@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:-M3N11P2M-V .

psr:-M3N11P2M-V
  skos:inScheme psr: ;
  skos:prefLabel "nombre pentagonal"@fr, "pentagonal number"@en ;
  skos:definition """A <b>pentagonal number</b> is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The <i>n</i>th pentagonal number <i>p<sub>n</sub></i> is the number of <i>distinct</i> dots in a pattern of dots consisting of the <i>outlines</i> of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex. For instance, the third one is formed from outlines comprising 1, 5 and 10 dots, but the 1, and 3 of the 5, coincide with 3 of the 10 – leaving 12 distinct dots, 10 in the form of a pentagon, and 2 inside. <i>p</i><sub>n</sub> 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 p_{n}={\\rac {3n^{2}-n}{2}}={\\inom {n}{1}}+3{\\inom {n}{2}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>p</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>         <mo>=</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mrow>               <mn>3</mn>               <msup>                 <mi>n</mi>                 <mrow class="MJX-TeXAtom-ORD">                   <mn>2</mn>                 </mrow>               </msup>               <mo>−<!-- − --></mo>               <mi>n</mi>             </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">               <mi>n</mi>               <mn>1</mn>             </mfrac>             <mrow class="MJX-TeXAtom-CLOSE">               <mo maxsize="2.047em" minsize="2.047em">)</mo>             </mrow>           </mrow>         </mrow>         <mo>+</mo>         <mn>3</mn>         <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>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_{n}={\\rac {3n^{2}-n}{2}}={\\inom {n}{1}}+3{\\inom {n}{2}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/199a87fd7ba4e879c91e450ef49cf179c04fbdc0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.089ex; width:30.992ex; height:6.343ex;" alt="{\\\\displaystyle p_{n}={\\rac {3n^{2}-n}{2}}={\\inom {n}{1}}+3{\\inom {n}{2}}}"></span></dd></dl> for <i>n</i> ≥ 1. The first few pentagonal numbers are: 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 3151, 3290, 3432, 3577, 3725, 3876, 4030, 4187... (sequence A000326 in the OEIS).  
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Pentagonal_number">https://en.wikipedia.org/wiki/Pentagonal_number</a>)"""@en, """En mathématiques, un <b>nombre pentagonal</b> est un nombre figuré qui compte des points régulièrement répartis dans un pentagone.  Pour tout entier <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> ≥ 1, d'après les formules générales pour les nombres polygonaux, à l'étape <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> où il y a <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> points dans chaque côté du pentagone, le nombre pentagonal est la somme des <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> premiers termes de la suite arithmétique de premier terme 1 et de raison 3</span> :  <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 P_{5,n}=1+4+\\\\dots +(3n-2)={n(3n-1) \\\\over 2}={\\rac {1}{3}}~P_{3,3n-1},}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>P</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>5</mn>             <mo>,</mo>             <mi>n</mi>           </mrow>         </msub>         <mo>=</mo>         <mn>1</mn>         <mo>+</mo>         <mn>4</mn>         <mo>+</mo>         <mo>⋯<!-- ⋯ --></mo>         <mo>+</mo>         <mo stretchy="false">(</mo>         <mn>3</mn>         <mi>n</mi>         <mo>−<!-- − --></mo>         <mn>2</mn>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mrow>               <mi>n</mi>               <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>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <mn>3</mn>           </mfrac>         </mrow>         <mtext> </mtext>         <msub>           <mi>P</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>3</mn>             <mo>,</mo>             <mn>3</mn>             <mi>n</mi>             <mo>−<!-- − --></mo>             <mn>1</mn>           </mrow>         </msub>         <mo>,</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle P_{5,n}=1+4+\\\\dots +(3n-2)={n(3n-1) \\\\over 2}={\\rac {1}{3}}~P_{3,3n-1},}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39191fe296c820c1cae02d304a35e9273f3e8b87" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:55.963ex; height:5.676ex;" alt="{\\\\displaystyle P_{5,n}=1+4+\\\\dots +(3n-2)={n(3n-1) \\\\over 2}={\\rac {1}{3}}~P_{3,3n-1},}"></span></center> soit le tiers du (3<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> – 1)-ième nombre triangulaire. Les dix premiers sont 1, 5, 12, 22, 35, 51, 70, 92, 117 et 145 (suite A000326 de l'OEIS). Les nombres pentagonaux sont importants dans la théorie des partitions d'entiers d'Euler et interviennent par exemple dans son théorème des nombres pentagonaux.  
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Nombre_pentagonal">https://fr.wikipedia.org/wiki/Nombre_pentagonal</a>)"""@fr ;
  dc:modified "2024-10-18"^^xsd:date ;
  a skos:Concept ;
  skos:broader psr:-X7NSSF7W-1 ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Pentagonal_number>, <https://fr.wikipedia.org/wiki/Nombre_pentagonal> .