@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:-XRXLKWR1-J
  skos:prefLabel "nombre polygonal centré"@fr, "centered polygonal numbers"@en ;
  a skos:Concept ;
  skos:narrower psr:-T7D13THP-7 .

psr:-T7D13THP-7
  skos:definition """Un <b>nombre ennéagonal centré</b> est un nombre figuré polygonal centré qui représente un ennéagone avec un point dans le centre, tous les autres points entourant le point central en faisant des ennéagones successifs. Le <i>n</i>-ième nombre ennéagonal 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_{9,n}=1+9T_{n-1}=1+9\\\\ {\\rac {n(n-1)}{2}}={(3n-1)(3n-2) \\\\over 2}=T_{3n-2}.}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>C</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>9</mn>             <mo>,</mo>             <mi>n</mi>           </mrow>         </msub>         <mo>=</mo>         <mn>1</mn>         <mo>+</mo>         <mn>9</mn>         <msub>           <mi>T</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>−<!-- − --></mo>             <mn>1</mn>           </mrow>         </msub>         <mo>=</mo>         <mn>1</mn>         <mo>+</mo>         <mn>9</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>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mrow>               <mo stretchy="false">(</mo>               <mn>3</mn>               <mi>n</mi>               <mo>−<!-- − --></mo>               <mn>1</mn>               <mo stretchy="false">)</mo>               <mo stretchy="false">(</mo>               <mn>3</mn>               <mi>n</mi>               <mo>−<!-- − --></mo>               <mn>2</mn>               <mo stretchy="false">)</mo>             </mrow>             <mn>2</mn>           </mfrac>         </mrow>         <mo>=</mo>         <msub>           <mi>T</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>3</mn>             <mi>n</mi>             <mo>−<!-- − --></mo>             <mn>2</mn>           </mrow>         </msub>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle C_{9,n}=1+9T_{n-1}=1+9\\\\ {\\rac {n(n-1)}{2}}={(3n-1)(3n-2) \\\\over 2}=T_{3n-2}.}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38c6096879884910e6b193f4493b30813626f1ef" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:65.299ex; height:5.676ex;" alt="{\\\\displaystyle C_{9,n}=1+9T_{n-1}=1+9\\\\ {\\rac {n(n-1)}{2}}={(3n-1)(3n-2) \\\\over 2}=T_{3n-2}.}"></span></center> Les nombres ennéagonaux centrés sont donc simplement les nombres triangulaires <i><span class="texhtml">T<sub>k</sub></span></i> pour <i>k</i> congru à 1 modulo 3. Les quinze premiers sont 1, 10, 28, 55, 91, 136, 190, 253, 325, 406, 496, 595, 703, 820 et 946 (suite A060544 de l'OEIS). 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Nombre_enn%C3%A9agonal_centr%C3%A9">https://fr.wikipedia.org/wiki/Nombre_enn%C3%A9agonal_centr%C3%A9</a>)"""@fr, """A <b>centered nonagonal number</b> (or <b>centered enneagonal number</b>) is a centered figurate number that represents a nonagon with a dot in the center and all other dots surrounding the center dot in successive nonagonal layers. The centered nonagonal number for <i>n</i> layers 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 Nc(n)={\\rac {(3n-2)(3n-1)}{2}}.}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>N</mi>         <mi>c</mi>         <mo stretchy="false">(</mo>         <mi>n</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mrow>               <mo stretchy="false">(</mo>               <mn>3</mn>               <mi>n</mi>               <mo>−<!-- − --></mo>               <mn>2</mn>               <mo stretchy="false">)</mo>               <mo stretchy="false">(</mo>               <mn>3</mn>               <mi>n</mi>               <mo>−<!-- − --></mo>               <mn>1</mn>               <mo stretchy="false">)</mo>             </mrow>             <mn>2</mn>           </mfrac>         </mrow>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle Nc(n)={\\rac {(3n-2)(3n-1)}{2}}.}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0892c341ac75228eca2cb5018879e1a071c2a16" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.595ex; height:5.676ex;" alt="{\\\\displaystyle Nc(n)={\\rac {(3n-2)(3n-1)}{2}}.}"></span></dd></dl> Multiplying the (<i>n</i> - 1)th triangular number by 9 and then adding 1 yields the <i>n</i>th centered nonagonal number, but centered nonagonal numbers have an even simpler relation to triangular numbers: every third triangular number (the 1st, 4th, 7th, etc.) is also a centered nonagonal number.  
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Centered_nonagonal_number">https://en.wikipedia.org/wiki/Centered_nonagonal_number</a>)"""@en ;
  skos:altLabel "centered enneagonal number"@en ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Nombre_enn%C3%A9agonal_centr%C3%A9>, <https://en.wikipedia.org/wiki/Centered_nonagonal_number> ;
  a skos:Concept ;
  skos:prefLabel "nombre ennéagonal centré"@fr, "centered nonagonal number"@en ;
  skos:inScheme psr: ;
  dc:modified "2024-10-18"^^xsd:date ;
  skos:broader psr:-XRXLKWR1-J .