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role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="开关目录" > <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">开关目录</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">三角形數</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="前往另一种语言写成的文章。47种语言可用" > <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-47" 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">47种语言</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D9%85%D8%AB%D9%84%D8%AB%D9%8A" title="عدد مثلثي – 阿拉伯语" lang="ar" hreflang="ar" data-title="عدد مثلثي" data-language-autonym="العربية" data-language-local-name="阿拉伯语" 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/%C3%9C%C3%A7bucaq_%C9%99d%C9%99dl%C9%99ri" title="Üçbucaq ədədləri – 阿塞拜疆语" lang="az" hreflang="az" data-title="Üçbucaq ədədləri" data-language-autonym="Azərbaycanca" data-language-local-name="阿塞拜疆语" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D1%8A%D0%B3%D1%8A%D0%BB%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Триъгълно число – 保加利亚语" lang="bg" hreflang="bg" data-title="Триъгълно число" data-language-autonym="Български" data-language-local-name="保加利亚语" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A4%E0%A7%8D%E0%A6%B0%E0%A6%BF%E0%A6%95%E0%A7%8B%E0%A6%A3_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="ত্রিকোণ সংখ্যা – 孟加拉语" lang="bn" hreflang="bn" data-title="ত্রিকোণ সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="孟加拉语" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_triangular" title="Nombre triangular – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Nombre triangular" data-language-autonym="Català" data-language-local-name="加泰罗尼亚语" 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_%D8%B3%DB%8E%DA%AF%DB%86%D8%B4%DB%95%DB%8C%DB%8C" title="ژمارەی سێگۆشەیی – 中库尔德语" lang="ckb" hreflang="ckb" data-title="ژمارەی سێگۆشەیی" data-language-autonym="کوردی" data-language-local-name="中库尔德语" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Troj%C3%BAheln%C3%ADkov%C3%A9_%C4%8D%C3%ADslo" title="Trojúhelníkové číslo – 捷克语" lang="cs" hreflang="cs" data-title="Trojúhelníkové číslo" data-language-autonym="Čeština" data-language-local-name="捷克语" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Trekanttal" title="Trekanttal – 丹麦语" lang="da" hreflang="da" data-title="Trekanttal" data-language-autonym="Dansk" data-language-local-name="丹麦语" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Dreieckszahl" title="Dreieckszahl – 德语" lang="de" hreflang="de" data-title="Dreieckszahl" data-language-autonym="Deutsch" data-language-local-name="德语" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A4%CF%81%CE%AF%CE%B3%CF%89%CE%BD%CE%BF%CE%B9_%CE%91%CF%81%CE%B9%CE%B8%CE%BC%CE%BF%CE%AF" title="Τρίγωνοι Αριθμοί – 希腊语" lang="el" hreflang="el" data-title="Τρίγωνοι Αριθμοί" data-language-autonym="Ελληνικά" data-language-local-name="希腊语" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/N%C3%B9mer_triangol%C3%A8r" title="Nùmer triangolèr – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Nùmer triangolèr" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Triangular_number" title="Triangular number – 英语" lang="en" hreflang="en" data-title="Triangular number" data-language-autonym="English" data-language-local-name="英语" 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/Triangula_nombro" title="Triangula nombro – 世界语" lang="eo" hreflang="eo" data-title="Triangula nombro" data-language-autonym="Esperanto" data-language-local-name="世界语" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_triangular" title="Número triangular – 西班牙语" lang="es" hreflang="es" data-title="Número triangular" data-language-autonym="Español" data-language-local-name="西班牙语" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D9%85%D8%AB%D9%84%D8%AB%DB%8C" title="عدد مثلثی – 波斯语" lang="fa" hreflang="fa" data-title="عدد مثلثی" data-language-autonym="فارسی" data-language-local-name="波斯语" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kolmioluku" title="Kolmioluku – 芬兰语" lang="fi" hreflang="fi" data-title="Kolmioluku" data-language-autonym="Suomi" data-language-local-name="芬兰语" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_triangulaire" title="Nombre triangulaire – 法语" lang="fr" hreflang="fr" data-title="Nombre triangulaire" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Triihuktaal" title="Triihuktaal – 北弗里西亚语" lang="frr" hreflang="frr" data-title="Triihuktaal" data-language-autonym="Nordfriisk" data-language-local-name="北弗里西亚语" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/N%C3%BAmero_triangular" title="Número triangular – 加利西亚语" lang="gl" hreflang="gl" data-title="Número triangular" data-language-autonym="Galego" data-language-local-name="加利西亚语" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%9E%D7%A9%D7%95%D7%9C%D7%A9%D7%99" title="מספר משולשי – 希伯来语" lang="he" hreflang="he" data-title="מספר משולשי" data-language-autonym="עברית" data-language-local-name="希伯来语" 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%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%8B%E0%A4%A3_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="त्रिकोण संख्या – 印地语" lang="hi" hreflang="hi" data-title="त्रिकोण संख्या" data-language-autonym="हिन्दी" data-language-local-name="印地语" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/H%C3%A1romsz%C3%B6gsz%C3%A1mok" title="Háromszögszámok – 匈牙利语" lang="hu" hreflang="hu" data-title="Háromszögszámok" data-language-autonym="Magyar" data-language-local-name="匈牙利语" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bilangan_segitiga" title="Bilangan segitiga – 印度尼西亚语" lang="id" hreflang="id" data-title="Bilangan segitiga" data-language-autonym="Bahasa Indonesia" data-language-local-name="印度尼西亚语" 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/%C3%9Er%C3%ADhyrningstala" title="Þríhyrningstala – 冰岛语" lang="is" hreflang="is" data-title="Þríhyrningstala" data-language-autonym="Íslenska" data-language-local-name="冰岛语" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_triangolare" title="Numero triangolare – 意大利语" lang="it" hreflang="it" data-title="Numero triangolare" data-language-autonym="Italiano" data-language-local-name="意大利语" 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/%E4%B8%89%E8%A7%92%E6%95%B0" title="三角数 – 日语" lang="ja" hreflang="ja" data-title="三角数" data-language-autonym="日本語" data-language-local-name="日语" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%82%BC%EA%B0%81%EC%88%98" title="삼각수 – 韩语" lang="ko" hreflang="ko" data-title="삼각수" data-language-autonym="한국어" data-language-local-name="韩语" 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/Numerus_triangularis" title="Numerus triangularis – 拉丁语" lang="la" hreflang="la" data-title="Numerus triangularis" data-language-autonym="Latina" data-language-local-name="拉丁语" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Driehoeksgetal" title="Driehoeksgetal – 荷兰语" lang="nl" hreflang="nl" data-title="Driehoeksgetal" data-language-autonym="Nederlands" data-language-local-name="荷兰语" 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/Trekanttall" title="Trekanttall – 书面挪威语" lang="nb" hreflang="nb" data-title="Trekanttall" data-language-autonym="Norsk bokmål" data-language-local-name="书面挪威语" 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%A4%E0%A8%BF%E0%A8%95%E0%A9%8B%E0%A8%A3%E0%A9%80%E0%A8%86_%E0%A8%B8%E0%A9%B0%E0%A8%96%E0%A8%BF%E0%A8%86" title="ਤਿਕੋਣੀਆ ਸੰਖਿਆ – 旁遮普语" lang="pa" hreflang="pa" data-title="ਤਿਕੋਣੀਆ ਸੰਖਿਆ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="旁遮普语" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_tr%C3%B3jk%C4%85tne" title="Liczby trójkątne – 波兰语" lang="pl" hreflang="pl" data-title="Liczby trójkątne" data-language-autonym="Polski" data-language-local-name="波兰语" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero_triangular" title="Número triangular – 葡萄牙语" lang="pt" hreflang="pt" data-title="Número triangular" data-language-autonym="Português" data-language-local-name="葡萄牙语" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Num%C4%83r_triunghiular" title="Număr triunghiular – 罗马尼亚语" lang="ro" hreflang="ro" data-title="Număr triunghiular" data-language-autonym="Română" data-language-local-name="罗马尼亚语" 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%A2%D1%80%D0%B5%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Треугольное число – 俄语" lang="ru" hreflang="ru" data-title="Треугольное число" data-language-autonym="Русский" data-language-local-name="俄语" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Triangular_number" title="Triangular number – Simple English" lang="en-simple" hreflang="en-simple" data-title="Triangular number" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Trojuholn%C3%ADkov%C3%A9_%C4%8D%C3%ADslo" title="Trojuholníkové číslo – 斯洛伐克语" lang="sk" hreflang="sk" data-title="Trojuholníkové číslo" data-language-autonym="Slovenčina" data-language-local-name="斯洛伐克语" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Trikotni%C5%A1ko_%C5%A1tevilo" title="Trikotniško število – 斯洛文尼亚语" lang="sl" hreflang="sl" data-title="Trikotniško število" data-language-autonym="Slovenščina" data-language-local-name="斯洛文尼亚语" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D1%80%D0%BE%D1%83%D0%B3%D0%B0%D0%BE%D0%BD%D0%B8_%D0%B1%D1%80%D0%BE%D1%98" title="Троугаони број – 塞尔维亚语" lang="sr" hreflang="sr" data-title="Троугаони број" data-language-autonym="Српски / srpski" data-language-local-name="塞尔维亚语" 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/Triangeltal" title="Triangeltal – 瑞典语" lang="sv" hreflang="sv" data-title="Triangeltal" data-language-autonym="Svenska" data-language-local-name="瑞典语" 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%AE%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%8B%E0%AE%A3_%E0%AE%8E%E0%AE%A3%E0%AF%8D" title="முக்கோண எண் – 泰米尔语" lang="ta" hreflang="ta" data-title="முக்கோண எண்" data-language-autonym="தமிழ்" data-language-local-name="泰米尔语" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B8%AA%E0%B8%B2%E0%B8%A1%E0%B9%80%E0%B8%AB%E0%B8%A5%E0%B8%B5%E0%B9%88%E0%B8%A2%E0%B8%A1" title="จำนวนสามเหลี่ยม – 泰语" lang="th" hreflang="th" data-title="จำนวนสามเหลี่ยม" data-language-autonym="ไทย" data-language-local-name="泰语" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%9C%C3%A7gensel_say%C4%B1" title="Üçgensel sayı – 土耳其语" lang="tr" hreflang="tr" data-title="Üçgensel sayı" data-language-autonym="Türkçe" data-language-local-name="土耳其语" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%BA%D1%83%D1%82%D0%BD%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Трикутне число – 乌克兰语" lang="uk" hreflang="uk" data-title="Трикутне число" data-language-autonym="Українська" data-language-local-name="乌克兰语" 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/Uchburchak_sonlari" title="Uchburchak sonlari – 乌兹别克语" lang="uz" hreflang="uz" data-title="Uchburchak sonlari" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="乌兹别克语" 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/S%E1%BB%91_tam_gi%C3%A1c" title="Số tam giác – 越南语" lang="vi" hreflang="vi" data-title="Số tam giác" data-language-autonym="Tiếng Việt" data-language-local-name="越南语" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%BD%A2%E6%95%B8" title="三角形數 – 粤语" lang="yue" hreflang="yue" data-title="三角形數" data-language-autonym="粵語" data-language-local-name="粤语" 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/Q245102#sitelinks-wikipedia" title="编辑跨语言链接" 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href="https://zh.wikipedia.org/w/index.php?title=%E4%B8%89%E8%A7%92%E5%BD%A2%E6%95%B8&action=edit">改善这篇條目</a>，更進一步的信息可能會在<a href="/wiki/Talk:%E4%B8%89%E8%A7%92%E5%BD%A2%E6%95%B8" title="Talk:三角形數">討論頁</a>或<a href="/wiki/Wikipedia:%E6%89%A9%E5%85%85%E8%AF%B7%E6%B1%82" title="Wikipedia:扩充请求">扩充请求</a>中找到。请在擴充條目後將此模板移除。</small></span><span class="hide-when-compact"></span></div></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r85100532">.mw-parser-output .hatnote{font-size:small}.mw-parser-output div.hatnote{padding-left:2em;margin-bottom:0.8em;margin-top:0.8em}.mw-parser-output .hatnote-notice-img::after{content:"\202f \202f \202f \202f "}.mw-parser-output .hatnote-notice-img-small::after{content:"\202f \202f "}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}body.skin-minerva .mw-parser-output .hatnote-notice-img,body.skin-minerva .mw-parser-output .hatnote-notice-img-small{display:none}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable"><span class="noviewer hatnote-notice-img" typeof="mw:File"><a href="/wiki/Wikipedia:%E6%B6%88%E6%AD%A7%E4%B9%89" title="Wikipedia:消歧义"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Confusion_grey.svg/25px-Confusion_grey.svg.png" decoding="async" width="25" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Confusion_grey.svg/38px-Confusion_grey.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Confusion_grey.svg/50px-Confusion_grey.svg.png 2x" data-file-width="260" data-file-height="200" /></a></span><b>提示</b>：此条目的主题不是<b><a href="/wiki/%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B8" class="mw-redirect" title="三角函數">三角函數</a></b>。</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:First_six_triangular_numbers.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/First_six_triangular_numbers.svg/240px-First_six_triangular_numbers.svg.png" decoding="async" width="240" height="201" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/First_six_triangular_numbers.svg/360px-First_six_triangular_numbers.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/First_six_triangular_numbers.svg/480px-First_six_triangular_numbers.svg.png 2x" data-file-width="374" data-file-height="314" /></a><figcaption>三角形數</figcaption></figure> <p>一定数目的点或圆在等距離的排列下可以形成一个等邊三角形，這樣的數被稱為<b>三角形數</b>。比如10個點可以組成一个<a href="/wiki/%E7%AD%89%E8%BE%B9%E4%B8%89%E8%A7%92%E5%BD%A2" class="mw-redirect" title="等边三角形">等邊三角形</a>，因此10是一個三角形數： </p><p>頭30個三角形數是<a href="/wiki/1" title="1">1</a>, <a href="/wiki/3" title="3">3</a>, <a href="/wiki/6" title="6">6</a>, <a href="/wiki/10" title="10">10</a>, <a href="/wiki/15" title="15">15</a>, <a href="/wiki/21" title="21">21</a>, <a href="/wiki/28" title="28">28</a>, <a href="/wiki/36" title="36">36</a>, <a href="/wiki/45" title="45">45</a>, <a href="/wiki/55" title="55">55</a>, <a href="/wiki/66" title="66">66</a>, <a href="/wiki/78" title="78">78</a>, <a href="/wiki/91" title="91">91</a>, <a href="/wiki/105" title="105">105</a>, <a href="/wiki/120" title="120">120</a>, <a href="/wiki/136" title="136">136</a>, <a href="/wiki/153" title="153">153</a>, <a href="/wiki/171" title="171">171</a>, <a href="/wiki/190" title="190">190</a>, <a href="/wiki/210" title="210">210</a>, <a href="/wiki/231" title="231">231</a>, <a href="/wiki/253" title="253">253</a>, <a href="/wiki/276" title="276">276</a>, <a href="/wiki/300" title="300">300</a>, <a href="/wiki/325" title="325">325</a>, 351, 378, 406, 435, 465, ...（<a href="/wiki/%E6%95%B4%E6%95%B8%E6%95%B8%E5%88%97%E7%B7%9A%E4%B8%8A%E5%A4%A7%E5%85%A8" title="整數數列線上大全">OEIS</a>數列<a href="//oeis.org/A000217" class="extiw" title="oeis:A000217">A000217</a>）。 </p><p>三角数的二倍的平方根取整，是这个三角数的序数。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="性質"><span id=".E6.80.A7.E8.B3.AA"></span>性質</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E4%B8%89%E8%A7%92%E5%BD%A2%E6%95%B8&action=edit&section=1" title="编辑章节：性質"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>第n个三角形數的公式是<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 {n(n+1)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {n(n+1)}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30fbb88f3c74182de3357dd19fbd2eb3e60dffc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.438ex; height:5.676ex;" alt="{\displaystyle {\frac {n(n+1)}{2}}}"></span>。</li> <li>第n个三角形數是從1开始的n个<a href="/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%80%A6" title="1 + 2 + 3 + 4 + …">自然数的和</a>。</li> <li>所有大于3的三角形數都不是<a href="/wiki/%E8%B4%A8%E6%95%B0" title="质数">质数</a>。</li> <li><style data-mw-deduplicate="TemplateStyles:r83946278">.mw-parser-output .template-facttext{background-color:var(--background-color-neutral,#eaecf0);color:inherit;margin:-.3em 0;padding:.3em 0}</style><mark class="template-facttext" title="需要提供文献来源">除了0，1，<a href="/wiki/3" title="3">3</a>，<a href="/wiki/21" title="21">21</a>，<a href="/wiki/55" title="55">55</a>以外，三角形數不可能是<a href="/wiki/%E8%B2%BB%E6%B3%A2%E9%82%A3%E5%A5%91%E6%95%B8" class="mw-redirect" title="費波那契數">費波那契數</a>。</mark><sup class="noprint Template-Fact"><a href="/wiki/Wikipedia:%E5%88%97%E6%98%8E%E6%9D%A5%E6%BA%90" title="Wikipedia:列明来源"><span style="white-space: nowrap;" title="来源请求开始于2019年4月21日。">[來源請求]</span></a></sup></li> <li>开始的n个<a href="/wiki/%E7%AB%8B%E6%96%B9%E6%95%B0" class="mw-redirect" title="立方数">立方数</a>的和是第n个三角形數的<a href="/wiki/%E5%B9%B3%E6%96%B9" title="平方">平方</a>（举例：1 + 8 + 27 + 64 = 100 = 10<sup>2</sup>）</li> <li>所有三角形數的<a href="/wiki/%E5%80%92%E6%95%B0" title="倒数">倒数</a>之和是2。</li> <li>任何三角形數乘以8再加1是一个<a href="/wiki/%E5%B9%B3%E6%96%B9%E6%95%B0" title="平方数">平方数</a>。</li> <li>三角數的個位數字不可能是2、4、7、9，<a href="/wiki/%E6%95%B8%E5%AD%97%E6%A0%B9" class="mw-redirect" title="數字根">數字根</a>不可能是2、4、5、7、8。</li> <li>一部分三角形數（3、10、21、36、55、78……）可以用以下这个<a href="/wiki/%E5%85%AC%E5%BC%8F" title="公式">公式</a>来表示：<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*(2n+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∗</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n*(2n+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4ea0f288b7aab69ff6b8a503ed5fb8b886ee7ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.959ex; height:2.843ex;" alt="{\displaystyle n*(2n+1)}"></span>；而剩下的另一部分（1、6、15、28、45、66……）则可以用<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*(2n-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∗</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n*(2n-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c94bafc98e38827fb029f639795d9adebacf0626" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.959ex; height:2.843ex;" alt="{\displaystyle n*(2n-1)}"></span>来表示。</li> <li>一种检验<a href="/wiki/%E6%AD%A3%E6%95%B4%E6%95%B0" class="mw-redirect" title="正整数">正整数</a><i>x</i>是否三角形数的方法，是<a href="/wiki/%E8%AE%A1%E7%AE%97" title="计算">计算</a>： <dl><dd><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 n={\frac {{\sqrt {8x+1}}-1}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>8</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n={\frac {{\sqrt {8x+1}}-1}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f0834bcdf48f209d552c6c3e58c5b666efffe43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.41ex; height:5.843ex;" alt="{\displaystyle n={\frac {{\sqrt {8x+1}}-1}{2}}.}"></span></dd></dl></dd> <dd>如果<i>n</i>是<a href="/wiki/%E6%95%B4%E6%95%B0" title="整数">整数</a>，那么<i>x</i>就是第<i>n</i>个<b>三角形数</b>。如果<i>n</i>不是<a href="/wiki/%E6%95%B4%E6%95%B0" title="整数">整数</a>，那么<i>x</i>不是三角形数。这个检验法是基于<a href="/wiki/%E6%81%92%E7%AD%89%E5%BC%8F" title="恒等式">恒等式</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 8T_{n}+1=S_{2n+1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo>=</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8T_{n}+1=S_{2n+1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4722b8e44b493dd15663536f9fb6769c99be1beb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.053ex; height:2.509ex;" alt="{\displaystyle 8T_{n}+1=S_{2n+1}.}"></span></dd></dl></li></ul> <div class="mw-heading mw-heading2"><h2 id="特殊的三角形數"><span id=".E7.89.B9.E6.AE.8A.E7.9A.84.E4.B8.89.E8.A7.92.E5.BD.A2.E6.95.B8"></span>特殊的三角形數</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E4%B8%89%E8%A7%92%E5%BD%A2%E6%95%B8&action=edit&section=2" title="编辑章节：特殊的三角形數"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>55、5,050、500,500、50,005,000……都是三角形數。</li> <li>第11个三角形數（66）、第1111个三角形數（617,716）、第111,111个三角形數（6,172,882,716）、第11,111,111个三角形數（61,728,399,382,716）都是<a href="/wiki/%E5%9B%9E%E6%96%87%E6%95%B0" title="回文数">回文</a>式的三角形數，但第111个、第11,111个和第1,111,111个三角形數不是。</li> <li>同時為三角形數及<a href="/wiki/%E6%99%AE%E6%B4%9B%E5%B0%BC%E5%85%8B%E6%95%B8" class="mw-redirect" title="普洛尼克數">普洛尼克數</a>的數（<a href="/wiki/%E4%B8%8D%E5%AE%9A%E6%96%B9%E7%A8%8B" class="mw-redirect" title="不定方程">不定方程</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 x(x+1)={\frac {y(y+1)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>y</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(x+1)={\frac {y(y+1)}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b261176b29fc5f98dd7a7daefd2d09f0e9d3eeb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.529ex; height:5.676ex;" alt="{\displaystyle x(x+1)={\frac {y(y+1)}{2}}}"></span>）：最小的幾個為<a href="/wiki/0" title="0">0</a>, <a href="/wiki/6" title="6">6</a>, <a href="/wiki/210" title="210">210</a>, 7140, 242556, 8239770,……<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>，對應的<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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>值分別為<a href="/wiki/0" title="0">0</a>, 2, <a href="/wiki/14" title="14">14</a>, <a href="/wiki/84" title="84">84</a>, <a href="/wiki/492" class="mw-redirect" title="492">492</a>, 2870,……（<a href="/wiki/%E6%95%B4%E6%95%B8%E6%95%B8%E5%88%97%E7%B7%9A%E4%B8%8A%E5%A4%A7%E5%85%A8" title="整數數列線上大全">OEIS</a>數列<a href="//oeis.org/A053141" class="extiw" title="oeis:A053141">A053141</a>），對應的<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 y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>值分別為0, <a href="/wiki/3" title="3">3</a>, <a href="/wiki/20" title="20">20</a>, <a href="/wiki/119" title="119">119</a>, 696, 4059,……（<a href="/wiki/%E6%95%B4%E6%95%B8%E6%95%B8%E5%88%97%E7%B7%9A%E4%B8%8A%E5%A4%A7%E5%85%A8" title="整數數列線上大全">OEIS</a>數列<a href="//oeis.org/A001652" class="extiw" title="oeis:A001652">A001652</a>）。</li></ul> <div class="mw-heading mw-heading2"><h2 id="它與其他數的關係"><span id=".E5.AE.83.E8.88.87.E5.85.B6.E4.BB.96.E6.95.B8.E7.9A.84.E9.97.9C.E4.BF.82"></span>它與其他數的關係</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E4%B8%89%E8%A7%92%E5%BD%A2%E6%95%B8&action=edit&section=3" title="编辑章节：它與其他數的關係"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>是否在相继出现的<a href="/wiki/%E4%B8%89%E8%A7%92%E5%BD%A2" title="三角形">三角形</a>数之间至少存在一个<a href="/wiki/%E7%B4%A0%E6%95%B0" class="mw-redirect" title="素数">素数</a>，在9000000以下的数目是正确的。</li> <li><a href="/wiki/%E5%9B%9B%E9%9D%A2%E9%AB%94%E6%95%B8" title="四面體數">四面體數</a>是三角形數在<a href="/wiki/%E7%AB%8B%E9%AB%94" class="mw-redirect" title="立體">立體</a>的推廣。</li> <li>两个相继的<a href="/wiki/%E4%B8%89%E8%A7%92%E5%BD%A2" title="三角形">三角形</a>數之和是<a href="/wiki/%E5%B9%B3%E6%96%B9%E6%95%B0" title="平方数">平方数</a>。</li> <li><a href="/wiki/%E4%B8%89%E8%A7%92%E5%B9%B3%E6%96%B9%E6%95%B8" title="三角平方數">三角平方數</a>是同時為三角形數和<a href="/wiki/%E5%B9%B3%E6%96%B9%E6%95%B8" class="mw-redirect" title="平方數">平方數</a>的數。</li> <li>三角形數屬於一種<a href="/wiki/%E5%A4%9A%E9%82%8A%E5%BD%A2%E6%95%B8" title="多邊形數">多邊形數</a>。</li> <li>所有<a href="/wiki/%E5%81%B6%E6%95%B0" class="mw-redirect" title="偶数">偶</a><a href="/wiki/%E5%AE%8C%E7%BE%8E%E6%95%B0" class="mw-redirect" title="完美数">完美数</a>都是三角形数。</li> <li>任何<a href="/wiki/%E8%87%AA%E7%84%B6%E6%95%B0" title="自然数">自然数</a>是最多三个三角形數的和。<a href="/wiki/%E9%AB%98%E6%96%AF" class="mw-redirect" title="高斯">高斯</a>发现了这个规律，他在1796年7月10日在<a href="/wiki/%E6%97%A5%E8%AE%B0" title="日记">日记</a>中写道：<a href="/wiki/%E5%B0%A4%E9%87%8C%E5%8D%A1_(%E8%AF%8D%E8%AF%AD)" title="尤里卡 (词语)">EYPHKA</a>! num = Δ + Δ + Δ</li></ul> <div class="mw-heading mw-heading2"><h2 id="外部連結"><span id=".E5.A4.96.E9.83.A8.E9.80.A3.E7.B5.90"></span>外部連結</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E4%B8%89%E8%A7%92%E5%BD%A2%E6%95%B8&action=edit&section=4" title="编辑章节：外部連結"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r82655521">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:#f9f9f9;display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><div class="side-box side-box-right plainlinks sistersitebox" style="font-size:small;"><style data-mw-deduplicate="TemplateStyles:r82655520">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist"><a href="/wiki/%E7%BB%B4%E5%9F%BA%E5%85%B1%E4%BA%AB%E8%B5%84%E6%BA%90" title="维基共享资源">维基共享资源</a>上的相关多媒体资源：<a href="https://commons.wikimedia.org/wiki/Category:triangular_numbers" class="extiw" title="commons:Category:triangular numbers"><span style="font-weight:bold;">三角形數</span></a></div></div> </div> <ul><li><cite id="CITEREFHazewinkel2001" class="citation">Hazewinkel, Michiel (编), <a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php?title=p/a013370">Arithmetic series</a>, <a href="/wiki/%E6%95%B0%E5%AD%A6%E7%99%BE%E7%A7%91%E5%85%A8%E4%B9%A6" title="数学百科全书">数学百科全书</a>, <a href="/wiki/Springer_Science%2BBusiness_Media" class="mw-redirect" title="Springer Science+Business Media">Springer</a>, 2001, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-1-55608-010-4" title="Special:网络书源/978-1-55608-010-4"><span title="国际标准书号">ISBN</span> 978-1-55608-010-4</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E4%B8%89%E8%A7%92%E5%BD%A2%E6%95%B8&rft.atitle=Arithmetic+series&rft.aufirst=Michiel&rft.aulast=Hazewinkel&rft.btitle=%E6%95%B0%E5%AD%A6%E7%99%BE%E7%A7%91%E5%85%A8%E4%B9%A6&rft.date=2001&rft.genre=bookitem&rft.isbn=978-1-55608-010-4&rft.pub=Springer&rft_id=http%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3Dp%2Fa013370&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/do_you_know/numbers.shtml#square">Triangular numbers</a>（<a rel="nofollow" class="external text" href="//web.archive.org/web/20100923231547/http://www.cut-the-knot.org/do_you_know/numbers.shtml#square">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>） at <a href="/wiki/Cut-the-knot" class="mw-redirect" title="Cut-the-knot">cut-the-knot</a></li> <li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/do_you_know/triSquare.shtml">There exist triangular numbers that are also square</a>（<a rel="nofollow" class="external text" href="//web.archive.org/web/20060427075825/http://www.cut-the-knot.org/do_you_know/triSquare.shtml">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>） at <a href="/wiki/Cut-the-knot" class="mw-redirect" title="Cut-the-knot">cut-the-knot</a></li> <li><span class="citation mathworld" id="Reference-Mathworld-Triangular_Number"><cite class="citation web"><a href="/wiki/%E5%9F%83%E9%87%8C%E5%85%8B%C2%B7%E9%9F%A6%E6%96%AF%E5%9D%A6%E5%9B%A0" title="埃里克·韦斯坦因">埃里克·韦斯坦因</a>. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/TriangularNumber.html">Triangular Number</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E4%B8%89%E8%A7%92%E5%BD%A2%E6%95%B8&rft.atitle=Triangular+Number&rft.au=%E5%9F%83%E9%87%8C%E5%85%8B%C2%B7%E9%9F%A6%E6%96%AF%E5%9D%A6%E5%9B%A0&rft.genre=unknown&rft.jtitle=MathWorld&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FTriangularNumber.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20130515042555/http://vihart.com/blog/gauss/">Triangular numbers via 12 days of Christmas</a> by <a href="/w/index.php?title=Vi_Hart&action=edit&redlink=1" class="new" title="Vi Hart（页面不存在）">Vi Hart</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160314071118/https://rhubbarb.wordpress.com/2009/04/23/hypertetrahedral-polytopic-roots/">Hypertetrahedral Polytopic Roots</a> by Rob Hubbard, including the generalisation to <i>triangular cube roots</i>, some higher dimensions, and some approximate formulae</li></ul> <div class="mw-heading mw-heading2"><h2 id="參考資料"><span id=".E5.8F.83.E8.80.83.E8.B3.87.E6.96.99"></span>參考資料</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E4%B8%89%E8%A7%92%E5%BD%A2%E6%95%B8&action=edit&section=5" title="编辑章节：參考資料"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><cite class="citation web"><a href="/wiki/Neil_Sloane" class="mw-redirect" title="Neil Sloane">Sloane, N.J.A.</a> (编). <a rel="nofollow" class="external text" href="https://oeis.org/A029549">Sequence A029549 (Triangular numbers that are also pronic numbers)</a>. The <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" class="mw-redirect" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a>. OEIS Foundation.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E4%B8%89%E8%A7%92%E5%BD%A2%E6%95%B8&rft.atitle=Sequence%26%23x20%3BA029549%26%23x20%3B%28Triangular+numbers+that+are+also+pronic+numbers%29&rft.aufirst=N.J.A.&rft.aulast=Sloane&rft.genre=unknown&rft.jtitle=The+On-Line+Encyclopedia+of+Integer+Sequences&rft_id=https%3A%2F%2Foeis.org%2FA029549&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="https://www.numbersaplenty.com/set/pronic_number">pronic numbers</a>. NUMBERS APLENTY. <span class="reference-accessdate"> [<span class="nowrap">2021-02-05</span>]</span>. （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20210225020753/https://www.numbersaplenty.com/set/pronic_number/">存档</a>于2021-02-25）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E4%B8%89%E8%A7%92%E5%BD%A2%E6%95%B8&rft.btitle=pronic+numbers&rft.genre=unknown&rft.pub=NUMBERS+APLENTY&rft_id=https%3A%2F%2Fwww.numbersaplenty.com%2Fset%2Fpronic_number&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r84265675">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output 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<li><a href="/wiki/%E4%B8%83%E9%82%8A%E5%BD%A2%E6%95%B8" title="七邊形數">七邊形數</a></li> <li><a href="/wiki/%E5%85%AB%E9%82%8A%E5%BD%A2%E6%95%B8" title="八邊形數">八邊形數</a></li> <li><a href="/wiki/%E4%B9%9D%E9%82%8A%E5%BD%A2%E6%95%B8" title="九邊形數">九邊形數</a></li> <li><a href="/wiki/%E5%8D%81%E9%82%8A%E5%BD%A2%E6%95%B8" title="十邊形數">十邊形數</a></li> <li><a href="/wiki/%E5%8D%81%E4%BA%8C%E9%82%8A%E5%BD%A2%E6%95%B8" title="十二邊形數">十二邊形數</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="9" style="width:1px;padding:0px 0px 0px 2px"><div><span typeof="mw:File"><a href="/wiki/File:Triangle10.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Triangle10.svg/100px-Triangle10.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Triangle10.svg/150px-Triangle10.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Triangle10.svg/200px-Triangle10.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">多面體數</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%9B%9B%E9%9D%A2%E9%AB%94%E6%95%B8" title="四面體數">四面體數</a></li> <li><a href="/wiki/%E7%AB%8B%E6%96%B9%E6%95%B8" title="立方數">六面體數</a></li> <li><a href="/wiki/%E5%85%AB%E9%9D%A2%E9%AB%94%E6%95%B8" title="八面體數">八面體數</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">錐體數</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%9B%9B%E9%9D%A2%E9%AB%94%E6%95%B8" title="四面體數">三角錐數</a></li> <li><a href="/wiki/%E5%9B%9B%E8%A7%92%E9%8C%90%E6%95%B8" title="四角錐數">四角錐數</a></li> <li><a href="/wiki/%E4%BA%94%E8%A7%92%E9%8C%90%E6%95%B8" title="五角錐數">五角錐數</a></li> <li><a href="/wiki/%E5%85%AD%E8%A7%92%E9%8C%90%E6%95%B8" title="六角錐數">六角錐數</a></li> <li><a href="/wiki/%E4%B8%83%E8%A7%92%E9%94%A5%E6%95%B0" title="七角锥数">七角錐數</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E4%B8%AD%E5%BF%83%E5%A4%9A%E9%82%8A%E5%BD%A2%E6%95%B8" title="中心多邊形數">中心多邊形數</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E4%B8%AD%E5%BF%83%E4%B8%89%E8%A7%92%E5%BD%A2%E6%95%B8" title="中心三角形數">中心三角形數</a></li> <li><a href="/wiki/%E4%B8%AD%E5%BF%83%E6%AD%A3%E6%96%B9%E5%BD%A2%E6%95%B8" title="中心正方形數">中心四邊形數</a></li> <li><a href="/wiki/%E4%B8%AD%E5%BF%83%E4%BA%94%E9%82%8A%E5%BD%A2%E6%95%B8" title="中心五邊形數">中心五邊形數</a></li> <li><a href="/wiki/%E4%B8%AD%E5%BF%83%E5%85%AD%E9%82%8A%E5%BD%A2%E6%95%B8" title="中心六邊形數">中心六邊形數</a></li> <li><a href="/wiki/%E4%B8%AD%E5%BF%83%E4%B8%83%E9%82%8A%E5%BD%A2%E6%95%B8" title="中心七邊形數">中心七邊形數</a></li> <li><a href="/wiki/%E6%98%9F%E6%95%B8" title="星數">中心十二邊形數</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">中心多面體數</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E4%B8%AD%E5%BF%83%E5%9B%9B%E9%9D%A2%E9%AB%94%E6%95%B8" title="中心四面體數">中心四面體數</a></li> <li><a href="/wiki/%E4%B8%AD%E5%BF%83%E7%AB%8B%E6%96%B9%E9%AB%94%E6%95%B8" title="中心立方體數">中心六面體數</a></li> <li><a href="/wiki/%E4%B8%AD%E5%BF%83%E5%85%AB%E9%9D%A2%E9%AB%94%E6%95%B8" title="中心八面體數">中心八面體數</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">多胞體數</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E8%87%AA%E7%84%B6%E6%95%B0" title="自然数">1-多胞體數</a></li> <li><a class="mw-selflink selflink">2-多胞體數</a></li> <li><a href="/wiki/%E5%9B%9B%E9%9D%A2%E9%AB%94%E6%95%B8" title="四面體數">3-多胞體數</a></li> <li><a href="/wiki/%E4%BA%94%E8%83%9E%E9%AB%94%E6%95%B8" title="五胞體數">4-多胞體數</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%98%9F%E6%95%B8" title="星數">星數</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E4%BA%94%E8%A7%92%E6%98%9F%E6%95%B8" class="mw-redirect" title="五角星數">五角星數</a></li> <li><a href="/wiki/%E6%98%9F%E6%95%B8" title="星數">六角星數</a></li> <li><a href="/wiki/%E4%B8%83%E8%A7%92%E6%98%9F%E6%95%B8" class="mw-redirect" title="七角星數">七角星數</a></li> <li><a href="/wiki/%E5%85%AB%E8%A7%92%E6%98%9F%E6%95%B8" class="mw-redirect" title="八角星數">八角星數</a></li> <li><a href="/wiki/%E5%85%AD%E8%A7%92%E6%98%9F%E8%B3%AA%E6%95%B8" class="mw-redirect" title="六角星質數">六角星質數</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">其他<a href="/wiki/%E6%9C%89%E5%BD%A2%E6%95%B8" title="有形數">有形數</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%B9%B3%E6%96%B9%E6%95%B0" title="平方数">平方数</a></li> <li><a href="/wiki/%E7%AB%8B%E6%96%B9%E6%95%B8" title="立方數">立方數</a></li> <li><a href="/wiki/%E5%9B%9B%E6%AC%A1%E6%96%B9%E6%95%B8" title="四次方數">四次方數</a></li> <li><a href="/wiki/%E4%BA%94%E6%AC%A1%E6%96%B9%E6%95%B8" title="五次方數">五次方數</a></li> <li><a href="/wiki/%E5%85%AD%E6%AC%A1%E6%96%B9%E6%95%B8" title="六次方數">六次方數</a></li> <li><a href="/wiki/%E4%B8%89%E8%A7%92%E5%B9%B3%E6%96%B9%E6%95%B8" title="三角平方數">三角平方數</a></li> <li><a href="/wiki/%E6%A2%AF%E5%BD%A2%E6%95%B8" class="mw-redirect" title="梯形數">梯形數</a></li> <li><a href="/wiki/%E6%99%AE%E6%B4%9B%E5%B0%BC%E5%85%8B%E6%95%B0" title="普洛尼克数">普洛尼克数</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">其他</th><td 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