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จำนวนฟีโบนัชชี - วิกิพีเดีย
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placeholder="ค้นหาใน วิกิพีเดีย" aria-label="ค้นหาใน วิกิพีเดีย" autocapitalize="sentences" title="ค้นหาวิกิ [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="พิเศษ:ค้นหา"> </div> <button class="cdx-button cdx-search-input__end-button">ค้นหา</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="เครื่องมือส่วนตัว"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="หน้าตา"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change 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</div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_th.wikipedia.org&uselang=th" class=""><span>บริจาคให้วิกิพีเดีย</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B8%AA%E0%B8%A3%E0%B9%89%E0%B8%B2%E0%B8%87%E0%B8%9A%E0%B8%B1%E0%B8%8D%E0%B8%8A%E0%B8%B5%E0%B8%9C%E0%B8%B9%E0%B9%89%E0%B9%83%E0%B8%8A%E0%B9%89%E0%B9%83%E0%B8%AB%E0%B8%A1%E0%B9%88&returnto=%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5" title="แนะนำให้คุณสร้างบัญชีและเข้าสู่ระบบ แต่ไม่บังคับ" class=""><span>สร้างบัญชี</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B8%A5%E0%B9%87%E0%B8%AD%E0%B8%81%E0%B8%AD%E0%B8%B4%E0%B8%99&returnto=%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5" title="แนะนำให้คุณเข้าสู่ระบบ แต่ไม่บังคับ [o]" accesskey="o" class=""><span>เข้าสู่ระบบ</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="ตัวเลือกเพิ่มเติม" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="เครื่องมือส่วนตัว" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">เครื่องมือส่วนตัว</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="เมนูผู้ใช้" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_th.wikipedia.org&uselang=th"><span>บริจาคให้วิกิพีเดีย</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B8%AA%E0%B8%A3%E0%B9%89%E0%B8%B2%E0%B8%87%E0%B8%9A%E0%B8%B1%E0%B8%8D%E0%B8%8A%E0%B8%B5%E0%B8%9C%E0%B8%B9%E0%B9%89%E0%B9%83%E0%B8%8A%E0%B9%89%E0%B9%83%E0%B8%AB%E0%B8%A1%E0%B9%88&returnto=%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5" title="แนะนำให้คุณสร้างบัญชีและเข้าสู่ระบบ แต่ไม่บังคับ"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>สร้างบัญชี</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B8%A5%E0%B9%87%E0%B8%AD%E0%B8%81%E0%B8%AD%E0%B8%B4%E0%B8%99&returnto=%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5" title="แนะนำให้คุณเข้าสู่ระบบ แต่ไม่บังคับ [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>เข้าสู่ระบบ</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> หน้าสำหรับผู้แก้ไขที่ออกจากระบบ <a href="/wiki/%E0%B8%A7%E0%B8%B4%E0%B8%98%E0%B8%B5%E0%B9%83%E0%B8%8A%E0%B9%89:%E0%B8%9A%E0%B8%97%E0%B8%99%E0%B8%B3" aria-label="เรียนรู้เพิ่มเกี่ยวกับการแก้ไข"><span>เรียนรู้เพิ่มเติม</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B9%80%E0%B8%A3%E0%B8%B7%E0%B9%88%E0%B8%AD%E0%B8%87%E0%B8%97%E0%B8%B5%E0%B9%88%E0%B8%89%E0%B8%B1%E0%B8%99%E0%B9%80%E0%B8%82%E0%B8%B5%E0%B8%A2%E0%B8%99" title="รายการการแก้ไขจากเลขที่อยู่ไอพีนี้ [y]" accesskey="y"><span>ส่วนร่วม</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B8%AB%E0%B8%99%E0%B9%89%E0%B8%B2%E0%B8%9E%E0%B8%B9%E0%B8%94%E0%B8%84%E0%B8%B8%E0%B8%A2%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%89%E0%B8%B1%E0%B8%99" title="อภิปรายเกี่ยวกับการแก้ไขจากเลขที่อยู่ไอพีนี้ [n]" accesskey="n"><span>คุย</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"><!-- CentralNotice --></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="ไซต์"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="สารบัญ" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">สารบัญ</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">ย้ายเมนูไปที่แถบด้านข้าง</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ซ่อน</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">บทนำ</div> </a> </li> <li id="toc-รูปปิด" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#รูปปิด"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>รูปปิด</span> </div> </a> <ul id="toc-รูปปิด-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-ความสัมพันธ์กับอัตราส่วนทอง" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#ความสัมพันธ์กับอัตราส่วนทอง"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>ความสัมพันธ์กับอัตราส่วนทอง</span> </div> </a> <ul id="toc-ความสัมพันธ์กับอัตราส่วนทอง-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-รูปเมทริกซ์(Matrix)" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#รูปเมทริกซ์(Matrix)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>รูปเมทริกซ์(Matrix)</span> </div> </a> <ul id="toc-รูปเมทริกซ์(Matrix)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-ลำดับฟีโบนัชชีในธรรมชาติ" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#ลำดับฟีโบนัชชีในธรรมชาติ"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>ลำดับฟีโบนัชชีในธรรมชาติ</span> </div> </a> <ul id="toc-ลำดับฟีโบนัชชีในธรรมชาติ-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-การนำไปใช้" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#การนำไปใช้"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>การนำไปใช้</span> </div> </a> <ul id="toc-การนำไปใช้-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-อ้างอิง" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#อ้างอิง"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>อ้างอิง</span> </div> </a> <ul id="toc-อ้างอิง-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="สารบัญ" 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="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">จำนวนฟีโบนัชชี</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="ไปที่บทความในภาษาอื่น ซึ่งมีใน 63 ภาษา" > <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-63" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">63 ภาษา</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar badge-Q70894304 mw-list-item" title=""><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D9%81%D9%8A%D8%A8%D9%88%D9%86%D8%A7%D8%AA%D8%B4%D9%8A" title="عدد فيبوناتشي – อาหรับ" lang="ar" hreflang="ar" data-title="عدد فيبوناتشي" data-language-autonym="العربية" data-language-local-name="อาหรับ" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Fibona%C3%A7%C3%A7i_%C9%99d%C9%99dl%C9%99ri" title="Fibonaççi ədədləri – อาเซอร์ไบจาน" lang="az" hreflang="az" data-title="Fibonaççi ədədləri" data-language-autonym="Azərbaycanca" data-language-local-name="อาเซอร์ไบจาน" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A4%D0%B8%D0%B1%D0%BE%D0%BD%D0%B0%D1%87%D1%87%D0%B8_%D2%BB%D0%B0%D0%BD%D0%B4%D0%B0%D1%80%D1%8B" title="Фибоначчи һандары – บัชคีร์" lang="ba" hreflang="ba" data-title="Фибоначчи һандары" data-language-autonym="Башҡортса" data-language-local-name="บัชคีร์" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9B%D1%96%D0%BA%D1%96_%D0%A4%D1%96%D0%B1%D0%B0%D0%BD%D0%B0%D1%87%D1%8B" title="Лікі Фібаначы – เบลารุส" lang="be" hreflang="be" data-title="Лікі Фібаначы" data-language-autonym="Беларуская" data-language-local-name="เบลารุส" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%B0_%D0%BD%D0%B0_%D0%A4%D0%B8%D0%B1%D0%BE%D0%BD%D0%B0%D1%87%D0%B8" title="Числа на Фибоначи – บัลแกเรีย" lang="bg" hreflang="bg" data-title="Числа на Фибоначи" data-language-autonym="Български" data-language-local-name="บัลแกเรีย" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Fibonaccijev_broj" title="Fibonaccijev broj – บอสเนีย" lang="bs" hreflang="bs" data-title="Fibonaccijev broj" data-language-autonym="Bosanski" data-language-local-name="บอสเนีย" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_de_Fibonacci" title="Nombre de Fibonacci – คาตาลัน" lang="ca" hreflang="ca" data-title="Nombre de Fibonacci" data-language-autonym="Català" data-language-local-name="คาตาลัน" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%98%D9%85%D8%A7%D8%B1%DB%95%DB%8C_%D9%81%DB%8C%D8%A8%DB%86%D9%86%D8%A7%DA%86%DB%8C" title="ژمارەی فیبۆناچی – เคิร์ดตอนกลาง" lang="ckb" hreflang="ckb" data-title="ژمارەی فیبۆناچی" data-language-autonym="کوردی" data-language-local-name="เคิร์ดตอนกลาง" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cu mw-list-item"><a href="https://cu.wikipedia.org/wiki/%D0%A4%D1%97%D0%B2%D0%BE%D0%BD%D0%B0%D0%BA%D1%97%D0%B8%D0%BD%D0%BE%D0%B2%D0%B8_%D1%87%D0%B8%D1%81%D0%BC%D1%94%D0%BD%D0%B0" title="Фївонакїинови чисмєна – เชอร์ชสลาวิก" lang="cu" hreflang="cu" data-title="Фївонакїинови чисмєна" data-language-autonym="Словѣньскъ / ⰔⰎⰑⰂⰡⰐⰠⰔⰍⰟ" data-language-local-name="เชอร์ชสลาวิก" class="interlanguage-link-target"><span>Словѣньскъ / ⰔⰎⰑⰂⰡⰐⰠⰔⰍⰟ</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A4%D0%B8%D0%B1%D0%BE%D0%BD%D0%B0%D1%87%D1%87%D0%B8_%D1%85%D0%B8%D1%81%D0%B5%D0%BF%C4%95" title="Фибоначчи хисепĕ – ชูวัช" lang="cv" hreflang="cv" data-title="Фибоначчи хисепĕ" data-language-autonym="Чӑвашла" data-language-local-name="ชูวัช" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Rhif_Fibonacci" title="Rhif Fibonacci – เวลส์" lang="cy" hreflang="cy" data-title="Rhif Fibonacci" data-language-autonym="Cymraeg" data-language-local-name="เวลส์" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Fibonacci-tal" title="Fibonacci-tal – เดนมาร์ก" lang="da" hreflang="da" data-title="Fibonacci-tal" data-language-autonym="Dansk" data-language-local-name="เดนมาร์ก" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Fibonaccizahl" title="Fibonaccizahl – เยอรมัน" lang="de" hreflang="de" data-title="Fibonaccizahl" data-language-autonym="Deutsch" data-language-local-name="เยอรมัน" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en badge-Q70893996 mw-list-item" title=""><a href="https://en.wikipedia.org/wiki/Fibonacci_number" title="Fibonacci number – อังกฤษ" lang="en" hreflang="en" data-title="Fibonacci number" data-language-autonym="English" data-language-local-name="อังกฤษ" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Fibona%C4%89i-nombro" title="Fibonaĉi-nombro – เอสเปรันโต" lang="eo" hreflang="eo" data-title="Fibonaĉi-nombro" data-language-autonym="Esperanto" data-language-local-name="เอสเปรันโต" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es badge-Q70894304 mw-list-item" title=""><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_de_Fibonacci" title="Número de Fibonacci – สเปน" lang="es" hreflang="es" data-title="Número de Fibonacci" data-language-autonym="Español" data-language-local-name="สเปน" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Fibonacci_jada" title="Fibonacci jada – เอสโตเนีย" lang="et" hreflang="et" data-title="Fibonacci jada" data-language-autonym="Eesti" data-language-local-name="เอสโตเนีย" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Fibonacciren_zenbakiak" title="Fibonacciren zenbakiak – บาสก์" lang="eu" hreflang="eu" data-title="Fibonacciren zenbakiak" data-language-autonym="Euskara" data-language-local-name="บาสก์" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%B9%D8%AF%D8%A7%D8%AF_%D9%81%DB%8C%D8%A8%D9%88%D9%86%D8%A7%DA%86%DB%8C" title="اعداد فیبوناچی – เปอร์เซีย" lang="fa" hreflang="fa" data-title="اعداد فیبوناچی" data-language-autonym="فارسی" data-language-local-name="เปอร์เซีย" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Fibonacci_arv" title="Fibonacci arv – โวโร" lang="vro" hreflang="vro" data-title="Fibonacci arv" data-language-autonym="Võro" data-language-local-name="โวโร" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-fr badge-Q70894304 mw-list-item" title=""><a href="https://fr.wikipedia.org/wiki/Nombre_de_Fibonacci" title="Nombre de Fibonacci – ฝรั่งเศส" lang="fr" hreflang="fr" data-title="Nombre de Fibonacci" data-language-autonym="Français" data-language-local-name="ฝรั่งเศส" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%AB%E0%AA%BF%E0%AA%AC%E0%AB%8B%E0%AA%A8%E0%AA%BE%E0%AA%95%E0%AA%BF" title="ફિબોનાકિ – คุชราต" lang="gu" hreflang="gu" data-title="ફિબોનાકિ" data-language-autonym="ગુજરાતી" data-language-local-name="คุชราต" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-he badge-Q70893996 mw-list-item" title=""><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%A4%D7%99%D7%91%D7%95%D7%A0%D7%90%D7%A6%27%D7%99" title="מספר פיבונאצ'י – ฮิบรู" lang="he" hreflang="he" data-title="מספר פיבונאצ'י" data-language-autonym="עברית" data-language-local-name="ฮิบรู" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B9%E0%A5%87%E0%A4%AE%E0%A4%9A%E0%A4%A8%E0%A5%8D%E0%A4%A6%E0%A5%8D%E0%A4%B0_%E0%A4%B6%E0%A5%8D%E0%A4%B0%E0%A5%87%E0%A4%A3%E0%A5%80" title="हेमचन्द्र श्रेणी – ฮินดี" lang="hi" hreflang="hi" data-title="हेमचन्द्र श्रेणी" data-language-autonym="हिन्दी" data-language-local-name="ฮินดี" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Fibonaccijev_broj" title="Fibonaccijev broj – โครเอเชีย" lang="hr" hreflang="hr" data-title="Fibonaccijev broj" data-language-autonym="Hrvatski" data-language-local-name="โครเอเชีย" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Fibonacci-sz%C3%A1mok" title="Fibonacci-számok – ฮังการี" lang="hu" hreflang="hu" data-title="Fibonacci-számok" data-language-autonym="Magyar" data-language-local-name="ฮังการี" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy badge-Q70893996 mw-list-item" title=""><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%A9%D5%BE%D5%A5%D6%80" title="Ֆիբոնաչիի թվեր – อาร์เมเนีย" lang="hy" hreflang="hy" data-title="Ֆիբոնաչիի թվեր" data-language-autonym="Հայերեն" data-language-local-name="อาร์เมเนีย" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hyw mw-list-item"><a href="https://hyw.wikipedia.org/wiki/%D5%96%D5%AB%D5%BA%D5%B8%D5%B6%D5%A1%D5%B9%D5%B9%D5%AB%D5%AB_%D5%A9%D5%AB%D6%82" title="Ֆիպոնաչչիի թիւ – Western Armenian" lang="hyw" hreflang="hyw" data-title="Ֆիպոնաչչիի թիւ" data-language-autonym="Արեւմտահայերէն" data-language-local-name="Western Armenian" class="interlanguage-link-target"><span>Արեւմտահայերէն</span></a></li><li class="interlanguage-link interwiki-id badge-Q70893996 mw-list-item" title=""><a href="https://id.wikipedia.org/wiki/Bilangan_Fibonacci" title="Bilangan Fibonacci – อินโดนีเซีย" lang="id" hreflang="id" data-title="Bilangan Fibonacci" data-language-autonym="Bahasa Indonesia" data-language-local-name="อินโดนีเซีย" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Fibonacci-runan" title="Fibonacci-runan – ไอซ์แลนด์" lang="is" hreflang="is" data-title="Fibonacci-runan" data-language-autonym="Íslenska" data-language-local-name="ไอซ์แลนด์" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it badge-Q70894304 mw-list-item" title=""><a href="https://it.wikipedia.org/wiki/Numero_di_Fibonacci" title="Numero di Fibonacci – อิตาลี" lang="it" hreflang="it" data-title="Numero di Fibonacci" data-language-autonym="Italiano" data-language-local-name="อิตาลี" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%95%E3%82%A3%E3%83%9C%E3%83%8A%E3%83%83%E3%83%81%E6%95%B0" title="フィボナッチ数 – ญี่ปุ่น" lang="ja" hreflang="ja" data-title="フィボナッチ数" data-language-autonym="日本語" data-language-local-name="ญี่ปุ่น" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/Fibonachchi_sanlar%C4%B1" title="Fibonachchi sanları – การา-กาลพาก" lang="kaa" hreflang="kaa" data-title="Fibonachchi sanları" data-language-autonym="Qaraqalpaqsha" data-language-local-name="การา-กาลพาก" class="interlanguage-link-target"><span>Qaraqalpaqsha</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A4%D0%B8%D0%B1%D0%BE%D0%BD%D0%B0%D1%87%D1%87%D0%B8_%D1%81%D0%B0%D0%BD%D0%B4%D0%B0%D1%80%D1%8B" title="Фибоначчи сандары – คาซัค" lang="kk" hreflang="kk" data-title="Фибоначчи сандары" data-language-autonym="Қазақша" data-language-local-name="คาซัค" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%94%BC%EB%B3%B4%EB%82%98%EC%B9%98_%EC%88%98" title="피보나치 수 – เกาหลี" lang="ko" hreflang="ko" data-title="피보나치 수" data-language-autonym="한국어" data-language-local-name="เกาหลี" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Numeri_Fibonacciani" title="Numeri Fibonacciani – ละติน" lang="la" hreflang="la" data-title="Numeri Fibonacciani" data-language-autonym="Latina" data-language-local-name="ละติน" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Fibona%C4%8Di_skait%C4%BCi" title="Fibonači skaitļi – ลัตเวีย" lang="lv" hreflang="lv" data-title="Fibonači skaitļi" data-language-autonym="Latviešu" data-language-local-name="ลัตเวีย" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A4%D0%B8%D0%B1%D0%BE%D0%BD%D0%B0%D1%87%D0%B8%D0%B5%D0%B2%D0%B0_%D0%BD%D0%B8%D0%B7%D0%B0" title="Фибоначиева низа – มาซิโดเนีย" lang="mk" hreflang="mk" data-title="Фибоначиева низа" data-language-autonym="Македонски" data-language-local-name="มาซิโดเนีย" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AB%E0%B4%BF%E0%B4%AC%E0%B4%A8%E0%B4%BE%E0%B4%9A%E0%B5%8D%E0%B4%9A%E0%B4%BF_%E0%B4%B6%E0%B5%8D%E0%B4%B0%E0%B5%87%E0%B4%A3%E0%B4%BF" title="ഫിബനാച്ചി ശ്രേണി – มาลายาลัม" lang="ml" hreflang="ml" data-title="ഫിബനാച്ചി ശ്രേണി" data-language-autonym="മലയാളം" data-language-local-name="มาลายาลัม" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A4%D0%B8%D0%B1%D0%BE%D0%BD%D0%B0%D1%87%D1%87%D0%B8%D0%B9%D0%BD_%D1%82%D0%BE%D0%BE" title="Фибоначчийн тоо – มองโกเลีย" lang="mn" hreflang="mn" data-title="Фибоначчийн тоо" data-language-autonym="Монгол" data-language-local-name="มองโกเลีย" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%AB%E0%A4%BF%E0%A4%AC%E0%A5%8B%E0%A4%A8%E0%A4%BE%E0%A4%9A%E0%A5%80_%E0%A4%B6%E0%A5%8D%E0%A4%B0%E0%A5%87%E0%A4%A3%E0%A5%80" title="फिबोनाची श्रेणी – มราฐี" lang="mr" hreflang="mr" data-title="फिबोनाची श्रेणी" data-language-autonym="मराठी" data-language-local-name="มราฐี" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-nl badge-Q70894304 mw-list-item" title=""><a href="https://nl.wikipedia.org/wiki/Fibonaccigetal" title="Fibonaccigetal – ดัตช์" lang="nl" hreflang="nl" data-title="Fibonaccigetal" data-language-autonym="Nederlands" data-language-local-name="ดัตช์" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Fibonaccitall" title="Fibonaccitall – นอร์เวย์บุคมอล" lang="nb" hreflang="nb" data-title="Fibonaccitall" data-language-autonym="Norsk bokmål" data-language-local-name="นอร์เวย์บุคมอล" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AB%E0%A8%BC%E0%A9%80%E0%A8%AC%E0%A9%8B%E0%A8%A8%E0%A8%BE%E0%A8%9A%E0%A9%80_%E0%A8%A4%E0%A8%B0%E0%A8%A4%E0%A9%80%E0%A8%AC" title="ਫ਼ੀਬੋਨਾਚੀ ਤਰਤੀਬ – ปัญจาบ" lang="pa" hreflang="pa" data-title="ਫ਼ੀਬੋਨਾਚੀ ਤਰਤੀਬ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="ปัญจาบ" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D9%81%DB%8C%D8%A8%D9%88%D9%86%D8%A7%DA%86%DB%90_%D8%A7%D8%B9%D8%AF%D8%A7%D8%AF" title="فیبوناچې اعداد – พัชโต" lang="ps" hreflang="ps" data-title="فیبوناچې اعداد" data-language-autonym="پښتو" data-language-local-name="พัชโต" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Num%C4%83r_Fibonacci" title="Număr Fibonacci – โรมาเนีย" lang="ro" hreflang="ro" data-title="Număr Fibonacci" data-language-autonym="Română" data-language-local-name="โรมาเนีย" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%B0_%D0%A4%D0%B8%D0%B1%D0%BE%D0%BD%D0%B0%D1%87%D1%87%D0%B8" title="Числа Фибоначчи – รัสเซีย" lang="ru" hreflang="ru" data-title="Числа Фибоначчи" data-language-autonym="Русский" data-language-local-name="รัสเซีย" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Fibonaccijev_niz" title="Fibonaccijev niz – เซอร์โบ-โครเอเชีย" lang="sh" hreflang="sh" data-title="Fibonaccijev niz" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="เซอร์โบ-โครเอเชีย" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B7%86%E0%B7%92%E0%B6%B6%E0%B7%9C%E0%B6%B1%E0%B7%8F%E0%B6%A0%E0%B7%8A%E0%B6%A0%E0%B7%92_%E0%B7%83%E0%B6%82%E0%B6%9B%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F" title="ෆිබොනාච්චි සංඛ්යා – สิงหล" lang="si" hreflang="si" data-title="ෆිබොනාච්චි සංඛ්යා" data-language-autonym="සිංහල" data-language-local-name="สิงหล" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Fibonacci_number" title="Fibonacci number – Simple English" lang="en-simple" hreflang="en-simple" data-title="Fibonacci number" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Fibonaccijevo_%C5%A1tevilo" title="Fibonaccijevo število – สโลวีเนีย" lang="sl" hreflang="sl" data-title="Fibonaccijevo število" data-language-autonym="Slovenščina" data-language-local-name="สโลวีเนีย" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Numrat_e_Fibonaccit" title="Numrat e Fibonaccit – แอลเบเนีย" lang="sq" hreflang="sq" data-title="Numrat e Fibonaccit" data-language-autonym="Shqip" data-language-local-name="แอลเบเนีย" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A4%D0%B8%D0%B1%D0%BE%D0%BD%D0%B0%D1%87%D0%B8%D1%98%D0%B5%D0%B2_%D0%BD%D0%B8%D0%B7" title="Фибоначијев низ – เซอร์เบีย" lang="sr" hreflang="sr" data-title="Фибоначијев низ" data-language-autonym="Српски / srpski" data-language-local-name="เซอร์เบีย" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Fibonaccital" title="Fibonaccital – สวีเดน" lang="sv" hreflang="sv" data-title="Fibonaccital" data-language-autonym="Svenska" data-language-local-name="สวีเดน" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AE%BF%E0%AE%AA%E0%AE%A9%E0%AE%BE%E0%AE%9A%E0%AF%8D%E0%AE%9A%E0%AE%BF_%E0%AE%8E%E0%AE%A3%E0%AF%8D%E0%AE%95%E0%AE%B3%E0%AF%8D" title="பிபனாச்சி எண்கள் – ทมิฬ" lang="ta" hreflang="ta" data-title="பிபனாச்சி எண்கள்" data-language-autonym="தமிழ்" data-language-local-name="ทมิฬ" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%AB%E0%B0%BF%E0%B0%AC%E0%B1%8B%E0%B0%A8%E0%B0%BE%E0%B0%9A%E0%B1%80_%E0%B0%B8%E0%B0%82%E0%B0%96%E0%B1%8D%E0%B0%AF%E0%B0%B2%E0%B1%81" title="ఫిబోనాచీ సంఖ్యలు – เตลูกู" lang="te" hreflang="te" data-title="ఫిబోనాచీ సంఖ్యలు" data-language-autonym="తెలుగు" data-language-local-name="เตลูกู" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Bilang_na_Fibonacci" title="Bilang na Fibonacci – ตากาล็อก" lang="tl" hreflang="tl" data-title="Bilang na Fibonacci" data-language-autonym="Tagalog" data-language-local-name="ตากาล็อก" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-uk badge-Q70894304 mw-list-item" title=""><a href="https://uk.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%B0_%D0%A4%D1%96%D0%B1%D0%BE%D0%BD%D0%B0%D1%87%D1%87%D1%96" title="Числа Фібоначчі – ยูเครน" lang="uk" hreflang="uk" data-title="Числа Фібоначчі" data-language-autonym="Українська" data-language-local-name="ยูเครน" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Fibonacci_sonlari" title="Fibonacci sonlari – อุซเบก" lang="uz" hreflang="uz" data-title="Fibonacci sonlari" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="อุซเบก" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/D%C3%A3y_Fibonacci" title="Dãy Fibonacci – เวียดนาม" lang="vi" hreflang="vi" data-title="Dãy Fibonacci" data-language-autonym="Tiếng Việt" data-language-local-name="เวียดนาม" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Ihap_Fibonacci" title="Ihap Fibonacci – วาเรย์" lang="war" hreflang="war" data-title="Ihap Fibonacci" data-language-autonym="Winaray" data-language-local-name="วาเรย์" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%96%90%E6%B3%A2%E9%82%A3%E5%A5%91%E6%95%B0" title="斐波那契数 – จีน" lang="zh" hreflang="zh" data-title="斐波那契数" data-language-autonym="中文" data-language-local-name="จีน" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%B2%BB%E6%B0%8F%E6%95%B8" title="費氏數 – กวางตุ้ง" lang="yue" hreflang="yue" data-title="費氏數" data-language-autonym="粵語" data-language-local-name="กวางตุ้ง" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q47577#sitelinks-wikipedia" title="แก้ไขลิงก์ข้ามภาษา" class="wbc-editpage">แก้ไขลิงก์</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="เนมสเปซ"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5" title="ดูหน้าเนื้อหา [c]" accesskey="c"><span>บทความ</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a 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mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="ดู"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5"><span>อ่าน</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5&action=edit" title="แก้ไขรหัสต้นฉบับของหน้านี้ [e]" accesskey="e"><span>แก้ไข</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5&action=history" title="แก้ไขเก่าของหน้านี้ [h]" accesskey="h"><span>ดูประวัติ</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="หน้าเครื่องมือ"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="เครื่องมือ" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">เครื่องมือ</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">เครื่องมือ</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">ย้ายเมนูไปที่แถบด้านข้าง</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">ซ่อน</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="ตัวเลือกเพิ่มเติม" > <div class="vector-menu-heading"> การกระทำ </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5"><span>อ่าน</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5&action=edit" title="แก้ไขรหัสต้นฉบับของหน้านี้ [e]" accesskey="e"><span>แก้ไข</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5&action=history"><span>ดูประวัติ</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> ทั่วไป </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B8%9A%E0%B8%97%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B8%97%E0%B8%B5%E0%B9%88%E0%B9%82%E0%B8%A2%E0%B8%87%E0%B8%A1%E0%B8%B2/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5" title="รายการหน้าวิกิทุกหน้าที่ลิงก์มาที่นี่ [j]" accesskey="j"><span>หน้าที่ลิงก์มา</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%9B%E0%B8%A3%E0%B8%B1%E0%B8%9A%E0%B8%9B%E0%B8%A3%E0%B8%B8%E0%B8%87%E0%B8%97%E0%B8%B5%E0%B9%88%E0%B9%82%E0%B8%A2%E0%B8%87%E0%B8%A1%E0%B8%B2/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5" rel="nofollow" title="รายการเปลี่ยนแปลงล่าสุดในหน้าที่ลิงก์จากหน้านี้ [k]" accesskey="k"><span>การเปลี่ยนแปลงที่เกี่ยวโยง</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/%E0%B8%A7%E0%B8%B4%E0%B8%81%E0%B8%B4%E0%B8%9E%E0%B8%B5%E0%B9%80%E0%B8%94%E0%B8%B5%E0%B8%A2:%E0%B8%AD%E0%B8%B1%E0%B8%9B%E0%B9%82%E0%B8%AB%E0%B8%A5%E0%B8%94" title="อัปโหลดไฟล์ [u]" accesskey="u"><span>อัปโหลดไฟล์</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B8%AB%E0%B8%99%E0%B9%89%E0%B8%B2%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9" title="รายการหน้าพิเศษทั้งหมด [q]" accesskey="q"><span>หน้าพิเศษ</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5&oldid=11910667" title="ลิงก์ถาวรมารุ่นนี้ของหน้านี้"><span>ลิงก์ถาวร</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5&action=info" title="ข้อมูลเพิ่มเติมเกี่ยวกับหน้านี้"><span>สารสนเทศหน้า</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B8%AD%E0%B9%89%E0%B8%B2%E0%B8%87%E0%B8%AD%E0%B8%B4%E0%B8%87&page=%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5&id=11910667&wpFormIdentifier=titleform" title="สารสนเทศเกี่ยวกับวิธีการอ้างอิงหน้านี้"><span>อ้างอิงบทความนี้</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:UrlShortener&url=https%3A%2F%2Fth.wikipedia.org%2Fwiki%2F%25E0%25B8%2588%25E0%25B8%25B3%25E0%25B8%2599%25E0%25B8%25A7%25E0%25B8%2599%25E0%25B8%259F%25E0%25B8%25B5%25E0%25B9%2582%25E0%25B8%259A%25E0%25B8%2599%25E0%25B8%25B1%25E0%25B8%258A%25E0%25B8%258A%25E0%25B8%25B5"><span>รับยูอาร์แอลแบบสั้น</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:QrCode&url=https%3A%2F%2Fth.wikipedia.org%2Fwiki%2F%25E0%25B8%2588%25E0%25B8%25B3%25E0%25B8%2599%25E0%25B8%25A7%25E0%25B8%2599%25E0%25B8%259F%25E0%25B8%25B5%25E0%25B9%2582%25E0%25B8%259A%25E0%25B8%2599%25E0%25B8%25B1%25E0%25B8%258A%25E0%25B8%258A%25E0%25B8%25B5"><span>ดาวน์โหลดคิวอาร์โค้ด</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> พิมพ์/ส่งออก </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B8%AB%E0%B8%99%E0%B8%B1%E0%B8%87%E0%B8%AA%E0%B8%B7%E0%B8%AD&bookcmd=book_creator&referer=%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5"><span>สร้างหนังสือ</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:DownloadAsPdf&page=%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5&action=show-download-screen"><span>ดาวน์โหลดเป็น PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5&printable=yes" title="รุ่นที่พร้อมพิมพ์ของหน้านี้ [p]" accesskey="p"><span>รุ่นพร้อมพิมพ์</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> ในโครงการอื่น </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Fibonacci_numbers" hreflang="en"><span>วิกิมีเดียคอมมอนส์</span></a></li><li class="wb-otherproject-link wb-otherproject-wikifunctions mw-list-item"><a href="https://www.wikifunctions.org/wiki/Z13835" hreflang="en"><span>วิกิฟังก์ชัน</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q47577" title="ลิงก์ไปยังสิ่งนี้ในคลังซึ่งเชื่อมโยงข้อมูลต่าง ๆ เข้าด้วยกัน [g]" accesskey="g"><span>สิ่งนี้ในวิกิสนเทศ</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="หน้าเครื่องมือ"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="หน้าตา"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">หน้าตา</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">ย้ายเมนูไปที่แถบด้านข้าง</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ซ่อน</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">จากวิกิพีเดีย สารานุกรมเสรี</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="th" dir="ltr"><figure typeof="mw:File/Thumb"><a href="/wiki/%E0%B9%84%E0%B8%9F%E0%B8%A5%E0%B9%8C:FibonacciBlocks.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/FibonacciBlocks.png/200px-FibonacciBlocks.png" decoding="async" width="200" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/8/83/FibonacciBlocks.png 1.5x" data-file-width="223" data-file-height="138" /></a><figcaption>การจัดเรียงสี่เหลี่ยมจัตุรัสที่มีความยาวด้านเท่ากับจำนวนฟีโบนัชชี</figcaption></figure> <p><b>จำนวนฟีโบนัชชี</b> หรือ <b>เลขจำนวนฟีโบนัชชี</b> (<a href="/wiki/%E0%B8%A0%E0%B8%B2%E0%B8%A9%E0%B8%B2%E0%B8%AD%E0%B8%B1%E0%B8%87%E0%B8%81%E0%B8%A4%E0%B8%A9" title="ภาษาอังกฤษ">อังกฤษ</a>: <span lang="en">Fibonacci number</span>) คือ<a href="/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B9%80%E0%B8%95%E0%B9%87%E0%B8%A1" title="จำนวนเต็ม">จำนวนเต็ม</a>ที่อยู่ใน<a href="/w/index.php?title=%E0%B8%A5%E0%B8%B3%E0%B8%94%E0%B8%B1%E0%B8%9A%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B9%80%E0%B8%95%E0%B9%87%E0%B8%A1&action=edit&redlink=1" class="new" title="ลำดับจำนวนเต็ม (ไม่มีหน้านี้)">ลำดับจำนวนเต็ม</a>ดังต่อไปนี้ </p> <dl><dd><a href="/wiki/0" title="0">0</a>, <a href="/wiki/1" title="1">1</a>, <a href="/wiki/1" title="1">1</a>, <a href="/wiki/2" title="2">2</a>, <a href="/wiki/3" title="3">3</a>, <a href="/wiki/5" title="5">5</a>, <a href="/wiki/8" title="8">8</a>, <a href="/wiki/13" title="13">13</a>, <a href="/wiki/21" title="21">21</a>, <a href="/wiki/34" title="34">34</a>, <a href="/wiki/55" title="55">55</a>, <a href="/wiki/89" title="89">89</a>, <a href="/w/index.php?title=144&action=edit&redlink=1" class="new" title="144 (ไม่มีหน้านี้)">144</a>, <a href="/w/index.php?title=233&action=edit&redlink=1" class="new" title="233 (ไม่มีหน้านี้)">233</a>, <a href="/wiki/300" title="300">377</a>, <a href="/wiki/600" title="600">610</a>, <a href="/wiki/900" title="900">987</a>, <a href="/wiki/1000" title="1000">1597</a>, <a href="/wiki/2000" title="2000">2584</a>, <a href="/wiki/4000" title="4000">4181</a>, <a href="/wiki/6000" title="6000">6765</a>, <a href="/wiki/10000" title="10000">10946</a> ... (ลำดับ <span class="nowrap"><span typeof="mw:File"><a href="/wiki/%E0%B8%AA%E0%B8%B2%E0%B8%A3%E0%B8%B2%E0%B8%99%E0%B8%B8%E0%B8%81%E0%B8%A3%E0%B8%A1%E0%B8%AD%E0%B8%AD%E0%B8%99%E0%B9%84%E0%B8%A5%E0%B8%99%E0%B9%8C%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%A5%E0%B8%B3%E0%B8%94%E0%B8%B1%E0%B8%9A%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B9%80%E0%B8%95%E0%B9%87%E0%B8%A1" title="OEIS"><img alt="OEIS" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/OEISicon_light.svg/11px-OEISicon_light.svg.png" decoding="async" width="11" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/OEISicon_light.svg/17px-OEISicon_light.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d8/OEISicon_light.svg/22px-OEISicon_light.svg.png 2x" data-file-width="409" data-file-height="556" /></a></span> <a href="//oeis.org/A000045" class="extiw" title="oeis:A000045">A000045</a>)</span></dd></dl> <p>โดยนิยามความสัมพันธ์ว่า จำนวนถัดไปเท่ากับผลบวกของจำนวนสองจำนวนก่อนหน้า และสองจำนวนแรกก็คือ 0 และ 1 ตามลำดับ ลำดับของจำนวนดังกล่าวเรียกว่า <b>ลำดับจำนวนฟีโบนัชชี</b> (<a href="/wiki/%E0%B8%A0%E0%B8%B2%E0%B8%A9%E0%B8%B2%E0%B8%AD%E0%B8%B1%E0%B8%87%E0%B8%81%E0%B8%A4%E0%B8%A9" title="ภาษาอังกฤษ">อังกฤษ</a>: <span lang="en">Fibonacci sequence</span>) </p><p>หากเขียนให้อยู่ในรูปของสัญลักษณ์ ลำดับ <i>F<sub>n</sub></i> ของจำนวนฟีโบนัชชีนิยามด้วย<a href="/wiki/%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B8%AA%E0%B8%B1%E0%B8%A1%E0%B8%9E%E0%B8%B1%E0%B8%99%E0%B8%98%E0%B9%8C%E0%B9%80%E0%B8%A7%E0%B8%B5%E0%B8%A2%E0%B8%99%E0%B9%80%E0%B8%81%E0%B8%B4%E0%B8%94" class="mw-redirect" title="ความสัมพันธ์เวียนเกิด">ความสัมพันธ์เวียนเกิด</a>ดังนี้ </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}=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> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}=F_{n-1}+F_{n-2}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f880da4d97e018b62feb5e2e8aa0c73808cd59e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:18.279ex; height:2.509ex;" alt="{\displaystyle F_{n}=F_{n-1}+F_{n-2}\!}"></span></dd></dl> <p>โดยกำหนดค่าเริ่มแรกให้ <sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{0}=0;\;F_{1}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>;</mo> <mspace width="thickmathspace" /> <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_{0}=0;\;F_{1}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf02289023419aa9e2c9d8a8acadbbccf58455bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.299ex; height:2.509ex;" alt="{\displaystyle F_{0}=0;\;F_{1}=1}"></span></dd></dl> <p>ชื่อของจำนวนฟีโบนัชชีตั้งขึ้นเพื่อเป็นเกียรติแก่นักคณิตศาสตร์ชาว<a href="/wiki/%E0%B8%AD%E0%B8%B4%E0%B8%95%E0%B8%B2%E0%B8%A5%E0%B8%B5" class="mw-redirect" title="อิตาลี">อิตาลี</a>ชื่อ เลโอนาร์โดแห่งปีซา (Leonardo de Pisa) ซึ่งเป็นที่รู้จักกันในนาม<a href="/wiki/%E0%B9%80%E0%B8%A5%E0%B9%82%E0%B8%AD%E0%B8%99%E0%B8%B2%E0%B8%A3%E0%B9%8C%E0%B9%82%E0%B8%94_%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5" title="เลโอนาร์โด ฟีโบนัชชี">ฟีโบนัชชี</a> (Fibonacci) ผู้ค้นพบจำนวนฟีโบนัชชีในต้นศตวรรษที่ 13 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="รูปปิด"><span id=".E0.B8.A3.E0.B8.B9.E0.B8.9B.E0.B8.9B.E0.B8.B4.E0.B8.94"></span>รูปปิด</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5&action=edit&section=1" title="แก้ไขส่วน: รูปปิด"><span>แก้</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>เนื่องจากลำดับฟีโบนัชชีเป็นลำดับที่นิยามด้วยความสัมพันธ์เวียนเกิดเชิงเส้น เราจึงสามารถหา<a href="/w/index.php?title=%E0%B8%A3%E0%B8%B9%E0%B8%9B%E0%B8%9B%E0%B8%B4%E0%B8%94&action=edit&redlink=1" class="new" title="รูปปิด (ไม่มีหน้านี้)">รูปปิด</a>ของจำนวนฟีโบนัชชีได้ โดยสมการแสดงรูปปิดของจำนวนฟีโบนัชชี มีชื่อเรียกว่า <i>สูตรของ<a href="/w/index.php?title=%E0%B8%88%E0%B8%B2%E0%B8%84_%E0%B8%9F%E0%B8%B4%E0%B8%A5%E0%B8%B4%E0%B8%9B%E0%B8%9B%E0%B9%8C_%E0%B8%A1%E0%B8%B2%E0%B8%A3%E0%B8%B5_%E0%B8%9A%E0%B8%B4%E0%B9%80%E0%B8%99%E0%B8%95%E0%B9%8C&action=edit&redlink=1" class="new" title="จาค ฟิลิปป์ มารี บิเนต์ (ไม่มีหน้านี้)">บิเนต์</a></i> มีดังต่อไปนี้ </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\left(n\right)={{\varphi ^{n}-(1-\varphi )^{n}} \over {\sqrt {5}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>φ<!-- φ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−<!-- − --></mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>φ<!-- φ --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\left(n\right)={{\varphi ^{n}-(1-\varphi )^{n}} \over {\sqrt {5}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbebdf401f651ec64f0d2482b46c1fab1f9cb7fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:23.396ex; height:6.676ex;" alt="{\displaystyle F\left(n\right)={{\varphi ^{n}-(1-\varphi )^{n}} \over {\sqrt {5}}}}"></span></dd></dl> <p>โดย <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =(1+{\sqrt {5}})/2\approx 1.618}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ<!-- φ --></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>≈<!-- ≈ --></mo> <mn>1.618</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =(1+{\sqrt {5}})/2\approx 1.618}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95baf222943d6bcff65c0ef7552661c9c9c659e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.249ex; height:3.009ex;" alt="{\displaystyle \varphi =(1+{\sqrt {5}})/2\approx 1.618}"></span> เป็นตัวเลขที่รู้จักกันโดยทั่วไปว่า<a href="/wiki/%E0%B8%AD%E0%B8%B1%E0%B8%95%E0%B8%A3%E0%B8%B2%E0%B8%AA%E0%B9%88%E0%B8%A7%E0%B8%99%E0%B8%97%E0%B8%AD%E0%B8%87" title="อัตราส่วนทอง">อัตราส่วนทอง</a> </p><p>พิจารณาสมการพหุนาม <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}=x+1}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}=x+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f05f1f6f113f4a1ae6c6e425c7cc3113de458980" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.815ex; height:2.843ex;" alt="{\displaystyle x^{2}=x+1}"></span> เมื่อคูณทั้งสองข้างด้วย <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c427ee189e46907f667b4f32462c6c2aa75c1983" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.649ex; height:2.676ex;" alt="{\displaystyle x^{n-1}}"></span> เราได้ว่า </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{n+1}=x^{n}+x^{n-1}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{n+1}=x^{n}+x^{n-1}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d634338ccd90f2588adebd2feec5ed64d599ea1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.171ex; height:2.843ex;" alt="{\displaystyle x^{n+1}=x^{n}+x^{n-1}\,}"></span></dd></dl> <p>ผลเฉลยของสมการ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}=x+1}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}=x+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f05f1f6f113f4a1ae6c6e425c7cc3113de458980" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.815ex; height:2.843ex;" alt="{\displaystyle x^{2}=x+1}"></span> ได้แก่ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ<!-- φ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> และ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−<!-- − --></mo> <mi>φ<!-- φ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-\varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2dc82a45504067381e9475683c53e89ba788ba3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.523ex; height:2.676ex;" alt="{\displaystyle 1-\varphi }"></span> ดังนั้น </p> <dl><dd><table> <tbody><tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{n+1}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>φ<!-- φ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ^{n+1}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af44a16a549d27b9e5b19f8863dbcc41b2c871f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.226ex; height:3.176ex;" alt="{\displaystyle \varphi ^{n+1}\,}"></span></td> <td>= <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{n}+\varphi ^{n-1}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>φ<!-- φ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>φ<!-- φ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ^{n}+\varphi ^{n-1}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66b408d19e3e93f89d4e3ff1d41c91b4f79273ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.805ex; height:3.176ex;" alt="{\displaystyle \varphi ^{n}+\varphi ^{n-1}\,}"></span> และ </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-\varphi )^{n+1}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>φ<!-- φ --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1-\varphi )^{n+1}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ca4cf40c4fa14c039e21b91c83ac570af9e77a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.038ex; height:3.176ex;" alt="{\displaystyle (1-\varphi )^{n+1}\,}"></span></td> <td>= <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-\varphi )^{n}+(1-\varphi )^{n-1}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>φ<!-- φ --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>φ<!-- φ --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1-\varphi )^{n}+(1-\varphi )^{n-1}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4c2f94ace35ff756d02a8cd20bbbd55089ff3bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.43ex; height:3.176ex;" alt="{\displaystyle (1-\varphi )^{n}+(1-\varphi )^{n-1}\,}"></span> </td></tr></tbody></table></dd></dl> <p>พิจารณาฟังก์ชัน </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{a,b}(n)=a\varphi ^{n}+b(1-\varphi )^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <msup> <mi>φ<!-- φ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>φ<!-- φ --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{a,b}(n)=a\varphi ^{n}+b(1-\varphi )^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d697c3d5cbc6cceda7809785d55ca650d3c53c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.419ex; height:3.009ex;" alt="{\displaystyle F_{a,b}(n)=a\varphi ^{n}+b(1-\varphi )^{n}}"></span> เมื่อ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> และ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> เป็นจำนวนจริงใดๆ</dd></dl> <p>เราได้ว่าฟังก์ชันเหล่านี้สอดคล้องกับความสัมพันธ์เวียนเกิดที่ใช้นิยมเลขฟีโบนัชชี </p> <dl><dd><table> <tbody><tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{a,b}(n+1)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{a,b}(n+1)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3771439877b0faaa982c9302d94f1b6b2cfc3a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.353ex; height:3.009ex;" alt="{\displaystyle F_{a,b}(n+1)\,}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =a\varphi ^{n+1}+b(1-\varphi )^{n+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>a</mi> <msup> <mi>φ<!-- φ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>φ<!-- φ --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =a\varphi ^{n+1}+b(1-\varphi )^{n+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3da203b7547a2683c5ae918614084a074e36efc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.011ex; height:3.176ex;" alt="{\displaystyle =a\varphi ^{n+1}+b(1-\varphi )^{n+1}}"></span> </td></tr> <tr> <td></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =a(\varphi ^{n}+\varphi ^{n-1})+b((1-\varphi )^{n}+(1-\varphi )^{n-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <msup> <mi>φ<!-- φ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>φ<!-- φ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>φ<!-- φ --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>φ<!-- φ --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =a(\varphi ^{n}+\varphi ^{n-1})+b((1-\varphi )^{n}+(1-\varphi )^{n-1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d183ae5f170f9da9a5bfb2ab5fbe43257a20352f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.6ex; height:3.176ex;" alt="{\displaystyle =a(\varphi ^{n}+\varphi ^{n-1})+b((1-\varphi )^{n}+(1-\varphi )^{n-1})}"></span> </td></tr> <tr> <td></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =a{\varphi ^{n}+b(1-\varphi )^{n}}+a{\varphi ^{n-1}+b(1-\varphi )^{n-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>φ<!-- φ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>φ<!-- φ --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mo>+</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>φ<!-- φ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>φ<!-- φ --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =a{\varphi ^{n}+b(1-\varphi )^{n}}+a{\varphi ^{n-1}+b(1-\varphi )^{n-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/754eb766938e090e64ae0bb72329f45403aa3f4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.209ex; height:3.176ex;" alt="{\displaystyle =a{\varphi ^{n}+b(1-\varphi )^{n}}+a{\varphi ^{n-1}+b(1-\varphi )^{n-1}}}"></span> </td></tr> <tr> <td></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =F_{a,b}(n)+F_{a,b}(n-1)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =F_{a,b}(n)+F_{a,b}(n-1)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83f56c61bb1c55c52da757b78f94bdd6c0bf9432" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.61ex; height:3.009ex;" alt="{\displaystyle =F_{a,b}(n)+F_{a,b}(n-1)\,}"></span> </td></tr></tbody></table></dd></dl> <p>เลือก <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=1/{\sqrt {5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=1/{\sqrt {5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c40a7e349bb62a1965219032aefe35d13b72c361" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.752ex; height:3.009ex;" alt="{\displaystyle a=1/{\sqrt {5}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=-1/{\sqrt {5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mo>−<!-- − --></mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=-1/{\sqrt {5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7874e32c7b89dbddfed2e1776ff4c11ba1b7ce5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.327ex; height:3.009ex;" alt="{\displaystyle b=-1/{\sqrt {5}}}"></span> เราได้ว่า </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{a,b}(0)={\frac {1}{\sqrt {5}}}-{\frac {1}{\sqrt {5}}}=0=F(0)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>5</mn> </msqrt> </mfrac> </mrow> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>5</mn> </msqrt> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{a,b}(0)={\frac {1}{\sqrt {5}}}-{\frac {1}{\sqrt {5}}}=0=F(0)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/081888ecd37fef77abdabc2b7357648372eaa250" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; margin-right: -0.387ex; width:32.998ex; height:6.176ex;" alt="{\displaystyle F_{a,b}(0)={\frac {1}{\sqrt {5}}}-{\frac {1}{\sqrt {5}}}=0=F(0)\,\!}"></span></dd></dl> <p>และ </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{a,b}(1)={\frac {\varphi }{\sqrt {5}}}-{\frac {(1-\varphi )}{\sqrt {5}}}={\frac {-1+2\varphi }{\sqrt {5}}}={\frac {-1+(1+{\sqrt {5}})}{\sqrt {5}}}=1=F(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>φ<!-- φ --></mi> <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<!-- φ --></mi> <mo stretchy="false">)</mo> </mrow> <msqrt> <mn>5</mn> </msqrt> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−<!-- − --></mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mi>φ<!-- φ --></mi> </mrow> <msqrt> <mn>5</mn> </msqrt> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−<!-- − --></mo> <mn>1</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> <msqrt> <mn>5</mn> </msqrt> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{a,b}(1)={\frac {\varphi }{\sqrt {5}}}-{\frac {(1-\varphi )}{\sqrt {5}}}={\frac {-1+2\varphi }{\sqrt {5}}}={\frac {-1+(1+{\sqrt {5}})}{\sqrt {5}}}=1=F(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c398e33df0179dec152da0aeffe8dba2b90687a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:67.929ex; height:7.009ex;" alt="{\displaystyle F_{a,b}(1)={\frac {\varphi }{\sqrt {5}}}-{\frac {(1-\varphi )}{\sqrt {5}}}={\frac {-1+2\varphi }{\sqrt {5}}}={\frac {-1+(1+{\sqrt {5}})}{\sqrt {5}}}=1=F(1)}"></span></dd></dl> <p>เราสามารถใช้ข้อความนี้เป็นขั้นฐานของ<a href="/w/index.php?title=%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%9E%E0%B8%B4%E0%B8%AA%E0%B8%B9%E0%B8%88%E0%B8%99%E0%B9%8C%E0%B9%81%E0%B8%9A%E0%B8%9A%E0%B8%AD%E0%B8%B8%E0%B8%9B%E0%B8%99%E0%B8%B1%E0%B8%A2%E0%B9%80%E0%B8%8A%E0%B8%B4%E0%B8%87%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C&action=edit&redlink=1" class="new" title="การพิสูจน์แบบอุปนัยเชิงคณิตศาสตร์ (ไม่มีหน้านี้)">การพิสูจน์แบบอุปนัยเชิงคณิตศาสตร์</a>ของข้อความ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{a,b}(n)=F(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{a,b}(n)=F(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ef7e7190e2c1c365337e15fe59e6eb47ce2efa3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.006ex; height:3.009ex;" alt="{\displaystyle F_{a,b}(n)=F(n)}"></span> และใช้เอกลักษณ์ของ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{a,b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{a,b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/121edc09ab04806710b05d76bc8dc5c903262c10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.759ex; height:2.843ex;" alt="{\displaystyle F_{a,b}}"></span> พิสูจน์ขั้นอุปนัยได้ เราจึงสามารถสรุปว่า </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(n)={{\varphi ^{n}-(1-\varphi )^{n}} \over {\sqrt {5}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>φ<!-- φ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−<!-- − --></mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>φ<!-- φ --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(n)={{\varphi ^{n}-(1-\varphi )^{n}} \over {\sqrt {5}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96417463376f55d260d7cd5a01fb8ee860b97fc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:23.009ex; height:6.676ex;" alt="{\displaystyle F(n)={{\varphi ^{n}-(1-\varphi )^{n}} \over {\sqrt {5}}}}"></span> สำหรับจำนวนเต็มที่ไม่เป็นลบ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> ทุกตัว</dd></dl> <p>เนื่องจาก <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |1-\varphi |^{n}/{\sqrt {5}}<1/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo>−<!-- − --></mo> <mi>φ<!-- φ --></mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo><</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |1-\varphi |^{n}/{\sqrt {5}}<1/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ddabf225af0093001ca900d0081d2ab54a7425e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.882ex; height:3.009ex;" alt="{\displaystyle |1-\varphi |^{n}/{\sqrt {5}}<1/2}"></span> สำหรับทุกๆ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>0\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>0</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>0\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62847c31704900f090986e3ce1caa2185fc9d598" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:6.043ex; height:2.176ex;" alt="{\displaystyle n>0\,\!}"></span> เราจึงได้ว่า <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a533a2589206ed698a0a63941eaf903b4698759" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:3.1ex; height:2.509ex;" alt="{\displaystyle F_{n}\,\!}"></span> จึงเป็นจำนวนเต็มที่ใกล้ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{n}/{\sqrt {5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>φ<!-- φ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ^{n}/{\sqrt {5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e44feeb18cba5ae7d9cef89d5bc2c4b016b71161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.999ex; height:3.009ex;" alt="{\displaystyle \varphi ^{n}/{\sqrt {5}}}"></span> ที่สุด หรือเขียนเป็นประโยคสัญลักษณ์โดยใช้<a href="/wiki/%E0%B8%9F%E0%B8%B1%E0%B8%87%E0%B8%81%E0%B9%8C%E0%B8%8A%E0%B8%B1%E0%B8%99%E0%B8%9E%E0%B8%B7%E0%B9%89%E0%B8%99" class="mw-redirect" title="ฟังก์ชันพื้น">ฟังก์ชันพื้น</a> (floor function) ได้ว่า </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(n)={\bigg \lfloor }{\frac {\varphi ^{n}}{\sqrt {5}}}+{\frac {1}{2}}{\bigg \rfloor }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">⌊</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>φ<!-- 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> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">⌋</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(n)={\bigg \lfloor }{\frac {\varphi ^{n}}{\sqrt {5}}}+{\frac {1}{2}}{\bigg \rfloor }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a45cab78b4fef898830a27b72ca61c782c17c5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.527ex; height:6.509ex;" alt="{\displaystyle F(n)={\bigg \lfloor }{\frac {\varphi ^{n}}{\sqrt {5}}}+{\frac {1}{2}}{\bigg \rfloor }}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="ความสัมพันธ์กับอัตราส่วนทอง"><span id=".E0.B8.84.E0.B8.A7.E0.B8.B2.E0.B8.A1.E0.B8.AA.E0.B8.B1.E0.B8.A1.E0.B8.9E.E0.B8.B1.E0.B8.99.E0.B8.98.E0.B9.8C.E0.B8.81.E0.B8.B1.E0.B8.9A.E0.B8.AD.E0.B8.B1.E0.B8.95.E0.B8.A3.E0.B8.B2.E0.B8.AA.E0.B9.88.E0.B8.A7.E0.B8.99.E0.B8.97.E0.B8.AD.E0.B8.87"></span>ความสัมพันธ์กับอัตราส่วนทอง</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5&action=edit&section=2" title="แก้ไขส่วน: ความสัมพันธ์กับอัตราส่วนทอง"><span>แก้</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%E0%B9%82%E0%B8%A2%E0%B8%AE%E0%B8%B1%E0%B8%99%E0%B9%80%E0%B8%99%E0%B8%AA_%E0%B9%80%E0%B8%84%E0%B8%9B%E0%B9%80%E0%B8%A5%E0%B8%AD%E0%B8%A3%E0%B9%8C" class="mw-redirect" title="โยฮันเนส เคปเลอร์">โยฮันเนส เคปเลอร์</a> ค้นพบว่าอัตราส่วนของจำนวนฟีโบนัชชีที่ติดกันลู่เข้าสู่<a href="/wiki/%E0%B8%AD%E0%B8%B1%E0%B8%95%E0%B8%A3%E0%B8%B2%E0%B8%AA%E0%B9%88%E0%B8%A7%E0%B8%99%E0%B8%97%E0%B8%AD%E0%B8%87" title="อัตราส่วนทอง">อัตราส่วนทอง</a> กล่าวคือ </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {F(n+1)}{F(n)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {F(n+1)}{F(n)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb116de38ed7c6e73adfcbd56ea794944beb48a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:9.784ex; height:6.509ex;" alt="{\displaystyle {\frac {F(n+1)}{F(n)}}}"></span> ลู่เข้าสู่<a href="/wiki/%E0%B8%AD%E0%B8%B1%E0%B8%95%E0%B8%A3%E0%B8%B2%E0%B8%AA%E0%B9%88%E0%B8%A7%E0%B8%99%E0%B8%97%E0%B8%AD%E0%B8%87" title="อัตราส่วนทอง">อัตราส่วนทอง</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ<!-- φ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span></dd></dl> <p><b>การพิสูจน์:</b> </p><p>สำหรับจำนวนจริง <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\neq 0,b\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≠<!-- ≠ --></mo> <mn>0</mn> <mo>,</mo> <mi>b</mi> <mo>≠<!-- ≠ --></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\neq 0,b\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1227a7c028ab4c1ae090739fc19f28a8394f3ec7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.783ex; height:2.676ex;" alt="{\displaystyle a\neq 0,b\neq 0}"></span> เราได้ว่า </p> <dl><dd><table> <tbody><tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }{\frac {F_{a,b}(n+1)}{F_{a,b}(n)}}}"> <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> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }{\frac {F_{a,b}(n+1)}{F_{a,b}(n)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d234ecf7b43b123dafdadd30ab10929a730e4904" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.462ex; height:6.509ex;" alt="{\displaystyle \lim _{n\to \infty }{\frac {F_{a,b}(n+1)}{F_{a,b}(n)}}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\lim _{n\to \infty }{\frac {a\varphi ^{n+1}-b(1-\varphi )^{n+1}}{a\varphi ^{n}-b(1-\varphi )^{n}}}}"> <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 class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <msup> <mi>φ<!-- φ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mi>b</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>φ<!-- φ --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mi>a</mi> <msup> <mi>φ<!-- φ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−<!-- − --></mo> <mi>b</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>φ<!-- φ --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\lim _{n\to \infty }{\frac {a\varphi ^{n+1}-b(1-\varphi )^{n+1}}{a\varphi ^{n}-b(1-\varphi )^{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c11c1e127895491d7309dedb34610c4811e3dd52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.507ex; height:6.676ex;" alt="{\displaystyle =\lim _{n\to \infty }{\frac {a\varphi ^{n+1}-b(1-\varphi )^{n+1}}{a\varphi ^{n}-b(1-\varphi )^{n}}}}"></span> </td></tr> <tr> <td></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\lim _{n\to \infty }{\frac {a\varphi -b(1-\varphi )({\frac {1-\varphi }{\varphi }})^{n}}{a-b({\frac {1-\varphi }{\varphi }})^{n}}}}"> <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 class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mi>φ<!-- φ --></mi> <mo>−<!-- − --></mo> <mi>b</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>φ<!-- φ --></mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <mi>φ<!-- φ --></mi> </mrow> <mi>φ<!-- φ --></mi> </mfrac> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mi>a</mi> <mo>−<!-- − --></mo> <mi>b</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <mi>φ<!-- φ --></mi> </mrow> <mi>φ<!-- φ --></mi> </mfrac> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\lim _{n\to \infty }{\frac {a\varphi -b(1-\varphi )({\frac {1-\varphi }{\varphi }})^{n}}{a-b({\frac {1-\varphi }{\varphi }})^{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/137a6362b92f24e4b6b29f7959b4f4411dd4a9c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:28.909ex; height:9.176ex;" alt="{\displaystyle =\lim _{n\to \infty }{\frac {a\varphi -b(1-\varphi )({\frac {1-\varphi }{\varphi }})^{n}}{a-b({\frac {1-\varphi }{\varphi }})^{n}}}}"></span> </td></tr> <tr> <td></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>φ<!-- φ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd146f6172fd7d3c32032c37449298a73b44c964" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.973ex; height:2.176ex;" alt="{\displaystyle =\varphi }"></span>, </td></tr></tbody></table></dd></dl> <p>เนื่องจาก <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|{\frac {1-\varphi }{\varphi }}\right|<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <mi>φ<!-- φ --></mi> </mrow> <mi>φ<!-- φ --></mi> </mfrac> </mrow> <mo>|</mo> </mrow> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{\frac {1-\varphi }{\varphi }}\right|<1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3985844665d6961603c4f7ecde8056e337f923d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.914ex; height:5.843ex;" alt="{\displaystyle \left|{\frac {1-\varphi }{\varphi }}\right|<1}"></span> ดังนั้น <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }({\frac {1-\varphi }{\varphi }})^{n}=0}"> <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> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−<!-- − --></mo> <mi>φ<!-- φ --></mi> </mrow> <mi>φ<!-- φ --></mi> </mfrac> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }({\frac {1-\varphi }{\varphi }})^{n}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a94571d24e411632ae062dc2bea5250e9f75146" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.92ex; height:5.843ex;" alt="{\displaystyle \lim _{n\to \infty }({\frac {1-\varphi }{\varphi }})^{n}=0}"></span> </p><p>เนื่องจากจำนวนฟีโบนัชชีคือ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{a,b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{a,b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/121edc09ab04806710b05d76bc8dc5c903262c10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.759ex; height:2.843ex;" alt="{\displaystyle F_{a,b}}"></span> เมื่อ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=1/{\sqrt {5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=1/{\sqrt {5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c40a7e349bb62a1965219032aefe35d13b72c361" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.752ex; height:3.009ex;" alt="{\displaystyle a=1/{\sqrt {5}}}"></span> และ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=-1/{\sqrt {5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mo>−<!-- − --></mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=-1/{\sqrt {5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7874e32c7b89dbddfed2e1776ff4c11ba1b7ce5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.327ex; height:3.009ex;" alt="{\displaystyle b=-1/{\sqrt {5}}}"></span> ลิมิตของอัตราส่วนของเลขฟีโบนัชชีที่ติดกันจึงสอดคล้องกับสมการข้างบนด้วย </p> <div class="mw-heading mw-heading2"><h2 id="รูปเมทริกซ์(Matrix)"><span id=".E0.B8.A3.E0.B8.B9.E0.B8.9B.E0.B9.80.E0.B8.A1.E0.B8.97.E0.B8.A3.E0.B8.B4.E0.B8.81.E0.B8.8B.E0.B9.8C.28Matrix.29"></span>รูปเมทริกซ์(Matrix)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5&action=edit&section=3" title="แก้ไขส่วน: รูปเมทริกซ์(Matrix)"><span>แก้</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>ระบบสมการความแตกต่างเชิงเส้นที่อธิบายลำดับฟีโบนัชชีได้คือ </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{F_{k+2} \choose F_{k+1}}&={\begin{pmatrix}1&1\\1&0\end{pmatrix}}{F_{k+1} \choose F_{k}}\\{\vec {F}}_{k+1}&=A{\vec {F}}_{k}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>A</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{F_{k+2} \choose F_{k+1}}&={\begin{pmatrix}1&1\\1&0\end{pmatrix}}{F_{k+1} \choose F_{k}}\\{\vec {F}}_{k+1}&=A{\vec {F}}_{k}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c20c408022e25f1f76196c75913b26156665fb07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:28.88ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}{F_{k+2} \choose F_{k+1}}&={\begin{pmatrix}1&1\\1&0\end{pmatrix}}{F_{k+1} \choose F_{k}}\\{\vec {F}}_{k+1}&=A{\vec {F}}_{k}\end{aligned}}}"></span></dd></dl> <p>และมีรูปปิดคือ </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}1&1\\1&0\end{pmatrix}}^{n}={\begin{pmatrix}F_{n+1}&F_{n}\\F_{n}&F_{n-1}\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1&1\\1&0\end{pmatrix}}^{n}={\begin{pmatrix}F_{n+1}&F_{n}\\F_{n}&F_{n-1}\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f90aae99d109a6d152d80d03d0353a5e849c560e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.227ex; margin-bottom: -0.278ex; width:29.906ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}1&1\\1&0\end{pmatrix}}^{n}={\begin{pmatrix}F_{n+1}&F_{n}\\F_{n}&F_{n-1}\end{pmatrix}}.}"></span></dd></dl> <p>ด้วยรูปปิดดังกล่าว การคำนวณค่าฟีโบนัชชีจึงสามารถคำนวณได้โดยใช้จำนวนการดำเนินการเลขคณิต O(log <i>n</i>) หรือใช้เวลา O(<i>M</i>(<i>n</i>) log(<i>n</i>)) โดยที่ <i>M</i>(<i>n</i>) คือเวลาในการคูณเลข <i>n</i> หลัก 2 ตัว<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> โดยใช้วิธี<a href="/w/index.php?title=%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%A2%E0%B8%81%E0%B8%81%E0%B8%B3%E0%B8%A5%E0%B8%B1%E0%B8%87%E0%B9%82%E0%B8%94%E0%B8%A2%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%A2%E0%B8%81%E0%B8%81%E0%B8%B3%E0%B8%A5%E0%B8%B1%E0%B8%87%E0%B8%AA%E0%B8%AD%E0%B8%87&action=edit&redlink=1" class="new" title="การยกกำลังโดยการยกกำลังสอง (ไม่มีหน้านี้)">ยกกำลังโดยการยกกำลังสอง</a> กล่าวคือ </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{n}={\begin{cases}x\,(x^{2})^{\frac {n-1}{2}},&{\mbox{if }}n{\mbox{ is odd}}\\(x^{2})^{\frac {n}{2}},&{\mbox{if }}n{\mbox{ is even}}.\end{cases}}}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>x</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> is odd</mtext> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </msup> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> is even</mtext> </mstyle> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{n}={\begin{cases}x\,(x^{2})^{\frac {n-1}{2}},&{\mbox{if }}n{\mbox{ is odd}}\\(x^{2})^{\frac {n}{2}},&{\mbox{if }}n{\mbox{ is even}}.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46fe9e68c70c04df4c3d22c469a57d4655b50539" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:32.015ex; height:7.843ex;" alt="{\displaystyle x^{n}={\begin{cases}x\,(x^{2})^{\frac {n-1}{2}},&{\mbox{if }}n{\mbox{ is odd}}\\(x^{2})^{\frac {n}{2}},&{\mbox{if }}n{\mbox{ is even}}.\end{cases}}}"></span></dd></dl> <p>เมื่อให้ <i>x</i> เป็นเมทริกซ์ จึงสามารถหาค่า <i>F</i><sub><i>n</i></sub> ได้ในเวลาที่กล่าวไว้แล้ว </p> <div class="mw-heading mw-heading2"><h2 id="ลำดับฟีโบนัชชีในธรรมชาติ"><span id=".E0.B8.A5.E0.B8.B3.E0.B8.94.E0.B8.B1.E0.B8.9A.E0.B8.9F.E0.B8.B5.E0.B9.82.E0.B8.9A.E0.B8.99.E0.B8.B1.E0.B8.8A.E0.B8.8A.E0.B8.B5.E0.B9.83.E0.B8.99.E0.B8.98.E0.B8.A3.E0.B8.A3.E0.B8.A1.E0.B8.8A.E0.B8.B2.E0.B8.95.E0.B8.B4"></span>ลำดับฟีโบนัชชีในธรรมชาติ</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5&action=edit&section=4" title="แก้ไขส่วน: ลำดับฟีโบนัชชีในธรรมชาติ"><span>แก้</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>สิ่งที่ปรากฏตามธรรมชาติมิได้มีแต่รูปร่างง่ายๆ เท่านั้น บางอย่างมีรูปร่างที่มีแบบแผนทางคณิตศาสตร์ที่ยุ่งยากขึ้นไปอีก ตัวอย่างที่น่าสนใจของธรรมชาติที่เป็นไปตามกฎเกณฑ์ของ คณิตศาสตร์ชั้นสูง ได้แก่ เส้นโค้งก้นหอย ซึ่งมีคุณสมบัติว่า ถ้าลากเส้นตรงจากจุดหลายของเกลียวข้างในสุดไปตัดกับเส้นโค้งแล้ว มุมที่เกิดจากเส้นตรงนั้นกับเส้นสัมผัสกับเส้นโค้ง ณ จุดตัดจะเท่ากันเสมอดังรูป มุม A = มุม B = มุม C เส้นโคังที่มีลักษณะเป็นก้นหอยจะพบได้ในหอยบางชนิด เช่น หอยทาก </p><p>นอกจากนี้ยังมีความโค้งของงาช้าง ความโค้งของเกสรดอกทานตะวัน ตาสับปะรดและตาลูกสน ก็มีลักษณะคล้ายส่วนของเส้นโค้งก้นหอยด้วย ยังมีเรื่องที่น่าสนใจในธรรมชาติที่เกี่ยวข้องกับคณิตศาสตร์อีก จากการศึกษาเส้นโค้งของตาลูกสน ตาสับปะรด และเกสรดอกทานตะวัน จะเห็นว่าเส้นโค้งที่หมุนตามเข็มนาฬิกาของตาลูกสนมีจำนวน 5 เส้น และหมุนทวนเข็มนาฬิกามีจำนวน 3 เส้น หรืออาจกล่าวได้ว่า จำนวนเส้นโค้งสองแบบมีอัตราส่วนเป็น 5 ต่อ 8 สำหรับตาสับปะรด เส้นโค้งตามเข็มนาฬิกาและทวนเข็มนาฬิกา มีอัตราส่วนเป็น 8 ต่อ 13 เส้นโค้งที่เกิดจากเกสรดอกทานตะวันตามเข็มนาฬิกา และทวนเข็มนาฬิกามีอัตราส่วนเป็น 21 ต่อ 34 ปรากฏการณ์นี้เป็นไปตามกฎเกณฑ์ของเลขฟีโบนัชชี </p> <div class="mw-heading mw-heading2"><h2 id="การนำไปใช้"><span 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href="/w/index.php?title=%E0%B8%97%E0%B8%A4%E0%B8%A9%E0%B8%8E%E0%B8%B5%E0%B8%9A%E0%B8%97%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%A1%E0%B8%B2%E0%B8%97%E0%B8%B4%E0%B8%A2%E0%B8%B2%E0%B9%80%E0%B8%8B%E0%B8%A7%E0%B8%B4%E0%B8%8A&action=edit&redlink=1" class="new" title="ทฤษฎีบทของมาทิยาเซวิช (ไม่มีหน้านี้)">ปัญหาข้อที่ 10 ของฮิลแบร์ท</a> </p><p>จำนวนเต็มทุกจำนวนสามารถเขียนอยู่ในรูปของผลบวกของจำนวนฟีโบนัชชีที่ไม่ติดกินได้เพียงแบบเดียวเท่านั้น ความจริงข้อนี้เป็นที่รู้จักกันในนาม<a href="/w/index.php?title=%E0%B8%97%E0%B8%A4%E0%B8%A9%E0%B8%8E%E0%B8%B5%E0%B8%9A%E0%B8%97%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B9%80%E0%B8%8B%E0%B8%84%E0%B9%80%E0%B8%84%E0%B8%99%E0%B8%94%E0%B8%AD%E0%B8%A3%E0%B9%8C%E0%B8%9F&action=edit&redlink=1" class="new" title="ทฤษฎีบทของเซคเคนดอร์ฟ (ไม่มีหน้านี้)">ทฤษฎีบทของเซคเคนดอร์ฟ</a> การเขียนจำนวนเต็มในรูปดังกล่าวเรียกว่า <i>การนำเสนอแบบเซคเคนดอร์ฟ</i> </p><p><a href="/w/index.php?title=%E0%B8%95%E0%B8%B1%E0%B8%A7%E0%B8%81%E0%B8%B3%E0%B9%80%E0%B8%99%E0%B8%B4%E0%B8%94%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%AA%E0%B8%B8%E0%B9%88%E0%B8%A1%E0%B9%80%E0%B8%97%E0%B8%B5%E0%B8%A2%E0%B8%A1&action=edit&redlink=1" class="new" title="ตัวกำเนิดจำนวนสุ่มเทียม (ไม่มีหน้านี้)">ตัวกำเนิดจำนวนสุ่มเทียม</a>บางตัวใช้จำนวนฟีโบนัชชีเป็นเครื่องมือในการสร้างเลขสุ่ม </p><p>จำนวนฟีโบนัชชีถูกใช้กำหนดความยาวของส่วนประกอบต่างๆ ของงานศิลปะ และถูกใช้ในการเทียบเสียงเครื่องดนตรี ผลงานเพลงที่มีความเกี่ยวข้องกับจำนวนฟีโบนัชชี ได้แก่ <i>เพลงสำหรับเครื่องสาย เครื่องประกอบจังหวะ และซีเลสตา</i> ของ <a href="/w/index.php?title=%E0%B9%80%E0%B8%9A%E0%B8%A5%E0%B8%B2_%E0%B8%9A%E0%B8%B2%E0%B8%97%E0%B9%87%E0%B8%AD%E0%B8%81&action=edit&redlink=1" class="new" title="เบลา บาท็อก (ไม่มีหน้านี้)">เบลา บาท็อก</a>, และเพลง<i><a href="/w/index.php?title=%E0%B9%81%E0%B8%A5%E0%B9%80%E0%B8%97%E0%B8%AD%E0%B8%A3%E0%B8%B2%E0%B8%97%E0%B8%B1%E0%B8%AA&action=edit&redlink=1" class="new" title="แลเทอราทัส (ไม่มีหน้านี้)">แลเทอราทัส</a></i> ของวง<a href="/wiki/%E0%B8%97%E0%B8%B9%E0%B8%A5_(%E0%B8%A7%E0%B8%87%E0%B8%94%E0%B8%99%E0%B8%95%E0%B8%A3%E0%B8%B5)" title="ทูล (วงดนตรี)">ทูล</a> ซึ่งมีจำนวนพยางค์ในวรรคของเนื้อร้องเท่ากับจำนวนฟีโบนัชชี ("Black/Then/White are/All I see/In my infancy/Red and yellow then came to be") </p> <div class="mw-heading mw-heading2"><h2 id="อ้างอิง"><span id=".E0.B8.AD.E0.B9.89.E0.B8.B2.E0.B8.87.E0.B8.AD.E0.B8.B4.E0.B8.87"></span>อ้างอิง</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5&action=edit&section=6" title="แก้ไขส่วน: อ้างอิง"><span>แก้</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Lucas p. 3</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r10205087">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:#d33}.mw-parser-output .cs1-visible-error{color:#d33}.mw-parser-output .cs1-maint{display:none;color:#3a3;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><cite id="CITEREFDijkstra1978" class="citation cs2"><a href="/w/index.php?title=Edsger_W._Dijkstra&action=edit&redlink=1" class="new" title="Edsger W. Dijkstra (ไม่มีหน้านี้)">Dijkstra, Edsger W.</a> (1978), <a rel="nofollow" class="external text" href="http://www.cs.utexas.edu/users/EWD/ewd06xx/EWD654.PDF"><i>In honour of Fibonacci</i></a> <span class="cs1-format">(PDF)</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=In+honour+of+Fibonacci&rft.date=1978&rft.aulast=Dijkstra&rft.aufirst=Edsger+W.&rft_id=http%3A%2F%2Fwww.cs.utexas.edu%2Fusers%2FEWD%2Fewd06xx%2FEWD654.PDF&rfr_id=info%3Asid%2Fth.wikipedia.org%3A%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5" class="Z3988"></span>.</span> </li> </ol></div></div> <style data-mw-deduplicate="TemplateStyles:r8938631">.mw-parser-output .refbegin{font-size:90%;margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{list-style-type:none;margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li,.mw-parser-output .refbegin-hanging-indents>dl>dd{margin-left:0;padding-left:3.2em;text-indent:-3.2em;list-style:none}.mw-parser-output .refbegin-100{font-size:100%}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r10205087"><cite id="CITEREFBall2003" class="citation book cs1">Ball, Keith M. (2003). "Chapter 8: Fibonacci's Rabbits Revisited". <i>Strange Curves, Counting Rabbits, and Other Mathematical Explorations</i>. <a href="/w/index.php?title=Princeton_University_Press&action=edit&redlink=1" class="new" title="Princeton University Press (ไม่มีหน้านี้)">Princeton University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B9%81%E0%B8%AB%E0%B8%A5%E0%B9%88%E0%B8%87%E0%B8%AB%E0%B8%99%E0%B8%B1%E0%B8%87%E0%B8%AA%E0%B8%B7%E0%B8%AD/0691113211" title="พิเศษ:แหล่งหนังสือ/0691113211"><bdi>0691113211</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+8%3A+Fibonacci%27s+Rabbits+Revisited&rft.btitle=Strange+Curves%2C+Counting+Rabbits%2C+and+Other+Mathematical+Explorations&rft.pub=Princeton+University+Press&rft.date=2003&rft.isbn=0691113211&rft.aulast=Ball&rft.aufirst=Keith+M.&rfr_id=info%3Asid%2Fth.wikipedia.org%3A%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r10205087"><cite id="CITEREFLucas1891" class="citation book cs1">Lucas, Édouard (1891). <i>Théorie des nombres</i>. Vol. 1. Gauthier-Villars.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Th%C3%A9orie+des+nombres&rft.pub=Gauthier-Villars&rft.date=1891&rft.aulast=Lucas&rft.aufirst=%C3%89douard&rfr_id=info%3Asid%2Fth.wikipedia.org%3A%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%9F%E0%B8%B5%E0%B9%82%E0%B8%9A%E0%B8%99%E0%B8%B1%E0%B8%8A%E0%B8%8A%E0%B8%B5" class="Z3988"></span></li></ul> </div> <ul><li>Arakelian, Hrant (2014), <i>Mathematics and History of the Golden Section</i>. Logos, 404 p. <a href="/wiki/%E0%B8%9E%E0%B8%B4%E0%B9%80%E0%B8%A8%E0%B8%A9:%E0%B9%81%E0%B8%AB%E0%B8%A5%E0%B9%88%E0%B8%87%E0%B8%AB%E0%B8%99%E0%B8%B1%E0%B8%87%E0%B8%AA%E0%B8%B7%E0%B8%AD/9785987046630" class="internal mw-magiclink-isbn">ISBN 978-5-98704-663-0</a>, (rus.)</li></ul> <!-- NewPP limit report Parsed by mw‐api‐int.codfw.main‐6c94b8d879‐h8xwv Cached time: 20241114004051 Cache expiry: 2592000 Reduced expiry: false Complications: [vary‐revision‐sha1, show‐toc] CPU time usage: 0.323 seconds Real time usage: 0.608 seconds Preprocessor visited node count: 535/1000000 Post‐expand include size: 6174/2097152 bytes Template argument size: 302/2097152 bytes Highest expansion depth: 8/100 Expensive parser function count: 0/500 Unstrip recursion depth: 1/20 Unstrip post‐expand size: 9059/5000000 bytes Lua time usage: 0.199/10.000 seconds Lua memory usage: 17138018/52428800 bytes Number of Wikibase entities loaded: 0/400 --> <!-- Transclusion expansion time report (%,ms,calls,template) 100.00% 288.320 1 -total 56.80% 163.768 2 แม่แบบ:Langx 32.12% 92.609 1 แม่แบบ:รายการอ้างอิง 27.51% 79.330 1 แม่แบบ:Citation 3.64% 10.505 2 แม่แบบ:Cite_book 3.44% 9.920 1 แม่แบบ:เริ่มอ้างอิง 1.61% 4.641 1 แม่แบบ:OEIS 1.09% 3.139 1 แม่แบบ:Main_other 1.01% 2.922 1 แม่แบบ:Nowrap 0.63% 1.816 1 แม่แบบ:จบอ้างอิง --> <!-- Saved in parser cache with key thwiki:pcache:idhash:36023-0!canonical and timestamp 20241114004051 and revision id 11910667. 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