Fibonacci[n] == (1/(5 LucasL[m])) (-LucasL[m + n] + 2 LucasL[m + n + 1] + (-1)^m (-LucasL[-m + n] + 2 LucasL[n - m + 1])) /; Element[m, Integers] && Element[n, Integers]

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mi> n </mi> </msub> <mo> &#63449; </mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <semantics> <mi> L </mi> <annotation encoding='Mathematica'> TagBox[&quot;L&quot;, LucasL] </annotation> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> - </mo> <msub> <semantics> <mi> L </mi> <annotation encoding='Mathematica'> TagBox[&quot;L&quot;, LucasL] </annotation> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <semantics> <mi> L </mi> <annotation encoding='Mathematica'> TagBox[&quot;L&quot;, LucasL] </annotation> </semantics> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> - </mo> <msub> <semantics> <mi> L </mi> <annotation encoding='Mathematica'> TagBox[&quot;L&quot;, LucasL] </annotation> </semantics> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> </msub> </mrow> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <msub> <semantics> <mi> L </mi> <annotation encoding='Mathematica'> TagBox[&quot;L&quot;, LucasL] </annotation> </semantics> <mi> m </mi> </msub> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> m </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Fibonacci </ci> <ci> n </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> LucasL </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> LucasL </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> LucasL </ci> <apply> <plus /> <ci> m </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> LucasL </ci> <apply> <plus /> <ci> m </ci> <ci> n </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <ci> LucasL </ci> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> m </ci> <integers /> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2007-05-02