@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:-X7NSSF7W-1
  skos:prefLabel "nombre polygonal"@fr, "polygonal number"@en ;
  a skos:Concept ;
  skos:narrower psr:-LRJ239K8-D .

psr:-LRJ239K8-D
  skos:definition """A <b>decagonal number</b> is a figurate number that extends the concept of triangular and square numbers to the decagon (a ten-sided polygon). However, unlike the triangular and square numbers, the patterns involved in the construction of decagonal numbers are not rotationally symmetrical. Specifically, the  <i>n</i>th decagonal numbers counts the number of dots in a pattern of <i>n</i> nested decagons, all sharing a common corner, where the <i>i</i>th decagon in the pattern has sides made of <i>i</i> dots spaced one unit apart from each other. The <i>n</i>-th decagonal number is given by the following 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 d_{n}=4n^{2}-3n.}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>d</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>         <mo>=</mo>         <mn>4</mn>         <msup>           <mi>n</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo>−<!-- − --></mo>         <mn>3</mn>         <mi>n</mi>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle d_{n}=4n^{2}-3n.}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/146b4172186999e5785585a0e48a6af70be48415" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.182ex; height:3.009ex;" alt="{\\\\displaystyle d_{n}=4n^{2}-3n.}"></span></dd></dl> The first few decagonal numbers are:  <dl><dd>0, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451, 540, 637, 742, 855, 976, 1105, 1242, 1387, 1540, 1701, 1870, 2047, 2232, 2425, 2626, 2835, 3052, 3277, 3510, 3751, 4000, 4257, 4522, 4795, 5076, 5365, 5662, 5967, 6280, 6601, 6930, 7267, 7612, 7965, 8326 (sequence A001107 in the OEIS)</dd></dl> The <i>n</i>th decagonal number can also be calculated by adding the square of <i>n</i> to thrice the (<i>n</i>−1)th pronic number or, to put it algebraically, as   <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 D_{n}=n^{2}+3\\\\left(n^{2}-n\\
ight).}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>D</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>         <mo>=</mo>         <msup>           <mi>n</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo>+</mo>         <mn>3</mn>         <mrow>           <mo>(</mo>           <mrow>             <msup>               <mi>n</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mn>2</mn>               </mrow>             </msup>             <mo>−<!-- − --></mo>             <mi>n</mi>           </mrow>           <mo>)</mo>         </mrow>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle D_{n}=n^{2}+3\\\\left(n^{2}-n\\
ight).}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a50c5f76421a91f5069637aeb535f0c8c53e4e5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.928ex; height:3.343ex;" alt="{\\\\displaystyle D_{n}=n^{2}+3\\\\left(n^{2}-n\\
ight).}"> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Decagonal_number">https://en.wikipedia.org/wiki/Decagonal_number</a>)"""@en, """En mathématiques, un <b>nombre décagonal</b> est un nombre figuré polygonal qui peut être représenté graphiquement par des points répartis dans un décagone. Le nombre décagonal d'ordre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle n}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>n</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle n}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\\\\displaystyle n}"></span> est donné par la formule </span> :  <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 D_{n}=n(4n-3).}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>D</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>         <mo>=</mo>         <mi>n</mi>         <mo stretchy="false">(</mo>         <mn>4</mn>         <mi>n</mi>         <mo>−<!-- − --></mo>         <mn>3</mn>         <mo stretchy="false">)</mo>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle D_{n}=n(4n-3).}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4e244588f8475e3d554eaed959fa9f00cf478c1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.652ex; height:2.843ex;" alt="{\\\\displaystyle D_{n}=n(4n-3).}"></span>.</dd></dl> Les onze premiers nombres décagonaux sont : 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451 (suite A001107 de l'OEIS).  
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Nombre_d%C3%A9cagonal">https://fr.wikipedia.org/wiki/Nombre_d%C3%A9cagonal</a>)"""@fr ;
  skos:prefLabel "nombre décagonal"@fr, "decagonal number"@en ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Decagonal_number>, <https://fr.wikipedia.org/wiki/Nombre_d%C3%A9cagonal> ;
  skos:inScheme psr: ;
  skos:broader psr:-X7NSSF7W-1 ;
  a skos:Concept ;
  dc:modified "2024-10-18"^^xsd:date .