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<li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <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 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Available in 15 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-15" 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">15 languages</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%D9%85%D8%B1%D9%83%D8%B2_%D9%85%D8%B1%D8%A8%D8%B9%D9%8A" title="عدد ممركز مربعي – Arabic" lang="ar" hreflang="ar" data-title="عدد ممركز مربعي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Zentrierte_Quadratzahl" title="Zentrierte Quadratzahl – German" lang="de" hreflang="de" data-title="Zentrierte Quadratzahl" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/N%C3%B9mer_quadr%C3%AA_sentr%C3%AA" title="Nùmer quadrê sentrê – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Nùmer quadrê sentrê" 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-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Centrita_kvadrata_nombro" title="Centrita kvadrata nombro – Esperanto" lang="eo" hreflang="eo" data-title="Centrita kvadrata nombro" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_carr%C3%A9_centr%C3%A9" title="Nombre carré centré – French" lang="fr" hreflang="fr" data-title="Nombre carré centré" data-language-autonym="Français" data-language-local-name="French" 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_quadrato_centrato" title="Numero quadrato centrato – Italian" lang="it" hreflang="it" data-title="Numero quadrato centrato" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/K%C3%B6z%C3%A9ppontos_n%C3%A9gyzetsz%C3%A1mok" title="Középpontos négyzetszámok – Hungarian" lang="hu" hreflang="hu" data-title="Középpontos négyzetszámok" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%B8%AD%E5%BF%83%E3%81%A4%E3%81%8D%E5%9B%9B%E8%A7%92%E6%95%B0" title="中心つき四角数 – Japanese" lang="ja" hreflang="ja" data-title="中心つき四角数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Num%C4%83r_centrat_p%C4%83tratic" title="Număr centrat pătratic – Romanian" lang="ro" hreflang="ro" data-title="Număr centrat pătratic" data-language-autonym="Română" data-language-local-name="Romanian" 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%A6%D0%B5%D0%BD%D1%82%D1%80%D0%B8%D1%80%D0%BE%D0%B2%D0%B0%D0%BD%D0%BD%D0%BE%D0%B5_%D0%BA%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%BD%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Центрированное квадратное число – Russian" lang="ru" hreflang="ru" data-title="Центрированное квадратное число" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Keskitetty_neli%C3%B6luku" title="Keskitetty neliöluku – Finnish" lang="fi" hreflang="fi" data-title="Keskitetty neliöluku" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Centrerat_kvadrattal" title="Centrerat kvadrattal – Swedish" lang="sv" hreflang="sv" data-title="Centrerat kvadrattal" data-language-autonym="Svenska" data-language-local-name="Swedish" 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%88%E0%AE%AF%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AE%9F%E0%AF%81%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AE%9F%E0%AF%8D%E0%AE%9F_%E0%AE%B5%E0%AE%B0%E0%AF%8D%E0%AE%95%E0%AF%8D%E0%AE%95_%E0%AE%8E%E0%AE%A3%E0%AF%8D" title="மையப்படுத்தப்பட்ட வர்க்க எண் – Tamil" lang="ta" hreflang="ta" data-title="மையப்படுத்தப்பட்ட வர்க்க எண்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A6%D0%B5%D0%BD%D1%82%D1%80%D0%BE%D0%B2%D0%B0%D0%BD%D0%B5_%D0%BA%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%BD%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Центроване квадратне число – Ukrainian" lang="uk" hreflang="uk" data-title="Центроване квадратне число" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/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 – Chinese" lang="zh" hreflang="zh" data-title="中心正方形數" data-language-autonym="中文" data-language-local-name="Chinese" 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/Q190900#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav 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number of dots in a square with a dot in the center</div> <p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-No_footnotes plainlinks metadata ambox ambox-style ambox-No_footnotes" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article includes a <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">list of references</a>, <a href="/wiki/Wikipedia:Further_reading" title="Wikipedia:Further reading">related reading</a>, or <a href="/wiki/Wikipedia:External_links" title="Wikipedia:External links">external links</a>, <b>but its sources remain unclear because it lacks <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Wikipedia:WikiProject_Fact_and_Reference_Check" class="mw-redirect" title="Wikipedia:WikiProject Fact and Reference Check">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">January 2016</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Elementary_number_theory" class="mw-redirect" title="Elementary number theory">elementary number theory</a>, a <b>centered square number</b> is a <a href="/wiki/Centered_polygonal_number" title="Centered polygonal number">centered</a> <a href="/wiki/Figurate_number" title="Figurate number">figurate number</a> that gives the number of dots in a <a href="/wiki/Square_(geometry)" class="mw-redirect" title="Square (geometry)">square</a> with a dot in the center and all other dots surrounding the center dot in successive square layers. That is, each centered square number equals the number of dots within a given <a href="/wiki/Taxicab_geometry" title="Taxicab geometry">city block distance</a> of the center dot on a regular <a href="/wiki/Square_lattice" title="Square lattice">square lattice</a>. While centered square numbers, like <a href="/wiki/Figurate_number" title="Figurate number">figurate numbers</a> in general, have few if any direct practical applications, they are sometimes studied in <a href="/wiki/Recreational_mathematics" title="Recreational mathematics">recreational mathematics</a> for their elegant geometric and arithmetic properties. </p><p>The figures for the first four centered square numbers are shown below: </p> <dl><dd><table> <tbody><tr align="center" valign="middle" style="line-height: 0;"> <td><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> </td> <td>    </td> <td>    </td> <td><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> </td> <td>    </td> <td>    </td> <td><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> </td> <td>    </td> <td>    </td> <td><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> </td></tr> <tr align="center" valign="top"> <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 C_{4,1}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{4,1}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac4419b039675f655fb3d5887664e7262b0aeddf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.256ex; height:2.843ex;" alt="{\displaystyle C_{4,1}=1}"></span> </td> <td>    </td> <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 C_{4,2}=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{4,2}=5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09be113fb487091a5c34a64f939e1ddaf0b04687" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.256ex; height:2.843ex;" alt="{\displaystyle C_{4,2}=5}"></span> </td> <td>    </td> <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 C_{4,3}=13}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>13</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{4,3}=13}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/995b316cca83b3142ddaf4a68dd96727e3c68961" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.419ex; height:2.843ex;" alt="{\displaystyle C_{4,3}=13}"></span> </td> <td>    </td> <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 C_{4,4}=25}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>,</mo> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mn>25</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{4,4}=25}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f40cab4c4234bee53f7d1626240700ad9629c725" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.419ex; height:2.843ex;" alt="{\displaystyle C_{4,4}=25}"></span> </td></tr></tbody></table></dd></dl> <p>Each centered square number is the sum of successive squares. Example: as shown in the following figure of <a href="/wiki/Floyd%27s_triangle" title="Floyd's triangle">Floyd's triangle</a>, 25 is a centered square number, and is the sum of the square 16 (yellow rhombus formed by shearing a square) and of the next smaller square, 9 (sum of two blue triangles): </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Centered_square_numbers_vs_triangular_numbers.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Centered_square_numbers_vs_triangular_numbers.svg/220px-Centered_square_numbers_vs_triangular_numbers.svg.png" decoding="async" width="220" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Centered_square_numbers_vs_triangular_numbers.svg/330px-Centered_square_numbers_vs_triangular_numbers.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Centered_square_numbers_vs_triangular_numbers.svg/440px-Centered_square_numbers_vs_triangular_numbers.svg.png 2x" data-file-width="512" data-file-height="288" /></a><figcaption>Centered square numbers (in red) are in the center of odd rows of Floyd's triangle.</figcaption></figure> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Relationships_with_other_figurate_numbers">Relationships with other figurate numbers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centered_square_number&action=edit&section=1" title="Edit section: Relationships with other figurate numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>C</i><sub><i>k</i>,<i>n</i></sub> generally represent the <i>n</i>th <a href="/wiki/Centered_polygonal_number" title="Centered polygonal number">centered <i>k</i>-gonal number</a>. The <i>n</i>th centered square number is given by the formula: </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 C_{4,n}=n^{2}+(n-1)^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{4,n}=n^{2}+(n-1)^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bae664a2f4dc0eae9eec9f60a6c01121524bc2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.455ex; height:3.343ex;" alt="{\displaystyle C_{4,n}=n^{2}+(n-1)^{2}.}"></span></dd></dl> <p>That is, the <i>n</i>th centered square number is the sum of the <i>n</i>th and the (<i>n</i> – 1)th <a href="/wiki/Square_number" title="Square number">square numbers</a>. The following pattern demonstrates this formula: </p> <dl><dd><table> <tbody><tr align="center" valign="middle" style="line-height: 0;"> <td><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> </td> <td>    </td> <td>    </td> <td><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> </td> <td>    </td> <td>    </td> <td><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> </td> <td>    </td> <td>    </td> <td><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> </td></tr> <tr align="center" valign="top"> <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 C_{4,1}=0+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{4,1}=0+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69932a89e379e5cd68a0cde4fac898a5d3a3a0af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.259ex; height:2.843ex;" alt="{\displaystyle C_{4,1}=0+1}"></span> </td> <td>    </td> <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 C_{4,2}=1+4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{4,2}=1+4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b90f026d2e806617eaa7be5cf76d38f1404cb48f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.259ex; height:2.843ex;" alt="{\displaystyle C_{4,2}=1+4}"></span> </td> <td>    </td> <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 C_{4,3}=4+9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>4</mn> <mo>+</mo> <mn>9</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{4,3}=4+9}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b80d215c527ff2c0904d1ae5a8ec78fd39f58ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.259ex; height:2.843ex;" alt="{\displaystyle C_{4,3}=4+9}"></span> </td> <td>    </td> <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 C_{4,4}=9+16}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>,</mo> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mn>9</mn> <mo>+</mo> <mn>16</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{4,4}=9+16}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d25ed1ff2a9de0749c5809f2e54533abdd22462" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.422ex; height:2.843ex;" alt="{\displaystyle C_{4,4}=9+16}"></span> </td></tr></tbody></table></dd></dl> <p>The formula can also be expressed as: </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 C_{4,n}={\frac {(2n-1)^{2}+1}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{4,n}={\frac {(2n-1)^{2}+1}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49a746b4ed9e77b122ed7cf8790756685bf6bce6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.167ex; height:5.843ex;" alt="{\displaystyle C_{4,n}={\frac {(2n-1)^{2}+1}{2}}.}"></span></dd></dl> <p>That is, the <i>n</i>th centered square number is half of the <i>n</i>th odd square number plus 1, as illustrated below: </p> <dl><dd><table> <tbody><tr align="center" valign="bottom" style="line-height: 0;"> <td><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> </td> <td>    </td> <td>    </td> <td><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:MissingDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/16px-MissingDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/24px-MissingDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/32px-MissingDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:MissingDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/16px-MissingDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/24px-MissingDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/32px-MissingDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:MissingDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/16px-MissingDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/24px-MissingDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/32px-MissingDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:MissingDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/16px-MissingDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/24px-MissingDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/32px-MissingDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> </td> <td>    </td> <td>    </td> <td><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:MissingDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/16px-MissingDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/24px-MissingDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/32px-MissingDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:MissingDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/16px-MissingDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/24px-MissingDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/32px-MissingDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br 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data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:MissingDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/16px-MissingDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/24px-MissingDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/32px-MissingDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:MissingDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/16px-MissingDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/24px-MissingDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/32px-MissingDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:MissingDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/16px-MissingDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/24px-MissingDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/32px-MissingDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:MissingDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/16px-MissingDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/24px-MissingDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/32px-MissingDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:MissingDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/16px-MissingDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/24px-MissingDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/32px-MissingDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:MissingDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/16px-MissingDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/24px-MissingDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/32px-MissingDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:MissingDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/16px-MissingDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/24px-MissingDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/32px-MissingDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:MissingDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/16px-MissingDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/24px-MissingDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/32px-MissingDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> </td> <td>    </td> <td>    </td> <td><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" 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class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/24px-MissingDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dc/MissingDot.svg/32px-MissingDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> </td></tr> <tr align="center" valign="top"> <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 C_{4,1}={\frac {1+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{4,1}={\frac {1+1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29dbd9d47831a79226a61e0f2b62cf7b163ca6fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.095ex; height:5.176ex;" alt="{\displaystyle C_{4,1}={\frac {1+1}{2}}}"></span> </td> <td>    </td> <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 C_{4,2}={\frac {9+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>9</mn> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{4,2}={\frac {9+1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d71daa74e28a3ef2a1a2205723525a8c1c3c609" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.095ex; height:5.176ex;" alt="{\displaystyle C_{4,2}={\frac {9+1}{2}}}"></span> </td> <td>    </td> <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 C_{4,3}={\frac {25+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>25</mn> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{4,3}={\frac {25+1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b750dcdf7b977b6147de46df5b174cb8e55a861" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.258ex; height:5.176ex;" alt="{\displaystyle C_{4,3}={\frac {25+1}{2}}}"></span> </td> <td>    </td> <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 C_{4,4}={\frac {49+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>,</mo> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>49</mn> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{4,4}={\frac {49+1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40c378f71aa70b484bb3359564a5025522f28aa5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.258ex; height:5.176ex;" alt="{\displaystyle C_{4,4}={\frac {49+1}{2}}}"></span> </td></tr></tbody></table></dd></dl> <p>Like all <a href="/wiki/Centered_polygonal_number" title="Centered polygonal number">centered polygonal numbers</a>, centered square numbers can also be expressed in terms of <a href="/wiki/Triangular_number" title="Triangular number">triangular numbers</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{4,n}=1+4\ T_{n-1}=1+2{n(n-1)},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>4</mn> <mtext> </mtext> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{4,n}=1+4\ T_{n-1}=1+2{n(n-1)},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cf83f52bd6dfb3aa353d8f8eebd50e52ee522c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.193ex; height:3.009ex;" alt="{\displaystyle C_{4,n}=1+4\ T_{n-1}=1+2{n(n-1)},}"></span></dd></dl> <p>where </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 T_{n}={\frac {n(n+1)}{2}}={\binom {n+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</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> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{n}={\frac {n(n+1)}{2}}={\binom {n+1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64b9aa9c71263041f11e4246c0e6a16f2bb8a2c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.029ex; height:6.343ex;" alt="{\displaystyle T_{n}={\frac {n(n+1)}{2}}={\binom {n+1}{2}}}"></span></dd></dl> <p>is the <i>n</i>th triangular number. This can be easily seen by removing the center dot and dividing the rest of the figure into four triangles, as below: </p> <dl><dd><table> <tbody><tr align="center" valign="middle" style="line-height: 0;"> <td><span typeof="mw:File"><a href="/wiki/File:BlackDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/BlackDot.svg/16px-BlackDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/BlackDot.svg/24px-BlackDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/BlackDot.svg/32px-BlackDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> </td> <td>    </td> <td>    </td> <td><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:BlackDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/BlackDot.svg/16px-BlackDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/BlackDot.svg/24px-BlackDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/BlackDot.svg/32px-BlackDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> </td> <td>    </td> <td>    </td> <td><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:BlackDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/BlackDot.svg/16px-BlackDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/BlackDot.svg/24px-BlackDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/BlackDot.svg/32px-BlackDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> </td> <td>    </td> <td>    </td> <td><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:BlackDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/BlackDot.svg/16px-BlackDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/BlackDot.svg/24px-BlackDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/BlackDot.svg/32px-BlackDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:RedDot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/16px-RedDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/24px-RedDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/RedDot.svg/32px-RedDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> </td></tr> <tr align="center" valign="top"> <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 C_{4,1}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{4,1}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac4419b039675f655fb3d5887664e7262b0aeddf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.256ex; height:2.843ex;" alt="{\displaystyle C_{4,1}=1}"></span> </td> <td>    </td> <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 C_{4,2}=1+4\times 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>4</mn> <mo>×</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{4,2}=1+4\times 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5149b7758665f89cea412086d37555a54eff426" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.262ex; height:2.843ex;" alt="{\displaystyle C_{4,2}=1+4\times 1}"></span> </td> <td>    </td> <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 C_{4,3}=1+4\times 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>4</mn> <mo>×</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{4,3}=1+4\times 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1799894eb256377151947578b4aa31e1d025883" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.262ex; height:2.843ex;" alt="{\displaystyle C_{4,3}=1+4\times 3}"></span> </td> <td>    </td> <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 C_{4,4}=1+4\times 6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>,</mo> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>4</mn> <mo>×</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{4,4}=1+4\times 6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8be18a59f9544f692dcd39ca7f1175a6e7cddabd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.262ex; height:2.843ex;" alt="{\displaystyle C_{4,4}=1+4\times 6}"></span> </td></tr></tbody></table></dd></dl> <p>The difference between two consecutive <a href="/wiki/Octahedral_number" title="Octahedral number">octahedral numbers</a> is a centered square number (Conway and Guy, p.50). </p><p>Another way the centered square numbers can be expressed is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{4,n}=1+4\dim(SO(n)),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>4</mn> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{4,n}=1+4\dim(SO(n)),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e906149b3d898106c0feaa72cecac7db0940bb66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.618ex; height:3.009ex;" alt="{\displaystyle C_{4,n}=1+4\dim(SO(n)),}"></span></dd></dl> <p>where </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 \dim(SO(n))={\frac {n(n-1)}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <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> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim(SO(n))={\frac {n(n-1)}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d9feb0e75439197ff12e101346e355ba5793d00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.344ex; height:5.676ex;" alt="{\displaystyle \dim(SO(n))={\frac {n(n-1)}{2}}.}"></span></dd></dl> <p>Yet another way the centered square numbers can be expressed is in terms of the <a href="/wiki/Centered_triangular_number" title="Centered triangular number">centered triangular numbers</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{4,n}={\frac {4C_{3,n}-1}{3}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{4,n}={\frac {4C_{3,n}-1}{3}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dc225edc737fe960117101a61065ceee482bf94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.066ex; height:5.676ex;" alt="{\displaystyle C_{4,n}={\frac {4C_{3,n}-1}{3}},}"></span></dd></dl> <p>where </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 C_{3,n}=1+3{\frac {n(n-1)}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>3</mn> <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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{3,n}=1+3{\frac {n(n-1)}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07128a518c8ea691982fb5db31efe6bda081a24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.508ex; height:5.676ex;" alt="{\displaystyle C_{3,n}=1+3{\frac {n(n-1)}{2}}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="List_of_centered_square_numbers">List of centered square numbers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centered_square_number&action=edit&section=2" title="Edit section: List of centered square numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The first centered square numbers (<i>C</i><sub>4,<i>n</i></sub> < 4500) are: </p> <dl><dd><a href="/wiki/1_(number)" class="mw-redirect" title="1 (number)">1</a>, <a href="/wiki/5_(number)" class="mw-redirect" title="5 (number)">5</a>, <a href="/wiki/13_(number)" title="13 (number)">13</a>, <a href="/wiki/25_(number)" title="25 (number)">25</a>, <a href="/wiki/41_(number)" title="41 (number)">41</a>, <a href="/wiki/61_(number)" title="61 (number)">61</a>, <a href="/wiki/85_(number)" title="85 (number)">85</a>, <a href="/wiki/113_(number)" title="113 (number)">113</a>, <a href="/wiki/145_(number)" title="145 (number)">145</a>, <a href="/wiki/181_(number)" title="181 (number)">181</a>, <a href="/wiki/221_(number)" title="221 (number)">221</a>, 265, <a href="/wiki/313_(number)" title="313 (number)">313</a>, <a href="/wiki/365_(number)" title="365 (number)">365</a>, 421, 481, 545, 613, 685, 761, 841, 925, 1013, 1105, 1201, 1301, 1405, 1513, 1625, 1741, 1861, 1985, 2113, 2245, 2381, 2521, 2665, 2813, 2965, 3121, 3281, 3445, 3613, 3785, 3961, 4141, 4325, … (sequence <span class="nowrap external"><a href="//oeis.org/A001844" class="extiw" title="oeis:A001844">A001844</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centered_square_number&action=edit&section=3" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>All centered square numbers are odd, and in base 10 one can notice the one's digit follows the pattern 1-5-3-5-1. </p><p>All centered square numbers and their divisors have a remainder of 1 when divided by 4. Hence all centered square numbers and their divisors end with digit 1 or 5 in base <a href="/wiki/Senary" title="Senary">6</a>, <a href="/wiki/Octal" title="Octal">8</a>, and <a href="/wiki/Duodecimal" title="Duodecimal">12</a>. </p><p>Every centered square number except 1 is the <a href="/wiki/Hypotenuse" title="Hypotenuse">hypotenuse</a> of a <a href="/wiki/Pythagorean_triple" title="Pythagorean triple">Pythagorean triple</a> (3-4-<b>5</b>, 5-12-<b>13</b>, 7-24-<b>25</b>, ...). This is exactly the sequence of Pythagorean triples where the two longest sides differ by 1. (Example: 5<sup>2</sup> + 12<sup>2</sup> = <b>13</b><sup>2</sup>.) </p><p>This is not to be confused with the relationship (<i>n</i> – 1)<sup>2</sup> + <i>n</i><sup>2</sup> = <i>C</i><sub>4,<i>n</i></sub>. (Example: 2<sup>2</sup> + 3<sup>2</sup> = <b>13</b>.) </p> <div class="mw-heading mw-heading2"><h2 id="Generating_function">Generating function</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centered_square_number&action=edit&section=4" title="Edit section: Generating function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The generating function that gives the centered square numbers is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {(x+1)^{2}}{(1-x)^{3}}}=1+5x+13x^{2}+25x^{3}+41x^{4}+~...~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>5</mn> <mi>x</mi> <mo>+</mo> <mn>13</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>25</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>41</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mtext> </mtext> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {(x+1)^{2}}{(1-x)^{3}}}=1+5x+13x^{2}+25x^{3}+41x^{4}+~...~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62c84e208a79257350d955fd436649a454f77f3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.024ex; height:6.676ex;" alt="{\displaystyle {\frac {(x+1)^{2}}{(1-x)^{3}}}=1+5x+13x^{2}+25x^{3}+41x^{4}+~...~.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centered_square_number&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAlfred1962" class="citation cs2">Alfred, U. (1962), "<span class="texhtml"><i>n</i></span> and <span class="texhtml"><i>n</i> + 1</span> consecutive integers with equal sums of squares", <i>Mathematics Magazine</i>, <b>35</b> (3): 155–164, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F0025570X.1962.11975326">10.1080/0025570X.1962.11975326</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2688938">2688938</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1571197">1571197</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=%3Cspan+class%3D%22texhtml+%22+%3En%3C%2Fspan%3E+and+%3Cspan+class%3D%22texhtml+%22+%3En+%2B+1%3C%2Fspan%3E+consecutive+integers+with+equal+sums+of+squares&rft.volume=35&rft.issue=3&rft.pages=155-164&rft.date=1962&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1571197%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2688938%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1080%2F0025570X.1962.11975326&rft.aulast=Alfred&rft.aufirst=U.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentered+square+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFApostol1976" class="citation cs2"><a href="/wiki/Tom_M._Apostol" title="Tom M. Apostol">Apostol, Tom M.</a> (1976), <i>Introduction to analytic number theory</i>, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90163-3" title="Special:BookSources/978-0-387-90163-3"><bdi>978-0-387-90163-3</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0434929">0434929</a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0335.10001">0335.10001</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+analytic+number+theory&rft.place=New+York-Heidelberg&rft.series=Undergraduate+Texts+in+Mathematics&rft.pub=Springer-Verlag&rft.date=1976&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0335.10001%23id-name%3DZbl&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0434929%23id-name%3DMR&rft.isbn=978-0-387-90163-3&rft.aulast=Apostol&rft.aufirst=Tom+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentered+square+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeiler1964" class="citation cs2">Beiler, A. H. (1964), <i>Recreations in the Theory of Numbers</i>, New York: Dover, p. 125</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Recreations+in+the+Theory+of+Numbers&rft.place=New+York&rft.pages=125&rft.pub=Dover&rft.date=1964&rft.aulast=Beiler&rft.aufirst=A.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentered+square+number" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFConwayGuy1996" class="citation cs2"><a href="/wiki/John_Horton_Conway" title="John Horton Conway">Conway, John H.</a>; <a href="/wiki/Richard_K._Guy" title="Richard K. Guy">Guy, Richard K.</a> (1996), <a rel="nofollow" class="external text" href="https://archive.org/details/bookofnumbers0000conw/page/41"><i>The Book of Numbers</i></a>, New York: Copernicus, pp. <a rel="nofollow" class="external text" href="https://archive.org/details/bookofnumbers0000conw/page/41">41–42</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-97993-X" title="Special:BookSources/0-387-97993-X"><bdi>0-387-97993-X</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1411676">1411676</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Book+of+Numbers&rft.place=New+York&rft.pages=41-42&rft.pub=Copernicus&rft.date=1996&rft.isbn=0-387-97993-X&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1411676%23id-name%3DMR&rft.aulast=Conway&rft.aufirst=John+H.&rft.au=Guy%2C+Richard+K.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fbookofnumbers0000conw%2Fpage%2F41&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentered+square+number" class="Z3988"></span>.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.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 .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl 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.navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Figurate_numbers" title="Template:Figurate numbers"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Figurate_numbers" title="Template talk:Figurate numbers"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Figurate_numbers" title="Special:EditPage/Template:Figurate numbers"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Figurate_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Figurate_number" title="Figurate number">Figurate numbers</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">2-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Centered_polygonal_number" title="Centered polygonal number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Centered_triangular_number" title="Centered triangular number">Centered triangular numbers</a></li> <li><a class="mw-selflink selflink">Centered square numbers</a></li> <li><a href="/wiki/Centered_pentagonal_number" title="Centered pentagonal number">Centered pentagonal numbers</a></li> <li><a href="/wiki/Centered_hexagonal_number" title="Centered hexagonal number">Centered hexagonal numbers</a></li> <li><a href="/wiki/Centered_heptagonal_number" title="Centered heptagonal number">Centered heptagonal numbers</a></li> <li><a href="/wiki/Centered_octagonal_number" title="Centered octagonal number">Centered octagonal numbers</a></li> <li><a href="/wiki/Centered_nonagonal_number" title="Centered nonagonal number">Centered nonagonal numbers</a></li> <li><a href="/wiki/Centered_decagonal_number" title="Centered decagonal number">Centered decagonal numbers</a></li> <li><a href="/wiki/Star_number" title="Star number">Star numbers</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Polygonal_number" title="Polygonal number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Triangular_number" title="Triangular number">Triangular numbers</a></li> <li><a href="/wiki/Square_number" title="Square number">Square numbers</a></li> <li><a href="/wiki/Pentagonal_number" title="Pentagonal number">Pentagonal numbers</a></li> <li><a href="/wiki/Hexagonal_number" title="Hexagonal number">Hexagonal numbers</a></li> <li><a href="/wiki/Heptagonal_number" title="Heptagonal number">Heptagonal numbers</a></li> <li><a href="/wiki/Octagonal_number" title="Octagonal number">Octagonal numbers</a></li> <li><a href="/wiki/Nonagonal_number" title="Nonagonal number">Nonagonal numbers</a></li> <li><a href="/wiki/Decagonal_number" title="Decagonal number">Decagonal numbers</a></li> <li><a href="/wiki/Dodecagonal_number" title="Dodecagonal number">Dodecagonal numbers</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Three-dimensional_space" title="Three-dimensional space">3-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Centered_polyhedral_number" title="Centered polyhedral number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Centered_tetrahedral_number" title="Centered tetrahedral number">Centered tetrahedral numbers</a></li> <li><a href="/wiki/Centered_cube_number" title="Centered cube number">Centered cube numbers</a></li> <li><a href="/wiki/Centered_octahedral_number" title="Centered octahedral number">Centered octahedral numbers</a></li> <li><a href="/wiki/Centered_dodecahedral_number" title="Centered dodecahedral number">Centered dodecahedral numbers</a></li> <li><a href="/wiki/Centered_icosahedral_number" title="Centered icosahedral number">Centered icosahedral numbers</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Polyhedral_number" class="mw-redirect" title="Polyhedral number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cube_(algebra)" title="Cube (algebra)">Cube numbers</a></li> <li><a href="/wiki/Octahedral_number" title="Octahedral number">Octahedral numbers</a></li> <li><a href="/wiki/Dodecahedral_number" title="Dodecahedral number">Dodecahedral numbers</a></li> <li><a href="/wiki/Icosahedral_number" title="Icosahedral number">Icosahedral numbers</a></li> <li><a href="/wiki/Stella_octangula_number" title="Stella octangula number">Stella octangula numbers</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Pyramidal_number" title="Pyramidal number">pyramidal</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tetrahedral_number" title="Tetrahedral number">Tetrahedral numbers</a></li> <li><a href="/wiki/Square_pyramidal_number" title="Square pyramidal number">Square pyramidal numbers</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Four-dimensional_space" title="Four-dimensional space">4-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">non-centered</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pentatope_number" title="Pentatope number">Pentatope numbers</a></li> <li><a href="/wiki/Squared_triangular_number" title="Squared triangular number">Squared triangular numbers</a></li> <li><a href="/wiki/Fourth_power" title="Fourth power">Tesseractic numbers</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Higher <a href="/wiki/Dimension" title="Dimension">dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="non-centered" scope="row" class="navbox-group" style="width:1%">non-centered</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fifth_power_(algebra)" title="Fifth power (algebra)">5-hypercube numbers</a></li> <li><a href="/wiki/Sixth_power" title="Sixth power">6-hypercube numbers</a></li> <li><a href="/wiki/Seventh_power" title="Seventh power">7-hypercube numbers</a></li> <li><a href="/wiki/Eighth_power" title="Eighth power">8-hypercube numbers</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Classes_of_natural_numbers" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Classes_of_natural_numbers" title="Template:Classes of natural numbers"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Classes_of_natural_numbers" title="Template talk:Classes of natural numbers"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Classes_of_natural_numbers" title="Special:EditPage/Template:Classes of natural numbers"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Classes_of_natural_numbers" style="font-size:114%;margin:0 4em">Classes of <a href="/wiki/Natural_number" title="Natural number">natural numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Powers_and_related_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Exponentiation" title="Exponentiation">Powers</a> and related numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Achilles_number" title="Achilles number">Achilles</a></li> <li><a href="/wiki/Power_of_two" title="Power of two">Power of 2</a></li> <li><a href="/wiki/Power_of_three" title="Power of three">Power of 3</a></li> <li><a href="/wiki/Power_of_10" title="Power of 10">Power of 10</a></li> <li><a href="/wiki/Square_number" title="Square number">Square</a></li> <li><a href="/wiki/Cube_(algebra)" title="Cube (algebra)">Cube</a></li> <li><a href="/wiki/Fourth_power" title="Fourth power">Fourth power</a></li> <li><a href="/wiki/Fifth_power_(algebra)" title="Fifth power (algebra)">Fifth power</a></li> <li><a href="/wiki/Sixth_power" title="Sixth power">Sixth power</a></li> <li><a href="/wiki/Seventh_power" title="Seventh power">Seventh power</a></li> <li><a href="/wiki/Eighth_power" title="Eighth power">Eighth power</a></li> <li><a href="/wiki/Perfect_power" title="Perfect power">Perfect power</a></li> <li><a href="/wiki/Powerful_number" title="Powerful number">Powerful</a></li> <li><a href="/wiki/Prime_power" title="Prime power">Prime power</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Of_the_form_a_×_2b_±_1" style="font-size:114%;margin:0 4em">Of the form <i>a</i> × 2<sup><i>b</i></sup> ± 1</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cullen_number" title="Cullen number">Cullen</a></li> <li><a href="/wiki/Double_Mersenne_number" title="Double Mersenne number">Double Mersenne</a></li> <li><a href="/wiki/Fermat_number" title="Fermat number">Fermat</a></li> <li><a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne</a></li> <li><a href="/wiki/Proth_number" class="mw-redirect" title="Proth number">Proth</a></li> <li><a href="/wiki/Thabit_number" title="Thabit number">Thabit</a></li> <li><a href="/wiki/Woodall_number" title="Woodall number">Woodall</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_polynomial_numbers" style="font-size:114%;margin:0 4em">Other polynomial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hilbert_number" title="Hilbert number">Hilbert</a></li> <li><a href="/wiki/Idoneal_number" title="Idoneal number">Idoneal</a></li> <li><a href="/wiki/Leyland_number" title="Leyland number">Leyland</a></li> <li><a href="/wiki/Loeschian_number" class="mw-redirect" title="Loeschian number">Loeschian</a></li> <li><a href="/wiki/Lucky_numbers_of_Euler" title="Lucky numbers of Euler">Lucky numbers of Euler</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Recursively_defined_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Recursion" title="Recursion">Recursively</a> defined numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fibonacci_sequence" title="Fibonacci sequence">Fibonacci</a></li> <li><a href="/wiki/Jacobsthal_number" title="Jacobsthal number">Jacobsthal</a></li> <li><a href="/wiki/Leonardo_number" title="Leonardo number">Leonardo</a></li> <li><a href="/wiki/Lucas_number" title="Lucas number">Lucas</a></li> <li><a href="/wiki/Supergolden_ratio#Narayana_sequence" title="Supergolden ratio">Narayana</a></li> <li><a href="/wiki/Padovan_sequence" title="Padovan sequence">Padovan</a></li> <li><a href="/wiki/Pell_number" title="Pell number">Pell</a></li> <li><a href="/wiki/Perrin_number" title="Perrin number">Perrin</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Possessing_a_specific_set_of_other_numbers" style="font-size:114%;margin:0 4em">Possessing a specific set of other numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amenable_number" title="Amenable number">Amenable</a></li> <li><a href="/wiki/Congruent_number" title="Congruent number">Congruent</a></li> <li><a href="/wiki/Kn%C3%B6del_number" title="Knödel number">Knödel</a></li> <li><a href="/wiki/Riesel_number" title="Riesel number">Riesel</a></li> <li><a href="/wiki/Sierpi%C5%84ski_number" title="Sierpiński number">Sierpiński</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Expressible_via_specific_sums" style="font-size:114%;margin:0 4em">Expressible via specific sums</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nonhypotenuse_number" title="Nonhypotenuse number">Nonhypotenuse</a></li> <li><a href="/wiki/Polite_number" title="Polite number">Polite</a></li> <li><a href="/wiki/Practical_number" title="Practical number">Practical</a></li> <li><a href="/wiki/Primary_pseudoperfect_number" title="Primary pseudoperfect number">Primary pseudoperfect</a></li> <li><a href="/wiki/Ulam_number" title="Ulam number">Ulam</a></li> <li><a href="/wiki/Wolstenholme_number" title="Wolstenholme number">Wolstenholme</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Figurate_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Figurate_number" title="Figurate number">Figurate numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">2-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Centered_polygonal_number" title="Centered polygonal number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Centered_triangular_number" title="Centered triangular number">Centered triangular</a></li> <li><a class="mw-selflink selflink">Centered square</a></li> <li><a href="/wiki/Centered_pentagonal_number" title="Centered pentagonal number">Centered pentagonal</a></li> <li><a href="/wiki/Centered_hexagonal_number" title="Centered hexagonal number">Centered hexagonal</a></li> <li><a href="/wiki/Centered_heptagonal_number" title="Centered heptagonal number">Centered heptagonal</a></li> <li><a href="/wiki/Centered_octagonal_number" title="Centered octagonal number">Centered octagonal</a></li> <li><a href="/wiki/Centered_nonagonal_number" title="Centered nonagonal number">Centered nonagonal</a></li> <li><a href="/wiki/Centered_decagonal_number" title="Centered decagonal number">Centered decagonal</a></li> <li><a href="/wiki/Star_number" title="Star number">Star</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Polygonal_number" title="Polygonal number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Triangular_number" title="Triangular number">Triangular</a></li> <li><a href="/wiki/Square_number" title="Square number">Square</a></li> <li><a href="/wiki/Square_triangular_number" title="Square triangular number">Square triangular</a></li> <li><a href="/wiki/Pentagonal_number" title="Pentagonal number">Pentagonal</a></li> <li><a href="/wiki/Hexagonal_number" title="Hexagonal number">Hexagonal</a></li> <li><a href="/wiki/Heptagonal_number" title="Heptagonal number">Heptagonal</a></li> <li><a href="/wiki/Octagonal_number" title="Octagonal number">Octagonal</a></li> <li><a href="/wiki/Nonagonal_number" title="Nonagonal number">Nonagonal</a></li> <li><a href="/wiki/Decagonal_number" title="Decagonal number">Decagonal</a></li> <li><a href="/wiki/Dodecagonal_number" title="Dodecagonal number">Dodecagonal</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Three-dimensional_space" title="Three-dimensional space">3-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Centered_polyhedral_number" title="Centered polyhedral number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Centered_tetrahedral_number" title="Centered tetrahedral number">Centered tetrahedral</a></li> <li><a href="/wiki/Centered_cube_number" title="Centered cube number">Centered cube</a></li> <li><a href="/wiki/Centered_octahedral_number" title="Centered octahedral number">Centered octahedral</a></li> <li><a href="/wiki/Centered_dodecahedral_number" title="Centered dodecahedral number">Centered dodecahedral</a></li> <li><a href="/wiki/Centered_icosahedral_number" title="Centered icosahedral number">Centered icosahedral</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Polyhedral_number" class="mw-redirect" title="Polyhedral number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tetrahedral_number" title="Tetrahedral number">Tetrahedral</a></li> <li><a href="/wiki/Cube_(algebra)" title="Cube (algebra)">Cubic</a></li> <li><a href="/wiki/Octahedral_number" title="Octahedral number">Octahedral</a></li> <li><a href="/wiki/Dodecahedral_number" title="Dodecahedral number">Dodecahedral</a></li> <li><a href="/wiki/Icosahedral_number" title="Icosahedral number">Icosahedral</a></li> <li><a href="/wiki/Stella_octangula_number" title="Stella octangula number">Stella octangula</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Pyramidal_number" title="Pyramidal number">pyramidal</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Square_pyramidal_number" title="Square pyramidal number">Square pyramidal</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Four-dimensional_space" title="Four-dimensional space">4-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">non-centered</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pentatope_number" title="Pentatope number">Pentatope</a></li> <li><a href="/wiki/Squared_triangular_number" title="Squared triangular number">Squared triangular</a></li> <li><a href="/wiki/Fourth_power" title="Fourth power">Tesseractic</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Combinatorial_numbers" style="font-size:114%;margin:0 4em">Combinatorial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell_number" title="Bell number">Bell</a></li> <li><a href="/wiki/Cake_number" title="Cake number">Cake</a></li> <li><a href="/wiki/Catalan_number" title="Catalan number">Catalan</a></li> <li><a href="/wiki/Dedekind_number" title="Dedekind number">Dedekind</a></li> <li><a href="/wiki/Delannoy_number" title="Delannoy number">Delannoy</a></li> <li><a href="/wiki/Euler_number" class="mw-redirect" title="Euler number">Euler</a></li> <li><a href="/wiki/Eulerian_number" title="Eulerian number">Eulerian</a></li> <li><a href="/wiki/Fuss%E2%80%93Catalan_number" title="Fuss–Catalan number">Fuss–Catalan</a></li> <li><a href="/wiki/Lah_number" title="Lah number">Lah</a></li> <li><a href="/wiki/Lazy_caterer%27s_sequence" title="Lazy caterer's sequence">Lazy caterer's sequence</a></li> <li><a href="/wiki/Lobb_number" title="Lobb number">Lobb</a></li> <li><a href="/wiki/Motzkin_number" title="Motzkin number">Motzkin</a></li> <li><a href="/wiki/Narayana_number" title="Narayana number">Narayana</a></li> <li><a href="/wiki/Ordered_Bell_number" title="Ordered Bell number">Ordered Bell</a></li> <li><a href="/wiki/Schr%C3%B6der_number" title="Schröder number">Schröder</a></li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Hipparchus_number" title="Schröder–Hipparchus number">Schröder–Hipparchus</a></li> <li><a href="/wiki/Stirling_numbers_of_the_first_kind" title="Stirling numbers of the first kind">Stirling first</a></li> <li><a href="/wiki/Stirling_numbers_of_the_second_kind" title="Stirling numbers of the second kind">Stirling second</a></li> <li><a href="/wiki/Telephone_number_(mathematics)" title="Telephone number (mathematics)">Telephone number</a></li> <li><a href="/wiki/Wedderburn%E2%80%93Etherington_number" title="Wedderburn–Etherington number">Wedderburn–Etherington</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Primes" style="font-size:114%;margin:0 4em"><a href="/wiki/Prime_number" title="Prime number">Primes</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Wieferich_prime#Wieferich_numbers" title="Wieferich prime">Wieferich</a></li> <li><a href="/wiki/Wall%E2%80%93Sun%E2%80%93Sun_prime" title="Wall–Sun–Sun prime">Wall–Sun–Sun</a></li> <li><a href="/wiki/Wolstenholme_prime" title="Wolstenholme prime">Wolstenholme prime</a></li> <li><a href="/wiki/Wilson_prime#Wilson_numbers" title="Wilson prime">Wilson</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Pseudoprimes" style="font-size:114%;margin:0 4em"><a href="/wiki/Pseudoprime" title="Pseudoprime">Pseudoprimes</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Carmichael_number" title="Carmichael number">Carmichael number</a></li> <li><a href="/wiki/Catalan_pseudoprime" title="Catalan pseudoprime">Catalan pseudoprime</a></li> <li><a href="/wiki/Elliptic_pseudoprime" title="Elliptic pseudoprime">Elliptic pseudoprime</a></li> <li><a href="/wiki/Euler_pseudoprime" title="Euler pseudoprime">Euler pseudoprime</a></li> <li><a href="/wiki/Euler%E2%80%93Jacobi_pseudoprime" title="Euler–Jacobi pseudoprime">Euler–Jacobi pseudoprime</a></li> <li><a href="/wiki/Fermat_pseudoprime" title="Fermat pseudoprime">Fermat pseudoprime</a></li> <li><a href="/wiki/Frobenius_pseudoprime" title="Frobenius pseudoprime">Frobenius pseudoprime</a></li> <li><a href="/wiki/Lucas_pseudoprime" title="Lucas pseudoprime">Lucas pseudoprime</a></li> <li><a href="/wiki/Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael number</a></li> <li><a href="/wiki/Perrin_number#Perrin_primality_test" title="Perrin number">Perrin pseudoprime</a></li> <li><a href="/wiki/Somer%E2%80%93Lucas_pseudoprime" title="Somer–Lucas pseudoprime">Somer–Lucas pseudoprime</a></li> <li><a href="/wiki/Strong_pseudoprime" title="Strong pseudoprime">Strong pseudoprime</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Arithmetic_functions_and_dynamics" style="font-size:114%;margin:0 4em"><a href="/wiki/Arithmetic_function" title="Arithmetic function">Arithmetic functions</a> and <a href="/wiki/Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Divisor_function" title="Divisor function">Divisor functions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abundant_number" title="Abundant number">Abundant</a></li> <li><a href="/wiki/Almost_perfect_number" title="Almost perfect number">Almost perfect</a></li> <li><a href="/wiki/Arithmetic_number" title="Arithmetic number">Arithmetic</a></li> <li><a href="/wiki/Betrothed_numbers" title="Betrothed numbers">Betrothed</a></li> <li><a href="/wiki/Colossally_abundant_number" title="Colossally abundant number">Colossally abundant</a></li> <li><a href="/wiki/Deficient_number" title="Deficient number">Deficient</a></li> <li><a href="/wiki/Descartes_number" title="Descartes number">Descartes</a></li> <li><a href="/wiki/Hemiperfect_number" title="Hemiperfect number">Hemiperfect</a></li> <li><a href="/wiki/Highly_abundant_number" title="Highly abundant number">Highly abundant</a></li> <li><a href="/wiki/Highly_composite_number" title="Highly composite number">Highly composite</a></li> <li><a href="/wiki/Hyperperfect_number" title="Hyperperfect number">Hyperperfect</a></li> <li><a href="/wiki/Multiply_perfect_number" title="Multiply perfect number">Multiply perfect</a></li> <li><a href="/wiki/Perfect_number" title="Perfect number">Perfect</a></li> <li><a href="/wiki/Practical_number" title="Practical number">Practical</a></li> <li><a href="/wiki/Primitive_abundant_number" title="Primitive abundant number">Primitive abundant</a></li> <li><a href="/wiki/Quasiperfect_number" title="Quasiperfect number">Quasiperfect</a></li> <li><a href="/wiki/Refactorable_number" title="Refactorable number">Refactorable</a></li> <li><a href="/wiki/Semiperfect_number" title="Semiperfect number">Semiperfect</a></li> <li><a href="/wiki/Sublime_number" title="Sublime number">Sublime</a></li> <li><a href="/wiki/Superabundant_number" title="Superabundant number">Superabundant</a></li> <li><a href="/wiki/Superior_highly_composite_number" title="Superior highly composite number">Superior highly composite</a></li> <li><a href="/wiki/Superperfect_number" title="Superperfect number">Superperfect</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Prime_omega_function" title="Prime omega function">Prime omega functions</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Almost_prime" title="Almost prime">Almost prime</a></li> <li><a href="/wiki/Semiprime" title="Semiprime">Semiprime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Euler%27s_totient_function" title="Euler's totient function">Euler's totient function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Highly_cototient_number" title="Highly cototient number">Highly cototient</a></li> <li><a href="/wiki/Highly_totient_number" title="Highly totient number">Highly totient</a></li> <li><a href="/wiki/Noncototient" title="Noncototient">Noncototient</a></li> <li><a href="/wiki/Nontotient" title="Nontotient">Nontotient</a></li> <li><a href="/wiki/Perfect_totient_number" title="Perfect totient number">Perfect totient</a></li> <li><a href="/wiki/Sparsely_totient_number" title="Sparsely totient number">Sparsely totient</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Aliquot_sequence" title="Aliquot sequence">Aliquot sequences</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amicable_numbers" title="Amicable numbers">Amicable</a></li> <li><a href="/wiki/Perfect_number" title="Perfect number">Perfect</a></li> <li><a href="/wiki/Sociable_numbers" class="mw-redirect" title="Sociable numbers">Sociable</a></li> <li><a href="/wiki/Untouchable_number" title="Untouchable number">Untouchable</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Primorial" title="Primorial">Primorial</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Euclid_number" title="Euclid number">Euclid</a></li> <li><a href="/wiki/Fortunate_number" title="Fortunate number">Fortunate</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_prime_factor_or_divisor_related_numbers" style="font-size:114%;margin:0 4em">Other <a href="/wiki/Prime_factor" class="mw-redirect" title="Prime factor">prime factor</a> or <a href="/wiki/Divisor" title="Divisor">divisor</a> related numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Blum_integer" title="Blum integer">Blum</a></li> <li><a href="/wiki/Cyclic_number_(group_theory)" title="Cyclic number (group theory)">Cyclic</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Nicolas_number" title="Erdős–Nicolas number">Erdős–Nicolas</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Woods_number" title="Erdős–Woods number">Erdős–Woods</a></li> <li><a href="/wiki/Friendly_number" title="Friendly number">Friendly</a></li> <li><a href="/wiki/Giuga_number" title="Giuga number">Giuga</a></li> <li><a href="/wiki/Harmonic_divisor_number" title="Harmonic divisor number">Harmonic divisor</a></li> <li><a href="/wiki/Jordan%E2%80%93P%C3%B3lya_number" title="Jordan–Pólya number">Jordan–Pólya</a></li> <li><a href="/wiki/Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael</a></li> <li><a href="/wiki/Pronic_number" title="Pronic number">Pronic</a></li> <li><a href="/wiki/Regular_number" title="Regular number">Regular</a></li> <li><a href="/wiki/Rough_number" title="Rough number">Rough</a></li> <li><a href="/wiki/Smooth_number" title="Smooth number">Smooth</a></li> <li><a href="/wiki/Sphenic_number" title="Sphenic number">Sphenic</a></li> <li><a href="/wiki/St%C3%B8rmer_number" title="Størmer number">Størmer</a></li> <li><a href="/wiki/Super-Poulet_number" title="Super-Poulet number">Super-Poulet</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Numeral_system-dependent_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Numeral_system" title="Numeral system">Numeral system</a>-dependent numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Arithmetic_function" title="Arithmetic function">Arithmetic functions</a> <br />and <a href="/wiki/Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Persistence_of_a_number" title="Persistence of a number">Persistence</a> <ul><li><a href="/wiki/Additive_persistence" class="mw-redirect" title="Additive persistence">Additive</a></li> <li><a href="/wiki/Multiplicative_persistence" class="mw-redirect" title="Multiplicative persistence">Multiplicative</a></li></ul></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Digit_sum" title="Digit sum">Digit sum</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Digit_sum" title="Digit sum">Digit sum</a></li> <li><a href="/wiki/Digital_root" title="Digital root">Digital root</a></li> <li><a href="/wiki/Self_number" title="Self number">Self</a></li> <li><a href="/wiki/Sum-product_number" title="Sum-product number">Sum-product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digit product</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Multiplicative_digital_root" title="Multiplicative digital root">Multiplicative digital root</a></li> <li><a href="/wiki/Sum-product_number" title="Sum-product number">Sum-product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Coding-related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Meertens_number" title="Meertens number">Meertens</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dudeney_number" title="Dudeney number">Dudeney</a></li> <li><a href="/wiki/Factorion" title="Factorion">Factorion</a></li> <li><a href="/wiki/Kaprekar_number" title="Kaprekar number">Kaprekar</a></li> <li><a href="/wiki/Kaprekar%27s_routine" title="Kaprekar's routine">Kaprekar's constant</a></li> <li><a href="/wiki/Keith_number" title="Keith number">Keith</a></li> <li><a href="/wiki/Lychrel_number" title="Lychrel number">Lychrel</a></li> <li><a href="/wiki/Narcissistic_number" title="Narcissistic number">Narcissistic</a></li> <li><a href="/wiki/Perfect_digit-to-digit_invariant" title="Perfect digit-to-digit invariant">Perfect digit-to-digit invariant</a></li> <li><a href="/wiki/Perfect_digital_invariant" title="Perfect digital invariant">Perfect digital invariant</a> <ul><li><a href="/wiki/Happy_number" title="Happy number">Happy</a></li></ul></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/P-adic_numbers" class="mw-redirect" title="P-adic numbers">P-adic numbers</a>-related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Automorphic_number" title="Automorphic number">Automorphic</a> <ul><li><a href="/wiki/Trimorphic_number" class="mw-redirect" title="Trimorphic number">Trimorphic</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Numerical_digit" title="Numerical digit">Digit</a>-composition related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Palindromic_number" title="Palindromic number">Palindromic</a></li> <li><a href="/wiki/Pandigital_number" title="Pandigital number">Pandigital</a></li> <li><a href="/wiki/Repdigit" title="Repdigit">Repdigit</a></li> <li><a href="/wiki/Repunit" title="Repunit">Repunit</a></li> <li><a href="/wiki/Self-descriptive_number" title="Self-descriptive number">Self-descriptive</a></li> <li><a href="/wiki/Smarandache%E2%80%93Wellin_number" title="Smarandache–Wellin number">Smarandache–Wellin</a></li> <li><a href="/wiki/Undulating_number" title="Undulating number">Undulating</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digit-<a href="/wiki/Permutation" title="Permutation">permutation</a> related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cyclic_number" title="Cyclic number">Cyclic</a></li> <li><a href="/wiki/Digit-reassembly_number" title="Digit-reassembly number">Digit-reassembly</a></li> <li><a href="/wiki/Parasitic_number" title="Parasitic number">Parasitic</a></li> <li><a href="/wiki/Primeval_number" title="Primeval number">Primeval</a></li> <li><a href="/wiki/Transposable_integer" title="Transposable integer">Transposable</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Divisor-related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Equidigital_number" title="Equidigital number">Equidigital</a></li> <li><a href="/wiki/Extravagant_number" title="Extravagant number">Extravagant</a></li> <li><a href="/wiki/Frugal_number" title="Frugal number">Frugal</a></li> <li><a href="/wiki/Harshad_number" title="Harshad number">Harshad</a></li> <li><a href="/wiki/Polydivisible_number" title="Polydivisible number">Polydivisible</a></li> <li><a href="/wiki/Smith_number" title="Smith number">Smith</a></li> <li><a href="/wiki/Vampire_number" title="Vampire number">Vampire</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Friedman_number" title="Friedman number">Friedman</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Binary_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Binary_number" title="Binary number">Binary numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Evil_number" title="Evil number">Evil</a></li> <li><a href="/wiki/Odious_number" title="Odious number">Odious</a></li> <li><a href="/wiki/Pernicious_number" title="Pernicious number">Pernicious</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Generated_via_a_sieve" style="font-size:114%;margin:0 4em">Generated via a <a href="/wiki/Sieve_theory" title="Sieve theory">sieve</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Lucky_number" title="Lucky number">Lucky</a></li> <li><a href="/wiki/Generation_of_primes" title="Generation of primes">Prime</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Sorting_related" style="font-size:114%;margin:0 4em"><a href="/wiki/Sorting_algorithm" title="Sorting algorithm">Sorting</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pancake_sorting" title="Pancake sorting">Pancake number</a></li> <li><a href="/wiki/Sorting_number" title="Sorting number">Sorting number</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Natural_language_related" style="font-size:114%;margin:0 4em"><a href="/wiki/Natural_language" title="Natural language">Natural language</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Aronson%27s_sequence" title="Aronson's sequence">Aronson's sequence</a></li> <li><a href="/wiki/Ban_number" title="Ban number">Ban</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Graphemics_related" style="font-size:114%;margin:0 4em"><a href="/wiki/Graphemics" title="Graphemics">Graphemics</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Strobogrammatic_number" title="Strobogrammatic number">Strobogrammatic</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td 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