@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:-F5CL35Z0-J .

psr: a skos:ConceptScheme .
psr:-F5CL35Z0-J
  skos:definition """Un <b>nombre octaédrique</b> est un nombre figuré polyédrique qui représente un octaèdre, ou deux pyramides placées ensemble, l'une placée sur l'autre renversée. Le <i>n</i>-ième nombre octaédrique <i>O<sub>n</sub></i> peut être obtenu en ajoutant deux nombres pyramidaux carrés consécutifs, ou en utilisant la formule suivante&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 O_{n}={n(2n^{2}+1) \\\\over 3}.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>O</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mi>n</mi>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mn>2</mn>
<br/>              <msup>
<br/>                <mi>n</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                </mrow>
<br/>              </msup>
<br/>              <mo>+</mo>
<br/>              <mn>1</mn>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>            <mn>3</mn>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle O_{n}={n(2n^{2}+1) \\\\over 3}.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91f806456c74992db1ca4d3381eb5cb7c7364871" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:18.391ex; height:5.843ex;" alt="{\\\\displaystyle O_{n}={n(2n^{2}+1) \\\\over 3}.}"></span></center>
<br/>Les dix premiers nombres octaédriques sont&nbsp;: 
<br/>
<br/><center>1, 6, 19, 44, 85, 146, 231, 344, 489, 670 (suite A005900 de l'OEIS).</center>
<br/>La série génératrice des nombres octaédriques est la fraction rationnelle
<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 {\\rac {z(z+1)^{2}}{(z-1)^{4}}}=\\\\sum _{n=1}^{\\\\infty }O_{n}z^{n}=z+6z^{2}+19z^{3}+\\\\cdots .}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mi>z</mi>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>z</mi>
<br/>              <mo>+</mo>
<br/>              <mn>1</mn>
<br/>              <msup>
<br/>                <mo stretchy="false">)</mo>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                </mrow>
<br/>              </msup>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>z</mi>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mn>1</mn>
<br/>              <msup>
<br/>                <mo stretchy="false">)</mo>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>4</mn>
<br/>                </mrow>
<br/>              </msup>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>            <mo>=</mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <msub>
<br/>          <mi>O</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <msup>
<br/>          <mi>z</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>=</mo>
<br/>        <mi>z</mi>
<br/>        <mo>+</mo>
<br/>        <mn>6</mn>
<br/>        <msup>
<br/>          <mi>z</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>+</mo>
<br/>        <mn>19</mn>
<br/>        <msup>
<br/>          <mi>z</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>3</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>+</mo>
<br/>        <mo>⋯<!-- ⋯ --></mo>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\rac {z(z+1)^{2}}{(z-1)^{4}}}=\\\\sum _{n=1}^{\\\\infty }O_{n}z^{n}=z+6z^{2}+19z^{3}+\\\\cdots .}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/500ed681b00e052be7917d902df176f41f772da1" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:46.261ex; height:7.009ex;" alt="{\\\\displaystyle {\\rac {z(z+1)^{2}}{(z-1)^{4}}}=\\\\sum _{n=1}^{\\\\infty }O_{n}z^{n}=z+6z^{2}+19z^{3}+\\\\cdots .}"> 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Nombre_octa%C3%A9drique">https://fr.wikipedia.org/wiki/Nombre_octa%C3%A9drique</a>)"""@fr, """In number theory, an <b>octahedral number</b> is a figurate number that represents the number of spheres in an octahedron formed from close-packed spheres. The <span class="nowrap"><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}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>n</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle n}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="n"></span>th</span> octahedral number <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 O_{n}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>O</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle O_{n}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4430e805a5f519b46eea4ec0d2dc04fc2fd19028" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.992ex; height:2.509ex;" alt="O_{n}"></span> can be obtained by the formula:
<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 O_{n}={n(2n^{2}+1) \\\\over 3}.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>O</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mi>n</mi>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mn>2</mn>
<br/>              <msup>
<br/>                <mi>n</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                </mrow>
<br/>              </msup>
<br/>              <mo>+</mo>
<br/>              <mn>1</mn>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>            <mn>3</mn>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle O_{n}={n(2n^{2}+1) \\\\over 3}.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91f806456c74992db1ca4d3381eb5cb7c7364871" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:18.391ex; height:5.843ex;" alt="O_{n}={n(2n^{2}+1) \\\\over 3}."></span></dd></dl>
<br/>The first few octahedral numbers are: 
<br/>
<br/><dl><dd>1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891 (sequence <span class="nowrap external">A005900</span> in the OEIS).</dd> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Octahedral_number">https://en.wikipedia.org/wiki/Octahedral_number</a>)"""@en ;
  a skos:Concept ;
  skos:inScheme psr: ;
  skos:broader psr:-ZLZPWC0Z-9 ;
  skos:prefLabel "nombre octaédrique"@fr, "octahedral number"@en ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Nombre_octa%C3%A9drique>, <https://en.wikipedia.org/wiki/Octahedral_number> .