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variants of this functions
Fibonacci




Mathematica Notation

Traditional Notation









Integer Functions > Fibonacci[nu] > Representations through equivalent functions > With Lucas numbers





http://functions.wolfram.com/04.11.27.0013.01









  


  








Input Form





Fibonacci[\[Nu]] == (-LucasL[2 \[Nu]] + 2 LucasL[2 \[Nu] + 1])/ (5 LucasL[\[Nu]]) - Sin[Pi \[Nu]]^2/(GoldenRatio^(2 \[Nu]) (Sqrt[5] LucasL[\[Nu]]))








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Fibonacci", "[", "\[Nu]", "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", RowBox[List["LucasL", "[", RowBox[List["2", "\[Nu]"]], "]"]]]], "+", RowBox[List["2", RowBox[List["LucasL", "[", RowBox[List[RowBox[List["2", "\[Nu]"]], "+", "1"]], "]"]]]]]], RowBox[List["5", RowBox[List["LucasL", "[", "\[Nu]", "]"]]]]], "-", FractionBox[RowBox[List[SuperscriptBox["GoldenRatio", RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], "2"]]], RowBox[List[SqrtBox["5"], RowBox[List["LucasL", "[", "\[Nu]", "]"]]]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mi> &#957; </mi> </msub> <mo> &#63449; </mo> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <semantics> <mi> L </mi> <annotation encoding='Mathematica'> TagBox[&quot;L&quot;, LucasL] </annotation> </semantics> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <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> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </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> &#957; </mi> </msub> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <msup> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, Function[List[], GoldenRatio]] </annotation> </semantics> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msqrt> <mn> 5 </mn> </msqrt> <mo> &#8290; </mo> <msub> <semantics> <mi> L </mi> <annotation encoding='Mathematica'> TagBox[&quot;L&quot;, LucasL] </annotation> </semantics> <mi> &#957; </mi> </msub> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Fibonacci </ci> <ci> &#957; </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> LucasL </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> LucasL </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <ci> LucasL </ci> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> GoldenRatio </ci> <apply> <times /> <cn type='integer'> -2 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <apply> <sin /> <apply> <times /> <pi /> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> LucasL </ci> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Fibonacci", "[", "\[Nu]_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", RowBox[List["LucasL", "[", RowBox[List["2", " ", "\[Nu]"]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["LucasL", "[", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], "]"]]]]]], RowBox[List["5", " ", RowBox[List["LucasL", "[", "\[Nu]", "]"]]]]], "-", FractionBox[RowBox[List[SuperscriptBox["GoldenRatio", RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], "2"]]], RowBox[List[SqrtBox["5"], " ", RowBox[List["LucasL", "[", "\[Nu]", "]"]]]]]]]]]]]


Date Added to functions.wolfram.com (modification date)





2007-05-02