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五胞体数 - Wikipedia
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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="サイト"> <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="目次" 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">目次</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">サイドバーに移動</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">非表示</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">ページ先頭</div> </a> </li> <li id="toc-関連項目" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#関連項目"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>関連項目</span> </div> </a> <ul id="toc-関連項目-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-外部リンク" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#外部リンク"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>外部リンク</span> </div> </a> <ul id="toc-外部リンク-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="目次" 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="目次の表示・非表示を切り替え" > <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">目次の表示・非表示を切り替え</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">五胞体数</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="特定の記事の別の言語版に移動します。 利用可能な言語10件" > <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-10" 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">10の言語版</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_pentat%C3%B2pic" title="カタロニア語: Nombre pentatòpic" lang="ca" hreflang="ca" data-title="Nombre pentatòpic" data-language-autonym="Català" data-language-local-name="カタロニア語" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Pentatopzahl" title="ドイツ語: Pentatopzahl" lang="de" hreflang="de" data-title="Pentatopzahl" data-language-autonym="Deutsch" data-language-local-name="ドイツ語" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Pentatope_number" title="英語: Pentatope number" lang="en" hreflang="en" data-title="Pentatope number" data-language-autonym="English" data-language-local-name="英語" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbaki_pentatopiko" title="バスク語: Zenbaki pentatopiko" lang="eu" hreflang="eu" data-title="Zenbaki pentatopiko" data-language-autonym="Euskara" data-language-local-name="バスク語" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_pentatopique" title="フランス語: Nombre pentatopique" lang="fr" hreflang="fr" data-title="Nombre pentatopique" data-language-autonym="Français" data-language-local-name="フランス語" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_pentatopico" title="イタリア語: Numero pentatopico" lang="it" hreflang="it" data-title="Numero pentatopico" data-language-autonym="Italiano" data-language-local-name="イタリア語" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Num%C4%83r_pentatopic" title="ルーマニア語: Număr pentatopic" lang="ro" hreflang="ro" data-title="Număr pentatopic" data-language-autonym="Română" data-language-local-name="ルーマニア語" 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%9F%D0%B5%D0%BD%D1%82%D0%B0%D1%82%D0%BE%D0%BF%D0%BD%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="ロシア語: Пентатопное число" lang="ru" hreflang="ru" data-title="Пентатопное число" data-language-autonym="Русский" data-language-local-name="ロシア語" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Pentatoptal" title="スウェーデン語: Pentatoptal" lang="sv" hreflang="sv" data-title="Pentatoptal" data-language-autonym="Svenska" data-language-local-name="スウェーデン語" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%BA%94%E8%83%9E%E9%AB%94%E6%95%B8" title="中国語: 五胞體數" lang="zh" hreflang="zh" data-title="五胞體數" data-language-autonym="中文" data-language-local-name="中国語" class="interlanguage-link-target"><span>中文</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/Q1403502#sitelinks-wikipedia" title="言語間リンクを編集" class="wbc-editpage">リンクを編集</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="名前空間"> <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/%E4%BA%94%E8%83%9E%E4%BD%93%E6%95%B0" title="本文を閲覧 [c]" accesskey="c"><span>ページ</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E3%83%8E%E3%83%BC%E3%83%88:%E4%BA%94%E8%83%9E%E4%BD%93%E6%95%B0&action=edit&redlink=1" rel="discussion" class="new" title="「本文ページについての議論」 (存在しないページ) [t]" accesskey="t"><span>ノート</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="別の言語に切り替える" > <label id="vector-variants-dropdown-label" for="vector-variants-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">日本語</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet 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="表示"> <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/%E4%BA%94%E8%83%9E%E4%BD%93%E6%95%B0"><span>閲覧</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E4%BA%94%E8%83%9E%E4%BD%93%E6%95%B0&action=edit" title="このページのソースコードを編集する [e]" accesskey="e"><span>編集</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E4%BA%94%E8%83%9E%E4%BD%93%E6%95%B0&action=history" title="このページの過去の版 [h]" accesskey="h"><span>履歴表示</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="ページツール"> <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="ツール" > <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">ツール</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">ツール</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">サイドバーに移動</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">非表示</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="その他の操作" > <div class="vector-menu-heading"> 操作 </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/%E4%BA%94%E8%83%9E%E4%BD%93%E6%95%B0"><span>閲覧</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E4%BA%94%E8%83%9E%E4%BD%93%E6%95%B0&action=edit" title="このページのソースコードを編集する [e]" accesskey="e"><span>編集</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E4%BA%94%E8%83%9E%E4%BD%93%E6%95%B0&action=history"><span>履歴表示</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> 全般 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%E7%89%B9%E5%88%A5:%E3%83%AA%E3%83%B3%E3%82%AF%E5%85%83/%E4%BA%94%E8%83%9E%E4%BD%93%E6%95%B0" title="ここにリンクしている全ウィキページの一覧 [j]" accesskey="j"><span>リンク元</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/%E7%89%B9%E5%88%A5:%E9%96%A2%E9%80%A3%E3%83%9A%E3%83%BC%E3%82%B8%E3%81%AE%E6%9B%B4%E6%96%B0%E7%8A%B6%E6%B3%81/%E4%BA%94%E8%83%9E%E4%BD%93%E6%95%B0" rel="nofollow" title="このページからリンクしているページの最近の更新 [k]" accesskey="k"><span>関連ページの更新状況</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:%E3%83%95%E3%82%A1%E3%82%A4%E3%83%AB%E3%81%AE%E3%82%A2%E3%83%83%E3%83%97%E3%83%AD%E3%83%BC%E3%83%89" title="ファイルをアップロードする [u]" accesskey="u"><span>ファイルをアップロード</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/%E7%89%B9%E5%88%A5:%E7%89%B9%E5%88%A5%E3%83%9A%E3%83%BC%E3%82%B8%E4%B8%80%E8%A6%A7" title="特別ページの一覧 [q]" accesskey="q"><span>特別ページ</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%E4%BA%94%E8%83%9E%E4%BD%93%E6%95%B0&oldid=87485979" title="このページのこの版への固定リンク"><span>この版への固定リンク</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%E4%BA%94%E8%83%9E%E4%BD%93%E6%95%B0&action=info" title="このページについての詳細情報"><span>ページ情報</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=%E7%89%B9%E5%88%A5:%E3%81%93%E3%81%AE%E3%83%9A%E3%83%BC%E3%82%B8%E3%82%92%E5%BC%95%E7%94%A8&page=%E4%BA%94%E8%83%9E%E4%BD%93%E6%95%B0&id=87485979&wpFormIdentifier=titleform" title="このページの引用方法"><span>このページを引用</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=%E7%89%B9%E5%88%A5:UrlShortener&url=https%3A%2F%2Fja.wikipedia.org%2Fwiki%2F%25E4%25BA%2594%25E8%2583%259E%25E4%25BD%2593%25E6%2595%25B0"><span>短縮URLを取得する</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=%E7%89%B9%E5%88%A5:QrCode&url=https%3A%2F%2Fja.wikipedia.org%2Fwiki%2F%25E4%25BA%2594%25E8%2583%259E%25E4%25BD%2593%25E6%2595%25B0"><span>QRコードをダウンロード</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"> 印刷/書き出し </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=%E7%89%B9%E5%88%A5:%E3%83%96%E3%83%83%E3%82%AF&bookcmd=book_creator&referer=%E4%BA%94%E8%83%9E%E4%BD%93%E6%95%B0"><span>ブックの新規作成</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=%E7%89%B9%E5%88%A5:DownloadAsPdf&page=%E4%BA%94%E8%83%9E%E4%BD%93%E6%95%B0&action=show-download-screen"><span>PDF 形式でダウンロード</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=%E4%BA%94%E8%83%9E%E4%BD%93%E6%95%B0&printable=yes" title="このページの印刷用ページ [p]" accesskey="p"><span>印刷用バージョン</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"> 他のプロジェクト </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1403502" title="関連付けられたデータリポジトリ項目へのリンク [g]" accesskey="g"><span>ウィキデータ項目</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="ページツール"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="表示"> <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">表示</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">サイドバーに移動</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">非表示</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">出典: フリー百科事典『ウィキペディア（Wikipedia）』</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="ja" dir="ltr"><figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/%E3%83%95%E3%82%A1%E3%82%A4%E3%83%AB:Pentatope_of_70_spheres_animation.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/0/0f/Pentatope_of_70_spheres_animation.gif" decoding="async" width="160" height="120" class="mw-file-element" data-file-width="160" data-file-height="120" /></a><figcaption>n=5のときの五胞体数である70個の<a href="/wiki/%E7%90%83" class="mw-disambig" title="球">球</a>。最初の5つの<a href="/wiki/%E4%B8%89%E8%A7%92%E9%8C%90%E6%95%B0" title="三角錐数">三角錐数</a>に等しい個数の球を順番に「3次元的な段」として重ねたものである</figcaption></figure> <p><b>五胞体数</b>（ごほうたいすう、<a href="/wiki/%E8%8B%B1%E8%AA%9E" title="英語">英</a>: <span lang="en">pentatope number</span>）は、点を右図のように<a href="/wiki/%E4%BA%94%E8%83%9E%E4%BD%93" title="五胞体">五胞体</a>の形に並べたとき、そこに含まれる点の総数にあたる<a href="/wiki/%E8%87%AA%E7%84%B6%E6%95%B0" title="自然数">自然数</a>である。<a href="/wiki/%E4%B8%89%E8%A7%92%E9%8C%90%E6%95%B0" title="三角錐数">三角錐数</a>を 1 から小さい順に加えた数と定義してもよい。例:15(=1 + 4 + 10)、70(=1 + 4 + 10 + 20 + 35) </p><p><i>n</i> 番目の五胞体数 P<sub><i>n</i></sub> は 1 から <i>n</i> 番目までの三角錐数 <style data-mw-deduplicate="TemplateStyles:r89142261">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span role="math" class="sfrac tion"><span class="num"><i>n</i>(<i>n</i> + 1)(<i>n</i> + 2)</span><span class="sr-only">/</span><span class="den">6</span></span> までの和に等しいので </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}P_{n}&=\sum _{k=1}^{n}{\frac {k(k+1)(k+2)}{6}}\\&={\frac {n(n+1)(n+2)(n+3)}{24}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mn>6</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <mn>24</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P_{n}&=\sum _{k=1}^{n}{\frac {k(k+1)(k+2)}{6}}\\&={\frac {n(n+1)(n+2)(n+3)}{24}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96386db5880c1fa863e5ce12f8669a9e3e4c46fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:30.412ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}P_{n}&=\sum _{k=1}^{n}{\frac {k(k+1)(k+2)}{6}}\\&={\frac {n(n+1)(n+2)(n+3)}{24}}\end{aligned}}}"></span></dd></dl> <p>また<a href="/wiki/%E7%B5%84%E3%81%BF%E5%90%88%E3%82%8F%E3%81%9B" class="mw-redirect" title="組み合わせ">組み合わせ</a>の記号を用いると <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 P_{n}={}_{n+3}{\rm {C}}_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>3</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{n}={}_{n+3}{\rm {C}}_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/896ae1580c4c3af49efb3e6e14c0895798a3385a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.861ex; height:2.509ex;" alt="{\displaystyle P_{n}={}_{n+3}{\rm {C}}_{4}}"></span> となる。 </p><p>五胞体数を小さい順に列記すると </p> <dl><dd><a href="/wiki/1" title="1">1</a>, <a href="/wiki/5" title="5">5</a>, <a href="/wiki/15" title="15">15</a>, <a href="/wiki/35" title="35">35</a>, <a href="/wiki/70" title="70">70</a>, <a href="/wiki/126" title="126">126</a>, <a href="/wiki/210" title="210">210</a>, <a href="/wiki/330" title="330">330</a>, <a href="/wiki/495" title="495">495</a>, <a href="/wiki/715" title="715">715</a>, <a href="/wiki/1001" title="1001">1001</a>, 1365, 1820, 2380, 3060, …（<a href="/wiki/%E3%82%AA%E3%83%B3%E3%83%A9%E3%82%A4%E3%83%B3%E6%95%B4%E6%95%B0%E5%88%97%E5%A4%A7%E8%BE%9E%E5%85%B8" title="オンライン整数列大辞典">オンライン整数列大辞典</a>の数列 <a rel="nofollow" class="external text" href="//oeis.org/A332">A332</a>）</dd></dl> <p>3つの連続する五胞体数のうち2つは<a href="/wiki/%E4%BA%94%E8%A7%92%E6%95%B0" title="五角数">五角数</a>である。なぜなら 3<i>n</i> − 2 番目の五胞体数は <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r89142261"><span role="math" class="sfrac tion"><span class="num">(3<i>n</i><sup>2</sup> − <i>n</i>)</span><span class="sr-only">/</span><span class="den">2</span></span> 番目の五角数であり、3<i>n</i> − 1 番目の五胞体数は <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r89142261"><span role="math" class="sfrac tion"><span class="num">(3<i>n</i><sup>2</sup> + <i>n</i>)</span><span class="sr-only">/</span><span class="den">2</span></span> 番目の五角数だからである。 </p><p><a href="/wiki/%E3%83%91%E3%82%B9%E3%82%AB%E3%83%AB%E3%81%AE%E4%B8%89%E8%A7%92%E5%BD%A2" title="パスカルの三角形">パスカルの三角形</a>では左上（または右上）から5列目の数が五胞体数にあたる。 </p><p>五胞体数の<a href="/wiki/%E9%80%86%E6%95%B0" title="逆数">逆数</a>の<a href="/wiki/%E7%B7%8F%E5%92%8C" title="総和">総和</a>は </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}\sum _{k=1}^{\infty }{\frac {1}{\frac {k(k+1)(k+2)(k+3)}{24}}}&=24\sum _{k=1}^{\infty }{\frac {1}{6}}\left({\frac {1}{k}}-{\frac {3}{k+1}}+{\frac {3}{k+2}}-{\frac {1}{k+3}}\right)\\&=4\left({\frac {1}{1}}-{\frac {3}{2}}+{\frac {1}{2}}+{\frac {1}{3}}\right)\\&={\frac {4}{3}}\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> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mfrac> <mrow> <mi>k</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <mn>24</mn> </mfrac> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>24</mn> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>3</mn> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>4</mn> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1</mn> </mfrac> </mrow> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sum _{k=1}^{\infty }{\frac {1}{\frac {k(k+1)(k+2)(k+3)}{24}}}&=24\sum _{k=1}^{\infty }{\frac {1}{6}}\left({\frac {1}{k}}-{\frac {3}{k+1}}+{\frac {3}{k+2}}-{\frac {1}{k+3}}\right)\\&=4\left({\frac {1}{1}}-{\frac {3}{2}}+{\frac {1}{2}}+{\frac {1}{3}}\right)\\&={\frac {4}{3}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23a35e9caecb0ef5d81c3fda31791e8016e3d36c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.972ex; margin-bottom: -0.199ex; width:63.81ex; height:19.509ex;" alt="{\displaystyle {\begin{aligned}\sum _{k=1}^{\infty }{\frac {1}{\frac {k(k+1)(k+2)(k+3)}{24}}}&=24\sum _{k=1}^{\infty }{\frac {1}{6}}\left({\frac {1}{k}}-{\frac {3}{k+1}}+{\frac {3}{k+2}}-{\frac {1}{k+3}}\right)\\&=4\left({\frac {1}{1}}-{\frac {3}{2}}+{\frac {1}{2}}+{\frac {1}{3}}\right)\\&={\frac {4}{3}}\end{aligned}}}"></span></dd></dl> <p>となる。 </p> <div class="mw-heading mw-heading2"><h2 id="関連項目"><span id=".E9.96.A2.E9.80.A3.E9.A0.85.E7.9B.AE"></span>関連項目</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E4%BA%94%E8%83%9E%E4%BD%93%E6%95%B0&action=edit&section=1" title="節を編集: 関連項目"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E5%9B%B3%E5%BD%A2%E6%95%B0" title="図形数">図形数</a></li> <li><a href="/wiki/%E4%BA%94%E8%83%9E%E4%BD%93" title="五胞体">五胞体</a></li> <li><a href="/wiki/%E4%B8%89%E8%A7%92%E9%8C%90%E6%95%B0" title="三角錐数">三角錐数</a></li> <li><a href="/wiki/%E4%BA%94%E8%A7%92%E6%95%B0" title="五角数">五角数</a></li> <li><a href="/wiki/%E3%83%91%E3%82%B9%E3%82%AB%E3%83%AB%E3%81%AE%E4%B8%89%E8%A7%92%E5%BD%A2" title="パスカルの三角形">パスカルの三角形</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="外部リンク"><span id=".E5.A4.96.E9.83.A8.E3.83.AA.E3.83.B3.E3.82.AF"></span>外部リンク</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E4%BA%94%E8%83%9E%E4%BD%93%E6%95%B0&action=edit&section=2" title="節を編集: 外部リンク"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Pentatope_Number"><style data-mw-deduplicate="TemplateStyles:r101121245">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation.cs-ja1 q,.mw-parser-output .citation.cs-ja2 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: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: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:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:var(--color-success,#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="CITEREFWeisstein" class="citation web cs1 cs1-prop-foreign-lang-source">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/PentatopeNumber.html">"Pentatope Number"</a>. <i>mathworld.wolfram.com</i> (英語).</cite><span 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