@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr: a skos:ConceptScheme .
psr:-KLQPNN0B-R
  skos:exactMatch <https://en.wikipedia.org/wiki/Centered_decagonal_number>, <https://fr.wikipedia.org/wiki/Nombre_d%C3%A9cagonal_centr%C3%A9> ;
  skos:prefLabel "nombre décagonal centré"@fr, "centered decagonal number"@en ;
  skos:broader psr:-XRXLKWR1-J ;
  dc:modified "2024-10-18"^^xsd:date ;
  a skos:Concept ;
  skos:definition """En mathématiques, un <b>nombre décagonal centré</b> est un nombre figuré polygonal centré qui représente un décagone avec point central, tous les points qui l'entourent formant des couches décagonales successives. Pour tout entier <i>n</i> ≥ 1, le <i>n</i>-ième nombre décagonal centré est donc  <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle C_{10,n}=1+10\\\\ {\\rac {n(n-1)}{2}}=5n^{2}-5n+1.}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>C</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>10</mn>             <mo>,</mo>             <mi>n</mi>           </mrow>         </msub>         <mo>=</mo>         <mn>1</mn>         <mo>+</mo>         <mn>10</mn>         <mtext> </mtext>         <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>         <mn>5</mn>         <msup>           <mi>n</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo>−<!-- − --></mo>         <mn>5</mn>         <mi>n</mi>         <mo>+</mo>         <mn>1.</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle C_{10,n}=1+10\\\\ {\\rac {n(n-1)}{2}}=5n^{2}-5n+1.}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da76e59d512e80444f6621ab97931fe4f86abfdd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:41.183ex; height:5.676ex;" alt="{\\\\displaystyle C_{10,n}=1+10\\\\ {\\rac {n(n-1)}{2}}=5n^{2}-5n+1.}"></span></center> Par conséquent, les nombres décagonaux centrés sont congrus à 1 modulo 10 (autrement dit : leur chiffre des unités en base dix est 1 — ils sont donc impairs). Ils forment la suite d'entiers  A062786 de l'OEIS : 1, 11, 31, 61, 101, 151, 211, 281, 361, 451, <abbr class="abbr" title="et cetera">etc.</abbr> 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Nombre_d%C3%A9cagonal_centr%C3%A9">https://fr.wikipedia.org/wiki/Nombre_d%C3%A9cagonal_centr%C3%A9</a>)"""@fr, """A <b>centered decagonal number</b> is a centered figurate number that represents a decagon with a dot in the center and all other dots surrounding the center dot in successive decagonal layers. The centered decagonal number for <i>n</i> is given by the formula  <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 5n^{2}-5n+1\\\\,}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mn>5</mn>         <msup>           <mi>n</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo>−<!-- − --></mo>         <mn>5</mn>         <mi>n</mi>         <mo>+</mo>         <mn>1</mn>         <mspace width="thinmathspace"></mspace>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle 5n^{2}-5n+1\\\\,}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed719241d5ae2a2050ac8798f7feaaef2a30ee1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.399ex; height:2.843ex;" alt="{\\\\displaystyle 5n^{2}-5n+1\\\\,}"></span></dd></dl> Thus, the first few centered decagonal numbers are  <dl><dd>1, 11, 31, 61, 101, 151, 211, 281, 361, 451, 551, 661, 781, 911, 1051, ... (sequence A062786 in the OEIS)</dd> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Centered_decagonal_number">https://en.wikipedia.org/wiki/Centered_decagonal_number</a>)"""@en ;
  skos:inScheme psr: .

psr:-XRXLKWR1-J
  skos:prefLabel "nombre polygonal centré"@fr, "centered polygonal numbers"@en ;
  a skos:Concept ;
  skos:narrower psr:-KLQPNN0B-R .