@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .

psr:-ZLZPWC0Z-9
  skos:prefLabel "nombre polyédrique"@fr, "polyhedral number"@en ;
  a skos:Concept ;
  skos:narrower psr:-ZLW9BKRS-W .

psr: a skos:ConceptScheme .
psr:-ZLW9BKRS-W
  skos:prefLabel "tetrahedral number"@en, "nombre tétraédrique"@fr ;
  skos:definition """En arithmétique géométrique, un <b>nombre tétraédrique</b>, ou <b>nombre pyramidal triangulaire</b>, est un nombre figuré qui peut être représenté par une pyramide de base triangulaire, c'est-à-dire un tétraèdre, dont chaque couche représente un nombre triangulaire. Pour tout entier naturel <i>n</i> non nul, le <i>n</i>-ième nombre pyramidal triangulaire, somme des <i>n</i> premiers nombres triangulaires, est donc&nbsp;: 
<br/>
<br/><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_{n}^{(3)}={\\rac {n(n+1)(n+2)}{6}}={n+2 \\\\choose 3},}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msubsup>
<br/>          <mi>P</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo stretchy="false">(</mo>
<br/>            <mn>3</mn>
<br/>            <mo stretchy="false">)</mo>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mi>n</mi>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>n</mi>
<br/>              <mo>+</mo>
<br/>              <mn>1</mn>
<br/>              <mo stretchy="false">)</mo>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>n</mi>
<br/>              <mo>+</mo>
<br/>              <mn>2</mn>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>            <mn>6</mn>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow>
<br/>            <mrow class="MJX-TeXAtom-OPEN">
<br/>              <mo maxsize="2.047em" minsize="2.047em">(</mo>
<br/>            </mrow>
<br/>            <mfrac linethickness="0">
<br/>              <mrow>
<br/>                <mi>n</mi>
<br/>                <mo>+</mo>
<br/>                <mn>2</mn>
<br/>              </mrow>
<br/>              <mn>3</mn>
<br/>            </mfrac>
<br/>            <mrow class="MJX-TeXAtom-CLOSE">
<br/>              <mo maxsize="2.047em" minsize="2.047em">)</mo>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle P_{n}^{(3)}={\\rac {n(n+1)(n+2)}{6}}={n+2 \\\\choose 3},}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83f82e83bcd9cc949f25863fafcba78cde3c6b4d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:36.462ex; height:6.343ex;" alt="{\\\\displaystyle P_{n}^{(3)}={\\rac {n(n+1)(n+2)}{6}}={n+2 \\\\choose 3},}"></span></center>
<br/>où <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 {i \\\\choose j}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow>
<br/>            <mrow class="MJX-TeXAtom-OPEN">
<br/>              <mo maxsize="2.047em" minsize="2.047em">(</mo>
<br/>            </mrow>
<br/>            <mfrac linethickness="0">
<br/>              <mi>i</mi>
<br/>              <mi>j</mi>
<br/>            </mfrac>
<br/>            <mrow class="MJX-TeXAtom-CLOSE">
<br/>              <mo maxsize="2.047em" minsize="2.047em">)</mo>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {i \\\\choose j}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ef7f9960c387940455f1485491ae81a94526f7" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:4.379ex; height:6.176ex;" alt="{i \\\\choose j}"></span> est le symbole du coefficient binomial. Les nombres tétraédriques sont donc ceux de la quatrième colonne du triangle de Pascal. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Nombre_t%C3%A9tra%C3%A9drique">https://fr.wikipedia.org/wiki/Nombre_t%C3%A9tra%C3%A9drique</a>)"""@fr, """A <b>tetrahedral number</b>, or <b>triangular pyramidal number</b>, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron. The <span class="texhtml mvar" style="font-style:italic;">n</span>th tetrahedral number, <span class="texhtml mvar" style="font-style:italic;">Te<sub>n</sub></span>, is the sum of the first <span class="texhtml mvar" style="font-style:italic;">n</span> triangular numbers, that is,
<br/>
<br/><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 Te_{n}=\\\\sum _{k=1}^{n}T_{k}=\\\\sum _{k=1}^{n}{\\rac {k(k+1)}{2}}=\\\\sum _{k=1}^{n}\\\\left(\\\\sum _{i=1}^{k}i\\
ight)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>T</mi>
<br/>        <msub>
<br/>          <mi>e</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>=</mo>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>k</mi>
<br/>            <mo>=</mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <msub>
<br/>          <mi>T</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>k</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>=</mo>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>k</mi>
<br/>            <mo>=</mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mi>k</mi>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>k</mi>
<br/>              <mo>+</mo>
<br/>              <mn>1</mn>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>            <mn>2</mn>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>k</mi>
<br/>            <mo>=</mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <mrow>
<br/>          <mo>(</mo>
<br/>          <mrow>
<br/>            <munderover>
<br/>              <mo>∑<!-- ∑ --></mo>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi>i</mi>
<br/>                <mo>=</mo>
<br/>                <mn>1</mn>
<br/>              </mrow>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi>k</mi>
<br/>              </mrow>
<br/>            </munderover>
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>          <mo>)</mo>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle Te_{n}=\\\\sum _{k=1}^{n}T_{k}=\\\\sum _{k=1}^{n}{\\rac {k(k+1)}{2}}=\\\\sum _{k=1}^{n}\\\\left(\\\\sum _{i=1}^{k}i\\
ight)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/642f439aa026076a3ec9ae4ef1de37b9f862ddfb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.171ex; width:44.203ex; height:7.509ex;" alt="{\\\\displaystyle Te_{n}=\\\\sum _{k=1}^{n}T_{k}=\\\\sum _{k=1}^{n}{\\rac {k(k+1)}{2}}=\\\\sum _{k=1}^{n}\\\\left(\\\\sum _{i=1}^{k}i\\
ight)}"></span></dd></dl>
<br/>The tetrahedral numbers are:
<br/>
<br/><dl><dd>1, 4, 10, 20, 35, 56, 84, 120, 165, 220, ... (sequence <span class="nowrap external">A000292</span> in the OEIS)</dd> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Tetrahedral_number">https://en.wikipedia.org/wiki/Tetrahedral_number</a>)"""@en ;
  skos:altLabel "nombre pyramidal triangulaire"@fr, "triangular pyramidal number"@en ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Nombre_t%C3%A9tra%C3%A9drique>, <https://en.wikipedia.org/wiki/Tetrahedral_number> ;
  a skos:Concept ;
  skos:inScheme psr: ;
  skos:broader psr:-ZLZPWC0Z-9 .