_figurato" title="Numero figurato">Numeri figurati</a> · <a href="/wiki/Numero_ettagonale" title="Numero ettagonale">Numeri ettagonali</a> · <a href="/wiki/Numero_esagonale" title="Numero esagonale">Numeri esagonali</a> · <a href="/wiki/Successione_di_Lucas" title="Successione di Lucas">Numeri di Lucas</a> · <a href="/w/index.php?title=Numero_di_Pell&action=edit&redlink=1" class="new" title="Numero di Pell (la pagina non esiste)">Numeri di Pell</a> · <a href="/wiki/Numero_pentagonale" title="Numero pentagonale">Numeri pentagonali</a> · <a href="/wiki/Numero_poligonale" title="Numero poligonale">Numero poligonale</a> · <a href="/wiki/Numero_triangolare" title="Numero triangolare">Numero triangolare</a></td></tr></tbody></table></td></tr><tr><th colspan="1" class="navbox_group">Proprietà delle successioni</th><td colspan="1" class="navbox_list navbox_even"><a href="/wiki/Successione_di_Cauchy" title="Successione di Cauchy">Successione di Cauchy</a> · <a href="/wiki/Funzione_monotona" title="Funzione monotona">Successione monotona</a> · <a href="/w/index.php?title=Successione_alternata&action=edit&redlink=1" class="new" title="Successione alternata (la pagina non esiste)">Successione alternata</a></td></tr><tr><th colspan="1" class="navbox_group">Proprietà delle serie</th><td colspan="1" class="navbox_list navbox_odd"><table class="subnavbox"><tbody><tr><th class="subnavbox_group" style="text-align:left; width:6em;">Tendenza</th><td colspan="1"><a href="/wiki/Serie_alternata" title="Serie alternata">Serie alternata</a> · <a href="/wiki/Serie_convergente" title="Serie convergente">Serie convergente</a> · <a href="/wiki/Serie_divergente" title="Serie divergente">Serie divergente</a> · <a href="/wiki/Serie_telescopica" title="Serie telescopica">Serie telescopica</a></td></tr><tr><th class="subnavbox_group" style="text-align:left; width:6em;">Convergenza</th><td colspan="1"><a href="/wiki/Serie_(matematica)#Convergenza_incondizionata" title="Serie (matematica)">Convergenza incondizionata</a> · <a href="/wiki/Serie_(matematica)#Convergenza_assoluta" title="Serie (matematica)">Convergenza assoluta</a> · <a href="/wiki/Convergenza_uniforme" class="mw-redirect" title="Convergenza uniforme">Convergenza uniforme</a></td></tr></tbody></table></td></tr><tr><th colspan="1" class="navbox_group">Serie esplicite</th><td colspan="1" class="navbox_list navbox_even"><table class="subnavbox"><tbody><tr><th class="subnavbox_group" style="text-align:lef; twidth:6em;">Convergenti</th><td colspan="1"><a href="/w/index.php?title=1/2_%E2%88%92_1/4_%2B_1/8_%E2%88%92_1/16_%2B_%E2%8B%AF&action=edit&redlink=1" class="new" title="1/2 − 1/4 + 1/8 − 1/16 + ⋯ (la pagina non esiste)">1/2 − 1/4 + 1/8 − 1/16 + ⋯</a> · <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> · <a href="/w/index.php?title=1/4_%2B_1/16_%2B_1/64_%2B_1/256_%2B_%E2%8B%AF&action=edit&redlink=1" class="new" title="1/4 + 1/16 + 1/64 + 1/256 + ⋯ (la pagina non esiste)">1/4 + 1/16 + 1/64 + 1/256 + ⋯</a> · <a href="/wiki/Funzione_zeta_di_Riemann" title="Funzione zeta di Riemann">1 + 1/2s+ 1/3s + ... (Riemann zeta)</a></td></tr><tr><th class="subnavbox_group" style="text-align:lef; twidth:6em;">Divergenti</th><td colspan="1"><a href="/wiki/Serie_sommativa_unitaria" title="Serie sommativa unitaria">1 + 1 + 1 + 1 + ⋯</a> · <a href="/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%C2%B7_%C2%B7_%C2%B7" title="1 + 2 + 3 + 4 + · · ·">1 + 2 + 3 + 4 + ⋯</a> · <a href="/wiki/1_%2B_2_%2B_4_%2B_8_%2B_..." title="1 + 2 + 4 + 8 + ...">1 + 2 + 4 + 8 + ⋯</a> · <a href="/wiki/Serie_di_Grandi" title="Serie di Grandi">1 − 1 + 1 − 1 + ⋯ (Grandi)</a> · <a href="/wiki/Progressione_aritmetica" title="Progressione aritmetica">a1 + (a1+d) + ⋯</a> · <a href="/wiki/1_%E2%88%92_2_%2B_3_%E2%88%92_4_%2B_%C2%B7_%C2%B7_%C2%B7" title="1 − 2 + 3 − 4 + · · ·">1 − 2 + 3 − 4 + ⋯</a> · <a href="/wiki/1_%E2%88%92_2_%2B_4_%E2%88%92_8_%2B_%C2%B7_%C2%B7_%C2%B7" title="1 − 2 + 4 − 8 + · · ·">1 − 2 + 4 − 8 + ⋯</a> · <a href="/wiki/Serie_armonica" title="Serie armonica">1 + 1/2 + 1/3 + 1/4 + ⋯ (armonica)</a> · <a href="/wiki/1_%E2%88%92_1_%2B_2_%E2%88%92_6_%2B_24_%E2%88%92_120_%2B_..." title="1 − 1 + 2 − 6 + 24 − 120 + ...">1 − 1 + 2 − 6 + 24 − 120 + ⋯ (fattoriale alternata)</a> · <a href="/wiki/Dimostrazione_della_divergenza_della_serie_dei_reciproci_dei_primi" title="Dimostrazione della divergenza della serie dei reciproci dei primi">1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inversi dei primi)</a></td></tr></tbody></table></td></tr><tr><th colspan="1" class="navbox_group">Tipi di serie</th><td colspan="1" class="navbox_list navbox_odd"><a href="/wiki/Serie_di_Taylor" title="Serie di Taylor">Serie di Taylor</a> · <a href="/wiki/Serie_di_potenze" title="Serie di potenze">Serie di potenze</a> · <a href="/wiki/Serie_formale_di_potenze" title="Serie formale di potenze">Serie formale di potenze</a> · <a href="/wiki/Serie_di_Laurent" title="Serie di Laurent">Serie di Laurent</a> · <a href="/w/index.php?title=Serie_di_Puiseux&action=edit&redlink=1" class="new" title="Serie di Puiseux (la pagina non esiste)">Serie di Puiseux</a> · <a href="/wiki/Serie_di_Dirichlet" title="Serie di Dirichlet">Serie di Dirichlet</a> · <a href="/wiki/Serie_trigonometrica" class="mw-redirect" title="Serie trigonometrica">Serie trigonometrica</a> · <a href="/wiki/Serie_di_Fourier" title="Serie di Fourier">Serie di Fourier</a> · <a href="/wiki/Funzione_generatrice" title="Funzione generatrice">Serie generatrice</a></td></tr><tr><th colspan="1" class="navbox_group"><a href="/wiki/Serie_ipergeometrica" title="Serie ipergeometrica">Serie Ipergeometrica</a></th><td colspan="1" class="navbox_list navbox_even"><a href="/w/index.php?title=Funzione_ipergeometrica_generalizzata&action=edit&redlink=1" class="new" title="Funzione ipergeometrica generalizzata (la pagina non esiste)">Serie ipergeometrica generalizzata</a> · <a href="/w/index.php?title=Funzione_ipergeometrica_di_un_argomento_di_matrice&action=edit&redlink=1" class="new" title="Funzione ipergeometrica di un argomento di matrice (la pagina non esiste)">Funzione ipergeometrica di un argomento di matrice</a> · <a href="/wiki/Funzioni_di_Lauricella" title="Funzioni di Lauricella">Serie di Lauricella</a> · <a href="/w/index.php?title=Serie_ipergeometrica_ellittica&action=edit&redlink=1" class="new" title="Serie ipergeometrica ellittica (la pagina non esiste)">Serie modulare</a> · <a href="/wiki/Equazione_di_Papperitz-Riemann" title="Equazione di Papperitz-Riemann">Serie ipergeometrica confluente</a> · <a href="/w/index.php?title=Serie_ipergeometrica_ellittica&action=edit&redlink=1" class="new" title="Serie ipergeometrica ellittica (la pagina non esiste)">Serie theta</a></td></tr><tr><th colspan="2" class="navbox_abovebelow"><a href="/wiki/Categoria:Serie_matematiche" title="Categoria:Serie matematiche"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Arrows-folder-categorize.svg/20px-Arrows-folder-categorize.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Arrows-folder-categorize.svg/30px-Arrows-folder-categorize.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/79/Arrows-folder-categorize.svg/40px-Arrows-folder-categorize.svg.png 2x" data-file-width="200" data-file-height="200" /></a></th></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r140554510">.mw-parser-output .CdA{border:1px solid #aaa;width:100%;margin:auto;font-size:90%;padding:2px}.mw-parser-output .CdA th{background-color:#f2f2f2;font-weight:bold;width:20%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .CdA{border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .CdA th{background-color:#202122}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .CdA{border-color:#54595D}html.skin-theme-clientpref-os .mw-parser-output .CdA th{background-color:#202122}}</style><table class="CdA"><tbody><tr><th><a href="/wiki/Aiuto:Controllo_di_autorit%C3%A0" title="Aiuto:Controllo di autorità">Controllo di autorità</a></th><td><a href="/wiki/Nuovo_soggettario" title="Nuovo soggettario">Thesaurus BNCF</a> <a rel="nofollow" class="external text" href="https://thes.bncf.firenze.sbn.it/termine.php?id=47110">47110</a> · <a href="/wiki/Library_of_Congress_Control_Number" title="Library of Congress Control Number">LCCN</a> (<abbr title="inglese">EN</abbr>) <a rel="nofollow" class="external text" href="http://id.loc.gov/authorities/subjects/sh85048028">sh85048028</a> · <a href="/wiki/Biblioteca_nazionale_di_Francia" title="Biblioteca nazionale di Francia">BNF</a> (<abbr title="francese">FR</abbr>) <a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb122868243">cb122868243</a> <a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb122868243">(data)</a> · <a href="/wiki/Biblioteca_nazionale_di_Israele" title="Biblioteca nazionale di Israele">J9U</a> (<abbr title="inglese">EN</abbr>, <abbr title="ebraico">HE</abbr>) <a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007531248505171">987007531248505171</a></td></tr></tbody></table> <div class="noprint" style="width:100%; padding: 3px 0; display: flex; flex-wrap: wrap; row-gap: 4px; column-gap: 8px; box-sizing: border-box;"><div style="flex-grow: 1"><style data-mw-deduplicate="TemplateStyles:r140555418">.mw-parser-output .itwiki-template-occhiello{width:100%;line-height:25px;border:1px solid #CCF;background-color:#F0EEFF;box-sizing:border-box}.mw-parser-output .itwiki-template-occhiello-progetto{background-color:#FAFAFA}@media screen{html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}</style><div class="itwiki-template-occhiello"><a href="/wiki/File:Crystal128-kmplot.svg" class="mw-file-description" title="Matematica"><img alt=" " src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/25px-Crystal128-kmplot.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/38px-Crystal128-kmplot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/50px-Crystal128-kmplot.svg.png 2x" data-file-width="245" data-file-height="244" /></a> <a href="/wiki/Portale:Matematica" title="Portale:Matematica">Portale Matematica</a>: accedi alle voci di Wikipedia che trattano di matematica</div></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Estratto da "<a dir="ltr" href="https://it.wikipedia.org/w/index.php?title=Successione_di_Fibonacci&oldid=142399047">https://it.wikipedia.org/w/index.php?title=Successione_di_Fibonacci&oldid=142399047</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Categoria:Categorie" title="Categoria:Categorie">Categorie</a>: <ul><li><a href="/wiki/Categoria:Combinatoria" title="Categoria:Combinatoria">Combinatoria</a></li><li><a href="/wiki/Categoria:Successioni_di_interi" title="Categoria:Successioni di interi">Successioni di interi</a></li><li><a href="/wiki/Categoria:Leonardo_Fibonacci" title="Categoria:Leonardo Fibonacci">Leonardo Fibonacci</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categorie nascoste: <ul><li><a href="/wiki/Categoria:Pagine_che_utilizzano_collegamenti_magici_ISBN" title="Categoria:Pagine che utilizzano collegamenti magici ISBN">Pagine che utilizzano collegamenti magici ISBN</a></li><li><a href="/wiki/Categoria:Controllare_-_matematica" title="Categoria:Controllare - matematica">Controllare - matematica</a></li><li><a href="/wiki/Categoria:Controllare_-_aprile_2010" title="Categoria:Controllare - aprile 2010">Controllare - aprile 2010</a></li><li><a href="/wiki/Categoria:Informazioni_senza_fonte" title="Categoria:Informazioni senza fonte">Informazioni senza fonte</a></li><li><a href="/wiki/Categoria:Voci_con_template_Collegamenti_esterni_senza_dati_da_Wikidata" title="Categoria:Voci con template Collegamenti esterni senza dati da Wikidata">Voci con template Collegamenti esterni senza dati da Wikidata</a></li><li><a href="/wiki/Categoria:Voci_con_codice_Thesaurus_BNCF" title="Categoria:Voci con codice Thesaurus BNCF">Voci con codice Thesaurus BNCF</a></li><li><a href="/wiki/Categoria:Voci_con_codice_LCCN" title="Categoria:Voci con codice LCCN">Voci con codice LCCN</a></li><li><a href="/wiki/Categoria:Voci_con_codice_BNF" title="Categoria:Voci con codice BNF">Voci con codice BNF</a></li><li><a href="/wiki/Categoria:Voci_con_codice_J9U" title="Categoria:Voci con codice J9U">Voci con codice J9U</a></li><li><a href="/wiki/Categoria:Voci_non_biografiche_con_codici_di_controllo_di_autorit%C3%A0" title="Categoria:Voci non biografiche con codici di controllo di autorità">Voci non biografiche con codici di controllo di autorità</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Questa pagina è stata modificata per l'ultima volta il 30 nov 2024 alle 22:02.</li> <li id="footer-info-copyright">Il testo è disponibile secondo la <a 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Successione di Fibonacci - Wikipedia