@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:-RF92126B-F
  skos:definition """En mathématiques, les suites de Lucas U(P, Q) et V(P, Q) associées à deux entiers P et Q sont deux suites récurrentes linéaires d'ordre 2 à valeurs entières qui généralisent respectivement la suite de Fibonacci et celle de Fibonacci-Lucas, correspondant aux valeurs P = 1 et Q = –1.  Elles doivent leur nom au mathématicien français Édouard Lucas. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Suite_de_Lucas">https://fr.wikipedia.org/wiki/Suite_de_Lucas</a>)"""@fr, """In mathematics, the <b>Lucas sequences</b> <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 U_{n}(P,Q)}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>U</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>         <mo stretchy="false">(</mo>         <mi>P</mi>         <mo>,</mo>         <mi>Q</mi>         <mo stretchy="false">)</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle U_{n}(P,Q)}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9acf94bbc2c8e51474c0b61e79469aa6049fcf0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.233ex; height:2.843ex;" alt="U_{n}(P,Q)"></span> and <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 V_{n}(P,Q)}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>V</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>         <mo stretchy="false">(</mo>         <mi>P</mi>         <mo>,</mo>         <mi>Q</mi>         <mo stretchy="false">)</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle V_{n}(P,Q)}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40dfdb812d2de843618bac1786e8887494aba43e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.001ex; height:2.843ex;" alt="V_{n}(P,Q)"></span> are certain constant-recursive integer sequences that satisfy the recurrence relation  <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 x_{n}=P\\\\cdot x_{n-1}-Q\\\\cdot x_{n-2}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>x</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>         <mo>=</mo>         <mi>P</mi>         <mo>⋅<!-- ⋅ --></mo>         <msub>           <mi>x</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>−<!-- − --></mo>             <mn>1</mn>           </mrow>         </msub>         <mo>−<!-- − --></mo>         <mi>Q</mi>         <mo>⋅<!-- ⋅ --></mo>         <msub>           <mi>x</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>−<!-- − --></mo>             <mn>2</mn>           </mrow>         </msub>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle x_{n}=P\\\\cdot x_{n-1}-Q\\\\cdot x_{n-2}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f001cb3047032efac2b4bf6ed7b4d5a5d7ec065" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.726ex; height:2.509ex;" alt="{\\\\displaystyle x_{n}=P\\\\cdot x_{n-1}-Q\\\\cdot x_{n-2}}"></span></dd></dl> where <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}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>P</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle P}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="P"></span> and <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 Q}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>Q</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle Q}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="Q"></span> are fixed integers. Any sequence satisfying this recurrence relation can be represented as a linear combination of the Lucas sequences <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 U_{n}(P,Q)}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>U</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>         <mo stretchy="false">(</mo>         <mi>P</mi>         <mo>,</mo>         <mi>Q</mi>         <mo stretchy="false">)</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle U_{n}(P,Q)}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9acf94bbc2c8e51474c0b61e79469aa6049fcf0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.233ex; height:2.843ex;" alt="U_{n}(P,Q)"></span> and <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 V_{n}(P,Q).}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>V</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>         <mo stretchy="false">(</mo>         <mi>P</mi>         <mo>,</mo>         <mi>Q</mi>         <mo stretchy="false">)</mo>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle V_{n}(P,Q).}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d676acac90fb321d4c12f3f1c2a286b8c517b761" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.648ex; height:2.843ex;" alt="{\\\\displaystyle V_{n}(P,Q).}"></span> More generally, Lucas sequences <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 U_{n}(P,Q)}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>U</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>         <mo stretchy="false">(</mo>         <mi>P</mi>         <mo>,</mo>         <mi>Q</mi>         <mo stretchy="false">)</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle U_{n}(P,Q)}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9acf94bbc2c8e51474c0b61e79469aa6049fcf0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.233ex; height:2.843ex;" alt="U_{n}(P,Q)"></span> and <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 V_{n}(P,Q)}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>V</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>         <mo stretchy="false">(</mo>         <mi>P</mi>         <mo>,</mo>         <mi>Q</mi>         <mo stretchy="false">)</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle V_{n}(P,Q)}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40dfdb812d2de843618bac1786e8887494aba43e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.001ex; height:2.843ex;" alt="V_{n}(P,Q)"></span> represent sequences of polynomials in <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}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>P</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle P}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="P"></span> and <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 Q}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>Q</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle Q}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="Q"></span> with integer coefficients. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Lucas_sequence">https://en.wikipedia.org/wiki/Lucas_sequence</a>)"""@en ;
  skos:prefLabel "suite de Lucas"@fr, "Lucas number"@en ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Suite_de_Lucas>, <https://en.wikipedia.org/wiki/Lucas_sequence> ;
  dc:created "2023-07-25"^^xsd:date ;
  a skos:Concept ;
  skos:broader psr:-CVDPQB0Q-M, psr:-FM1M1PDT-5 ;
  skos:altLabel "Lucas sequence"@en ;
  dc:modified "2024-10-18"^^xsd:date ;
  skos:inScheme psr: .

psr:-CVDPQB0Q-M
  skos:prefLabel "natural numbers"@en, "entier naturel"@fr ;
  a skos:Concept ;
  skos:narrower psr:-RF92126B-F .

psr:-FM1M1PDT-5
  skos:prefLabel "suite d'entiers"@fr, "integer sequence"@en ;
  a skos:Concept ;
  skos:narrower psr:-RF92126B-F .