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Available in 25 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-25" 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">25 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_%D8%B4%D9%83%D9%84%D9%8A" title="عدد شكلي – Arabic" lang="ar" hreflang="ar" data-title="عدد شكلي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_figurat" title="Nombre figurat – Catalan" lang="ca" hreflang="ca" data-title="Nombre figurat" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Figurierte_Zahl" title="Figurierte Zahl – German" lang="de" hreflang="de" data-title="Figurierte Zahl" 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_endecagon%C3%A8l" title="Nùmer endecagonèl – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Nùmer endecagonèl" 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-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_figurado" title="Número figurado – Spanish" lang="es" hreflang="es" data-title="Número figurado" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Figuriga_nombro" title="Figuriga nombro – Esperanto" lang="eo" hreflang="eo" data-title="Figuriga 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_figur%C3%A9" title="Nombre figuré – French" lang="fr" hreflang="fr" data-title="Nombre figuré" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/N%C3%BAmero_figurado" title="Número figurado – Galician" lang="gl" hreflang="gl" data-title="Número figurado" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bilangan_bergambar" title="Bilangan bergambar – Indonesian" lang="id" hreflang="id" data-title="Bilangan bergambar" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_figurato" title="Numero figurato – Italian" lang="it" hreflang="it" data-title="Numero figurato" 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/Figur%C3%A1lis_sz%C3%A1mok" title="Figurális számok – Hungarian" lang="hu" hreflang="hu" data-title="Figurális számok" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Figuratief_getal" title="Figuratief getal – Dutch" lang="nl" hreflang="nl" data-title="Figuratief getal" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%9B%B3%E5%BD%A2%E6%95%B0" title="図形数 – Japanese" lang="ja" hreflang="ja" data-title="図形数" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Figurtall" title="Figurtall – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Figurtall" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Figurtal" title="Figurtal – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Figurtal" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero_figurado" title="Número figurado – Portuguese" lang="pt" hreflang="pt" data-title="Número figurado" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Num%C4%83r_figurativ" title="Număr figurativ – Romanian" lang="ro" hreflang="ro" data-title="Număr figurativ" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ru.wikipedia.org/wiki/%D0%A4%D0%B8%D0%B3%D1%83%D1%80%D0%BD%D1%8B%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Фигурные числа – Russian" lang="ru" hreflang="ru" data-title="Фигурные числа" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A4%D0%B8%D0%B3%D1%83%D1%80%D0%B0%D1%82%D0%B8%D0%B2%D0%BD%D0%B8_%D0%B1%D1%80%D0%BE%D1%98" title="Фигуративни број – Serbian" lang="sr" hreflang="sr" data-title="Фигуративни број" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kuvioluku" title="Kuvioluku – Finnish" lang="fi" hreflang="fi" data-title="Kuvioluku" 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/Figurtal" title="Figurtal – Swedish" lang="sv" hreflang="sv" data-title="Figurtal" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li 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src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Pascal_triangle_simplex_numbers.svg/220px-Pascal_triangle_simplex_numbers.svg.png" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Pascal_triangle_simplex_numbers.svg/330px-Pascal_triangle_simplex_numbers.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Pascal_triangle_simplex_numbers.svg/440px-Pascal_triangle_simplex_numbers.svg.png 2x" data-file-width="512" data-file-height="341" /></a><figcaption>Derivation of <a href="/wiki/Hyperpyramid" title="Hyperpyramid">hyperpyramidal</a> figurate numbers from a left-justified <a href="/wiki/Pascal%27s_triangle" title="Pascal's triangle">Pascal's triangle</a>. <style data-mw-deduplicate="TemplateStyles:r981673959">.mw-parser-output .legend{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .legend-color{display:inline-block;min-width:1.25em;height:1.25em;line-height:1.25;margin:1px 0;text-align:center;border:1px solid black;background-color:transparent;color:black}.mw-parser-output .legend-text{}</style><div class="legend"><span class="legend-color mw-no-invert" style="background-color:#a67726; color:black;-webkit-print-color-adjust: exact; print-color-adjust: exact;"> </span> <a href="/wiki/Natural_number" title="Natural number">Natural numbers</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959"><div class="legend"><span class="legend-color mw-no-invert" style="background-color:#e238fc; color:black;-webkit-print-color-adjust: exact; print-color-adjust: exact;"> </span> <a href="/wiki/Triangular_number" title="Triangular number">Triangular numbers</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959"><div class="legend"><span class="legend-color mw-no-invert" style="background-color:#ff731c; color:black;-webkit-print-color-adjust: exact; print-color-adjust: exact;"> </span> <a href="/wiki/Tetrahedral_number" title="Tetrahedral number">Tetrahedral numbers</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959"><div class="legend"><span class="legend-color mw-no-invert" style="background-color:yellow; color:black;-webkit-print-color-adjust: exact; print-color-adjust: exact;"> </span> <a href="/wiki/Pentatope_number" title="Pentatope number">Pentatope numbers</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959"><div class="legend"><span class="legend-color mw-no-invert" style="background-color:lime; color:black;-webkit-print-color-adjust: exact; print-color-adjust: exact;"> </span> <a href="/wiki/5-simplex" title="5-simplex">5-simplex</a> numbers</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959"><div class="legend"><span class="legend-color mw-no-invert" style="background-color:#036bfc; color:white;-webkit-print-color-adjust: exact; print-color-adjust: exact;"> </span> <a href="/wiki/6-simplex" title="6-simplex">6-simplex</a> numbers</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959"><div class="legend"><span class="legend-color mw-no-invert" style="background-color:#7f03fc; color:white;-webkit-print-color-adjust: exact; print-color-adjust: exact;"> </span> <a href="/wiki/7-simplex" title="7-simplex">7-simplex</a> numbers</div> </figcaption></figure> <p>The term <b>figurate number</b> is used by different writers for members of different sets of numbers, generalizing from <a href="/wiki/Triangular_numbers" class="mw-redirect" title="Triangular numbers">triangular numbers</a> to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean </p> <ul><li><a href="/wiki/Polygonal_number" title="Polygonal number">polygonal number</a></li> <li>a number represented as a discrete <span class="texhtml mvar" style="font-style:italic;">r</span>-dimensional regular <a href="/wiki/Geometry" title="Geometry">geometric</a> pattern of <span class="texhtml mvar" style="font-style:italic;">r</span>-dimensional <a href="/wiki/Ball_(mathematics)" title="Ball (mathematics)">balls</a> such as a polygonal number (for <span class="texhtml"><i>r</i> = 2</span>) or a <b>polyhedral number</b> (for <span class="texhtml"><i>r</i> = 3</span>).</li> <li>a member of the subset of the sets above containing only triangular numbers, <a href="/wiki/Pyramidal_numbers" class="mw-redirect" title="Pyramidal numbers">pyramidal numbers</a>, and their analogs in other dimensions.<sup id="cite_ref-Dickson_1-0" class="reference"><a href="#cite_note-Dickson-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li></ul> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Terminology">Terminology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Figurate_number&action=edit&section=1" title="Edit section: Terminology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some kinds of figurate number were discussed in the 16th and 17th centuries under the name "figural number".<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>In historical works about <a href="/wiki/Greek_mathematics" title="Greek mathematics">Greek mathematics</a> the preferred term used to be <i>figured number</i>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>In a use going back to <a href="/wiki/Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a>'s <a href="/wiki/Ars_Conjectandi" title="Ars Conjectandi">Ars Conjectandi</a>,<sup id="cite_ref-Dickson_1-1" class="reference"><a href="#cite_note-Dickson-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> the term <i>figurate number</i> is used for <a href="/wiki/Triangular_number" title="Triangular number">triangular numbers</a> made up of successive <a href="/wiki/Integer" title="Integer">integers</a>, <a href="/wiki/Tetrahedral_number" title="Tetrahedral number">tetrahedral numbers</a> made up of successive triangular numbers, etc. These turn out to be the <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficients</a>. In this usage the <a href="/wiki/Square_number" title="Square number">square numbers</a> (4, 9, 16, 25, ...) would not be considered figurate numbers when viewed as arranged in a square. </p><p>A number of other sources use the term <i>figurate number</i> as synonymous for the <a href="/wiki/Polygonal_number" title="Polygonal number">polygonal numbers</a>, either just the usual kind or both those and the <a href="/wiki/Centered_polygonal_number" title="Centered polygonal number">centered polygonal numbers</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Figurate_number&action=edit&section=2" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The mathematical study of figurate numbers is said to have originated with <a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a>, possibly based on Babylonian or Egyptian precursors. Generating whichever class of figurate numbers the Pythagoreans studied using <a href="/wiki/Gnomon_(figure)" title="Gnomon (figure)">gnomons</a> is also attributed to Pythagoras. Unfortunately, there is no trustworthy source for these claims, because all surviving writings about the Pythagoreans<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> are from centuries later.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Speusippus" title="Speusippus">Speusippus</a> is the earliest source to expose the view that ten, as the fourth triangular number, was in fact the <a href="/wiki/Tetractys" title="Tetractys">tetractys</a>, supposed to be of great importance for <a href="/wiki/Pythagoreanism" title="Pythagoreanism">Pythagoreanism</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Figurate numbers were a concern of the Pythagorean worldview. It was well understood that some numbers could have many figurations, e.g. <a href="/wiki/36_(number)" title="36 (number)">36</a> is a both a square and a triangle and also various rectangles. </p><p>The modern study of figurate numbers goes back to <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a>, specifically the <a href="/wiki/Fermat_polygonal_number_theorem" title="Fermat polygonal number theorem">Fermat polygonal number theorem</a>. Later, it became a significant topic for <a href="/wiki/Euler" class="mw-redirect" title="Euler">Euler</a>, who gave an explicit formula for all <a href="/wiki/Square_triangular_number" title="Square triangular number">triangular numbers that are also perfect squares</a>, among many other discoveries relating to figurate numbers. </p><p>Figurate numbers have played a significant role in modern recreational mathematics.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> In research mathematics, figurate numbers are studied by way of the <a href="/wiki/Ehrhart_polynomial" title="Ehrhart polynomial">Ehrhart polynomials</a>, <a href="/wiki/Polynomial" title="Polynomial">polynomials</a> that count the number of integer points in a polygon or polyhedron when it is expanded by a given factor.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Triangular_numbers_and_their_analogs_in_higher_dimensions">Triangular numbers and their analogs in higher dimensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Figurate_number&action=edit&section=3" title="Edit section: Triangular numbers and their analogs in higher dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Triangular_number" title="Triangular number">triangular numbers</a> for <span class="texhtml"><i>n</i> = 1, 2, 3, ...</span> are the result of the juxtaposition of the linear numbers (linear gnomons) for <span class="texhtml"><i>n</i> = 1, 2, 3, ...</span>: </p> <table cellpadding="10"> <tbody><tr align="center" valign="bottom"> <td style="display: inline-block; line-height: 0;"><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> </td> <td style="display: inline-block; line-height: 0;"><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" 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data-file-width="20" data-file-height="17" /></a></span> </td> <td style="display: inline-block; line-height: 0;"><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, 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data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> </td> <td style="display: inline-block; line-height: 0;"><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> </td> <td style="display: inline-block; line-height: 0;"><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span><span typeof="mw:File"><a href="/wiki/File:GrayDotX.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/16px-GrayDotX.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/24px-GrayDotX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/GrayDotX.svg/32px-GrayDotX.svg.png 2x" data-file-width="20" data-file-height="17" /></a></span> </td></tr></tbody></table> <p>These are the binomial coefficients <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 \textstyle {\binom {n+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</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="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\binom {n+1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7934f2939b063b4a985e9e21240f978164a079f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.217ex; height:3.343ex;" alt="{\displaystyle \textstyle {\binom {n+1}{2}}}"></span>. This is the case <span class="texhtml"><i>r</i> = 2</span> of the fact that the <span class="texhtml mvar" style="font-style:italic;">r</span>th diagonal of <a href="/wiki/Pascal%27s_triangle#Patterns_and_properties" title="Pascal's triangle">Pascal's triangle</a> for <span class="texhtml"><i>r</i> ≥ 0</span> consists of the figurate numbers for the <span class="texhtml mvar" style="font-style:italic;">r</span>-dimensional analogs of triangles (<span class="texhtml mvar" style="font-style:italic;">r</span>-dimensional <a href="/wiki/Simplex" title="Simplex">simplices</a>). </p><p>The simplicial polytopic numbers for <span class="texhtml"><i>r</i> = 1, 2, 3, 4, ...</span> are: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}(n)={\frac {n}{1}}={\binom {n+0}{1}}={\binom {n}{1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>1</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>0</mn> </mrow> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </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"> <mi>n</mi> <mn>1</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 P_{1}(n)={\frac {n}{1}}={\binom {n+0}{1}}={\binom {n}{1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a0860410fbf45119921d4ae3f626b4224ab4ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.911ex; height:6.176ex;" alt="{\displaystyle P_{1}(n)={\frac {n}{1}}={\binom {n+0}{1}}={\binom {n}{1}}}"></span> (linear numbers),</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{2}(n)={\frac {n(n+1)}{2}}={\binom {n+1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <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> <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 P_{2}(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/a7a394d54ef738605541cccf4f0e2b7b84d67457" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.204ex; height:6.343ex;" alt="{\displaystyle P_{2}(n)={\frac {n(n+1)}{2}}={\binom {n+1}{2}}}"></span> (<a href="/wiki/Triangular_number" title="Triangular number">triangular numbers</a>),</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{3}(n)={\frac {n(n+1)(n+2)}{6}}={\binom {n+2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <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> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mn>6</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>2</mn> </mrow> <mn>3</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 P_{3}(n)={\frac {n(n+1)(n+2)}{6}}={\binom {n+2}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61e6100dca09dfc501d4cc7b34eb144326cac199" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.411ex; height:6.343ex;" alt="{\displaystyle P_{3}(n)={\frac {n(n+1)(n+2)}{6}}={\binom {n+2}{3}}}"></span> (<a href="/wiki/Tetrahedral_number" title="Tetrahedral number">tetrahedral numbers</a>),</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{4}(n)={\frac {n(n+1)(n+2)(n+3)}{24}}={\binom {n+3}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <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> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <mn>24</mn> </mfrac> </mrow> <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>3</mn> </mrow> <mn>4</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 P_{4}(n)={\frac {n(n+1)(n+2)(n+3)}{24}}={\binom {n+3}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cffc5c6cbcb6241bfbbadad88e50328e25c883b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.618ex; height:6.343ex;" alt="{\displaystyle P_{4}(n)={\frac {n(n+1)(n+2)(n+3)}{24}}={\binom {n+3}{4}}}"></span> (pentachoric numbers, <a href="/wiki/Pentatopic_number" class="mw-redirect" title="Pentatopic number">pentatopic numbers</a>, 4-simplex numbers),</li></ul> <p><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 \qquad \vdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> <mo>⋮</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad \vdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df187a85775c68fcd9530ab6fdb13c0c1c554a2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.292ex; height:3.676ex;" alt="{\displaystyle \qquad \vdots }"></span> </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{r}(n)={\frac {n(n+1)(n+2)\cdots (n+r-1)}{r!}}={\binom {n+(r-1)}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <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> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>r</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>r</mi> <mo>!</mo> </mrow> </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> <mo stretchy="false">(</mo> <mi>r</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mi>r</mi> </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 P_{r}(n)={\frac {n(n+1)(n+2)\cdots (n+r-1)}{r!}}={\binom {n+(r-1)}{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2566277e182862bc7f8c155a4f19837eec5ab5cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:57.622ex; height:6.343ex;" alt="{\displaystyle P_{r}(n)={\frac {n(n+1)(n+2)\cdots (n+r-1)}{r!}}={\binom {n+(r-1)}{r}}}"></span> (<span class="texhtml mvar" style="font-style:italic;">r</span>-topic numbers, <span class="texhtml mvar" style="font-style:italic;">r</span>-<a href="/wiki/Simplex" title="Simplex">simplex</a> numbers).</li></ul> <p>The terms <i><a href="/wiki/Square_number" title="Square number">square number</a></i> and <i><a href="/wiki/Cubic_number" class="mw-redirect" title="Cubic number">cubic number</a></i> derive from their geometric representation as a <a href="/wiki/Square_(geometry)" class="mw-redirect" title="Square (geometry)">square</a> or <a href="/wiki/Cube_(geometry)" class="mw-redirect" title="Cube (geometry)">cube</a>. The difference of two positive triangular numbers is a <a href="/wiki/Trapezoidal_number" class="mw-redirect" title="Trapezoidal number">trapezoidal number</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Gnomon">Gnomon</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Figurate_number&action=edit&section=4" title="Edit section: Gnomon"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Gnomon_(figure)#Building_figurate_numbers" title="Gnomon (figure)">Gnomon (figure) § Building figurate numbers</a></div> <p>The <b>gnomon</b> is the piece added to a figurate number to transform it to the next larger one. </p><p>For example, the gnomon of the square number is the <a href="/wiki/Odd_number" class="mw-redirect" title="Odd number">odd number</a>, of the general form <span class="texhtml">2<i>n</i> + 1</span>, <span class="texhtml"><i>n</i> = 0, 1, 2, 3, ...</span>. The square of size 8 composed of gnomons looks like this: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}1&2&3&4&5&6&7&8\\2&2&3&4&5&6&7&8\\3&3&3&4&5&6&7&8\\4&4&4&4&5&6&7&8\\5&5&5&5&5&6&7&8\\6&6&6&6&6&6&7&8\\7&7&7&7&7&7&7&8\\8&8&8&8&8&8&8&8\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mn>5</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mn>6</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mn>7</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mn>8</mn> </mtd> <mtd> <mn>8</mn> </mtd> <mtd> <mn>8</mn> </mtd> <mtd> <mn>8</mn> </mtd> <mtd> <mn>8</mn> </mtd> <mtd> <mn>8</mn> </mtd> <mtd> <mn>8</mn> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}1&2&3&4&5&6&7&8\\2&2&3&4&5&6&7&8\\3&3&3&4&5&6&7&8\\4&4&4&4&5&6&7&8\\5&5&5&5&5&6&7&8\\6&6&6&6&6&6&7&8\\7&7&7&7&7&7&7&8\\8&8&8&8&8&8&8&8\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9222f74cc636e94417cce30a499c73c73bf956a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.171ex; width:26.309ex; height:25.509ex;" alt="{\displaystyle {\begin{matrix}1&2&3&4&5&6&7&8\\2&2&3&4&5&6&7&8\\3&3&3&4&5&6&7&8\\4&4&4&4&5&6&7&8\\5&5&5&5&5&6&7&8\\6&6&6&6&6&6&7&8\\7&7&7&7&7&7&7&8\\8&8&8&8&8&8&8&8\end{matrix}}}"></span></dd></dl> <p>To transform from the <i><span class="texhtml mvar" style="font-style:italic;">n</span>-square</i> (the square of size <span class="texhtml mvar" style="font-style:italic;">n</span>) to the <span class="texhtml">(<i>n</i> + 1)</span>-square, one adjoins <span class="texhtml">2<i>n</i> + 1</span> elements: one to the end of each row (<span class="texhtml mvar" style="font-style:italic;">n</span> elements), one to the end of each column (<span class="texhtml mvar" style="font-style:italic;">n</span> elements), and a single one to the corner. For example, when transforming the 7-square to the 8-square, we add 15 elements; these adjunctions are the 8s in the above figure. </p><p>This gnomonic technique also provides a <a href="/wiki/Mathematical_proof" title="Mathematical proof">mathematical proof</a> that the sum of the first <span class="texhtml mvar" style="font-style:italic;">n</span> odd numbers is <span class="texhtml"><i>n</i><sup>2</sup></span>; the figure illustrates <span class="nowrap">1 + 3 + 5 + 7 + 9 + 11 + 13 + 15</span> = 64 = 8<sup>2</sup>. </p><p>There is a similar <b>gnomon</b> with <a href="/wiki/Centered_hexagonal_number" title="Centered hexagonal number">centered hexagonal numbers</a> adding up to make cubes of each integer number. </p> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Figurate_number&action=edit&section=5" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Dickson-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Dickson_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Dickson_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><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="CITEREFDickson1919" class="citation book cs1"><a href="/wiki/Leonard_Eugene_Dickson" title="Leonard Eugene Dickson">Dickson, L. E.</a> (1919). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://archive.org/details/historyoftheoryo02dick"><i>History of the Theory of Numbers</i></a></span>. Vol. 2. p. 3. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8284-0086-2" title="Special:BookSources/978-0-8284-0086-2"><bdi>978-0-8284-0086-2</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2021-08-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+of+the+Theory+of+Numbers&rft.pages=3&rft.date=1919&rft.isbn=978-0-8284-0086-2&rft.aulast=Dickson&rft.aufirst=L.+E.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryoftheoryo02dick&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFigurate+number" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSimpsonWeiner1992" class="citation encyclopaedia cs1">Simpson, J. A.; Weiner, E. S. C., eds. (1992). "Figural number". <i>The Compact Oxford English Dictionary</i> (2nd ed.). Oxford, England: Clarendon Press. p. 587.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Figural+number&rft.btitle=The+Compact+Oxford+English+Dictionary&rft.place=Oxford%2C+England&rft.pages=587&rft.edition=2nd&rft.pub=Clarendon+Press&rft.date=1992&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFigurate+number" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeath1921" class="citation book cs1"><a href="/wiki/Thomas_Little_Heath" class="mw-redirect" title="Thomas Little Heath">Heath, Sir Thomas</a> (1921). <i><a href="/wiki/A_History_of_Greek_Mathematics" title="A History of Greek Mathematics">A History of Greek Mathematics</a></i>. Vol. 1. Oxford at the Clarendon Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Greek+Mathematics&rft.pub=Oxford+at+the+Clarendon+Press&rft.date=1921&rft.aulast=Heath&rft.aufirst=Sir+Thomas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFigurate+number" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMaziarzGreenwood1968" class="citation book cs1">Maziarz, Edward A.; Greenwood, Thomas (1968). <i>Greek Mathematical Philosophy</i>. 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In: L. Schimmelpfennig (ed.): <i>Number and Letter Systems in the Service of Religious Education</i>. Tübingen: Seraphim, 2019. p.25-45</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKraitchik2006" class="citation book cs1">Kraitchik, Maurice (2006). <i>Mathematical Recreations</i> (2nd revised ed.). <a href="/wiki/Dover_Books" class="mw-redirect" title="Dover Books">Dover Books</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-45358-3" title="Special:BookSources/978-0-486-45358-3"><bdi>978-0-486-45358-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Recreations&rft.edition=2nd+revised&rft.pub=Dover+Books&rft.date=2006&rft.isbn=978-0-486-45358-3&rft.aulast=Kraitchik&rft.aufirst=Maurice&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFigurate+number" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeckDe_LoeraDevelinPfeifle2005" class="citation book cs1">Beck, M.; <a href="/wiki/Jes%C3%BAs_A._De_Loera" title="Jesús A. De Loera">De Loera, J. A.</a>; Develin, M.; Pfeifle, J.; <a href="/wiki/Richard_P._Stanley" title="Richard P. Stanley">Stanley, R. P.</a> (2005). "Coefficients and roots of Ehrhart polynomials". <i>Integer points in polyhedra—geometry, number theory, algebra, optimization</i>. Contemp. Math. Vol. 374. Providence, RI: Amer. Math. Soc. pp. <span class="nowrap">15–</span>36. <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=2134759">2134759</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Coefficients+and+roots+of+Ehrhart+polynomials&rft.btitle=Integer+points+in+polyhedra%E2%80%94geometry%2C+number+theory%2C+algebra%2C+optimization&rft.place=Providence%2C+RI&rft.series=Contemp.+Math.&rft.pages=%3Cspan+class%3D%22nowrap%22%3E15-%3C%2Fspan%3E36&rft.pub=Amer.+Math.+Soc.&rft.date=2005&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2134759%23id-name%3DMR&rft.aulast=Beck&rft.aufirst=M.&rft.au=De+Loera%2C+J.+A.&rft.au=Develin%2C+M.&rft.au=Pfeifle%2C+J.&rft.au=Stanley%2C+R.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFigurate+number" class="Z3988"></span></span> </li> </ol></div></div> <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=Figurate_number&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGazalé1999" class="citation cs2">Gazalé, Midhat J. (1999), "Gnomon: From Pharaohs to Fractals", <i>European Journal of Physics</i>, <b>20</b> (6), <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>: 523, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1999EJPh...20..523G">1999EJPh...20..523G</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0143-0807%2F20%2F6%2F501">10.1088/0143-0807/20/6/501</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-00514-0" title="Special:BookSources/978-0-691-00514-0"><bdi>978-0-691-00514-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=European+Journal+of+Physics&rft.atitle=Gnomon%3A+From+Pharaohs+to+Fractals&rft.volume=20&rft.issue=6&rft.pages=523&rft.date=1999&rft_id=info%3Adoi%2F10.1088%2F0143-0807%2F20%2F6%2F501&rft_id=info%3Abibcode%2F1999EJPh...20..523G&rft.isbn=978-0-691-00514-0&rft.aulast=Gazal%C3%A9&rft.aufirst=Midhat+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFigurate+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDezaDeza2012" class="citation cs2"><a href="/wiki/Elena_Deza" title="Elena Deza">Deza, Elena</a>; Deza, Michel Marie (2012), <i>Figurate Numbers, First Edition</i>, <a href="/wiki/World_Scientific" title="World Scientific">World Scientific</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4355-48-3" title="Special:BookSources/978-981-4355-48-3"><bdi>978-981-4355-48-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Figurate+Numbers%2C+First+Edition&rft.pub=World+Scientific&rft.date=2012&rft.isbn=978-981-4355-48-3&rft.aulast=Deza&rft.aufirst=Elena&rft.au=Deza%2C+Michel+Marie&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFigurate+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeath2000" class="citation cs2">Heath, Thomas Little (2000), <i>A history of Greek Mathematics: Volume 1. From Thales to Euclid</i>, <a href="/w/index.php?title=Adamant_Media_Corporation&action=edit&redlink=1" class="new" title="Adamant Media Corporation (page does not exist)">Adamant Media Corporation</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-543-97448-8" title="Special:BookSources/978-0-543-97448-8"><bdi>978-0-543-97448-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+history+of+Greek+Mathematics%3A+Volume+1.+From+Thales+to+Euclid&rft.pub=Adamant+Media+Corporation&rft.date=2000&rft.isbn=978-0-543-97448-8&rft.aulast=Heath&rft.aufirst=Thomas+Little&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFigurate+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeath2000" class="citation cs2">Heath, Thomas Little (2000), <i>A history of Greek Mathematics: Volume 2. From Aristarchus to Diophantus</i>, <a href="/w/index.php?title=Adamant_Media_Corporation&action=edit&redlink=1" class="new" title="Adamant Media Corporation (page does not exist)">Adamant Media Corporation</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-543-96877-7" title="Special:BookSources/978-0-543-96877-7"><bdi>978-0-543-96877-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+history+of+Greek+Mathematics%3A+Volume+2.+From+Aristarchus+to+Diophantus&rft.pub=Adamant+Media+Corporation&rft.date=2000&rft.isbn=978-0-543-96877-7&rft.aulast=Heath&rft.aufirst=Thomas+Little&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFigurate+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDickson1923" class="citation cs2"><a href="/wiki/L._E._Dickson" class="mw-redirect" title="L. E. Dickson">Dickson, Leonard Eugene</a> (1923), <i><a href="/wiki/History_of_the_Theory_of_Numbers" title="History of the Theory of Numbers">History of the Theory of Numbers</a></i>, <a href="/w/index.php?title=Chelsea_Publishing_Company,_Inc.&action=edit&redlink=1" class="new" title="Chelsea Publishing Company, Inc. (page does not exist)">Chelsea Publishing Co</a>, <a href="/wiki/ASIN_(identifier)" class="mw-redirect" title="ASIN (identifier)">ASIN</a> <a rel="nofollow" class="external text" href="https://www.amazon.com/dp/B000OKO3TK">B000OKO3TK</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+of+the+Theory+of+Numbers&rft.pub=Chelsea+Publishing+Co&rft.date=1923&rft_id=https%3A%2F%2Fwww.amazon.com%2Fdp%2FB000OKO3TK%23id-name%3DASIN&rft.aulast=Dickson&rft.aufirst=Leonard+Eugene&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFigurate+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoyerMerzbach" class="citation cs2">Boyer, Carl B.; Merzbach, Uta C., <i>A History of Mathematics</i> (2nd ed.)</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematics&rft.edition=2nd&rft.aulast=Boyer&rft.aufirst=Carl+B.&rft.au=Merzbach%2C+Uta+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFigurate+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 dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul 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title="Centered square number">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-centered12" 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_numbers743" 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_numbers743" 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_numbers743" 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_±_1743" 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_numbers743" 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_numbers743" 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_numbers743" 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_sums743" 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_numbers743" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">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 href="/wiki/Centered_square_number" title="Centered square number">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_numbers743" 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="Primes743" 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="Pseudoprimes743" 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_dynamics743" 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_numbers743" 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_numbers743" 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_numbers743" 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_sieve743" 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_related743" 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_related743" 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_related743" 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 class="navbox-abovebelow hlist" colspan="2" style="font-weight:bold;"><div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Symbol_portal_class.svg" class="mw-file-description" title="Portal"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/16px-Symbol_portal_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/23px-Symbol_portal_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/31px-Symbol_portal_class.svg.png 2x" data-file-width="180" data-file-height="185" /></a></span> <a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></li></ul> </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="Sequences_and_series331" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><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:Series_(mathematics)" title="Template:Series (mathematics)"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Series_(mathematics)" title="Template talk:Series (mathematics)"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Series_(mathematics)" title="Special:EditPage/Template:Series (mathematics)"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Sequences_and_series331" style="font-size:114%;margin:0 4em"><a href="/wiki/Sequence" title="Sequence">Sequences</a> and <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Integer_sequence" title="Integer sequence">Integer sequences</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%;text-align:left">Basic</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/Arithmetic_progression" title="Arithmetic progression">Arithmetic progression</a></li> <li><a href="/wiki/Geometric_progression" title="Geometric progression">Geometric progression</a></li> <li><a href="/wiki/Harmonic_progression_(mathematics)" title="Harmonic progression (mathematics)">Harmonic progression</a></li> <li><a href="/wiki/Square_number" title="Square number">Square number</a></li> <li><a href="/wiki/Cube_(algebra)" title="Cube (algebra)">Cubic number</a></li> <li><a href="/wiki/Factorial" title="Factorial">Factorial</a></li> <li><a href="/wiki/Power_of_two" title="Power of two">Powers of two</a></li> <li><a href="/wiki/Power_of_three" title="Power of three">Powers of three</a></li> <li><a href="/wiki/Power_of_10" title="Power of 10">Powers of 10</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Advanced <span class="nobold">(<a href="/wiki/List_of_OEIS_sequences" class="mw-redirect" title="List of OEIS sequences">list</a>)</span></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/Complete_sequence" title="Complete sequence">Complete sequence</a></li> <li><a href="/wiki/Fibonacci_sequence" title="Fibonacci sequence">Fibonacci sequence</a></li> <li><a class="mw-selflink selflink">Figurate number</a></li> <li><a href="/wiki/Heptagonal_number" title="Heptagonal number">Heptagonal number</a></li> <li><a href="/wiki/Hexagonal_number" title="Hexagonal number">Hexagonal number</a></li> <li><a href="/wiki/Lucas_number" title="Lucas number">Lucas number</a></li> <li><a href="/wiki/Pell_number" title="Pell number">Pell number</a></li> <li><a href="/wiki/Pentagonal_number" title="Pentagonal number">Pentagonal number</a></li> <li><a href="/wiki/Polygonal_number" title="Polygonal number">Polygonal number</a></li> <li><a href="/wiki/Triangular_number" title="Triangular number">Triangular number</a> <ul><li><a href="/wiki/Triangular_array" title="Triangular array">array</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/Fibonacci_sequence" title="Fibonacci sequence"><img alt="Fibonacci spiral with square sizes up to 34." src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Fibonacci_spiral_34.svg/80px-Fibonacci_spiral_34.svg.png" decoding="async" width="80" height="51" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Fibonacci_spiral_34.svg/120px-Fibonacci_spiral_34.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Fibonacci_spiral_34.svg/160px-Fibonacci_spiral_34.svg.png 2x" data-file-width="915" data-file-height="579" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties of sequences</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cauchy_sequence" title="Cauchy sequence">Cauchy sequence</a></li> <li><a href="/wiki/Monotonic_function" title="Monotonic function">Monotonic function</a></li> <li><a href="/wiki/Periodic_sequence" title="Periodic sequence">Periodic sequence</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties of series</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%;text-align:left">Series</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/Alternating_series" title="Alternating series">Alternating</a></li> <li><a href="/wiki/Convergent_series" title="Convergent series">Convergent</a></li> <li><a href="/wiki/Divergent_series" title="Divergent series">Divergent</a></li> <li><a href="/wiki/Telescoping_series" title="Telescoping series">Telescoping</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Convergence</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/Absolute_convergence" title="Absolute convergence">Absolute</a></li> <li><a href="/wiki/Conditional_convergence" title="Conditional convergence">Conditional</a></li> <li><a href="/wiki/Uniform_convergence" title="Uniform convergence">Uniform</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Explicit series</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%;text-align:left">Convergent</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/1/2_%E2%88%92_1/4_%2B_1/8_%E2%88%92_1/16_%2B_%E2%8B%AF" title="1/2 − 1/4 + 1/8 − 1/16 + ⋯">1/2 − 1/4 + 1/8 − 1/16 + ⋯</a></li> <li><a href="/wiki/1/2_%2B_1/4_%2B_1/8_%2B_1/16_%2B_%E2%8B%AF" title="1/2 + 1/4 + 1/8 + 1/16 + ⋯">1/2 + 1/4 + 1/8 + 1/16 + ⋯</a></li> <li><a href="/wiki/1/4_%2B_1/16_%2B_1/64_%2B_1/256_%2B_%E2%8B%AF" title="1/4 + 1/16 + 1/64 + 1/256 + ⋯">1/4 + 1/16 + 1/64 + 1/256 + ⋯</a></li> <li><a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">1 + 1/2<sup><i>s</i></sup> + 1/3<sup><i>s</i></sup> + ... (Riemann zeta function)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Divergent</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/1_%2B_1_%2B_1_%2B_1_%2B_%E2%8B%AF" title="1 + 1 + 1 + 1 + ⋯">1 + 1 + 1 + 1 + ⋯</a></li> <li><a href="/wiki/Grandi%27s_series" title="Grandi's series">1 − 1 + 1 − 1 + ⋯ (Grandi's series)</a></li> <li><a href="/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF" title="1 + 2 + 3 + 4 + ⋯">1 + 2 + 3 + 4 + ⋯</a></li> <li><a href="/wiki/1_%E2%88%92_2_%2B_3_%E2%88%92_4_%2B_%E2%8B%AF" title="1 − 2 + 3 − 4 + ⋯">1 − 2 + 3 − 4 + ⋯</a></li> <li><a href="/wiki/1_%2B_2_%2B_4_%2B_8_%2B_%E2%8B%AF" title="1 + 2 + 4 + 8 + ⋯">1 + 2 + 4 + 8 + ⋯</a></li> <li><a href="/wiki/1_%E2%88%92_2_%2B_4_%E2%88%92_8_%2B_%E2%8B%AF" title="1 − 2 + 4 − 8 + ⋯">1 − 2 + 4 − 8 + ⋯</a></li> <li><a href="/wiki/Infinite_arithmetic_series" class="mw-redirect" title="Infinite arithmetic series">Infinite arithmetic series</a></li> <li><a href="/wiki/1_%E2%88%92_1_%2B_2_%E2%88%92_6_%2B_24_%E2%88%92_120_%2B_..." class="mw-redirect" title="1 − 1 + 2 − 6 + 24 − 120 + ...">1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)</a></li> <li><a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)</a></li> <li><a href="/wiki/Divergence_of_the_sum_of_the_reciprocals_of_the_primes" title="Divergence of the sum of the reciprocals of the primes">1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Kinds of series</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a></li> <li><a href="/wiki/Power_series" title="Power series">Power series</a></li> <li><a href="/wiki/Formal_power_series" title="Formal power series">Formal power series</a></li> <li><a href="/wiki/Laurent_series" title="Laurent series">Laurent series</a></li> <li><a href="/wiki/Puiseux_series" title="Puiseux series">Puiseux series</a></li> <li><a href="/wiki/Dirichlet_series" title="Dirichlet series">Dirichlet series</a></li> <li><a href="/wiki/Trigonometric_series" title="Trigonometric series">Trigonometric series</a></li> <li><a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a></li> <li><a href="/wiki/Generating_series" class="mw-redirect" title="Generating series">Generating series</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Hypergeometric_function" title="Hypergeometric function">Hypergeometric series</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Generalized_hypergeometric_series" class="mw-redirect" title="Generalized hypergeometric series">Generalized hypergeometric series</a></li> <li><a href="/wiki/Hypergeometric_function_of_a_matrix_argument" title="Hypergeometric function of a matrix argument">Hypergeometric function of a matrix argument</a></li> <li><a href="/wiki/Lauricella_hypergeometric_series" title="Lauricella hypergeometric series">Lauricella hypergeometric series</a></li> <li><a href="/wiki/Modular_hypergeometric_series" class="mw-redirect" title="Modular hypergeometric series">Modular hypergeometric series</a></li> <li><a href="/wiki/Riemann%27s_differential_equation" title="Riemann's differential equation">Riemann's differential equation</a></li> <li><a href="/wiki/Theta_hypergeometric_series" class="mw-redirect" title="Theta hypergeometric series">Theta hypergeometric series</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="3"><div> <ul><li><span class="noviewer" 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