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variants of this functions
Fibonacci




Mathematica Notation

Traditional Notation









Integer Functions > Fibonacci[nu] > Transformations > Products, sums, and powers of the direct function > Powers of the direct function





http://functions.wolfram.com/04.11.16.0033.01









  


  








Input Form





Fibonacci[n]^4 == (1/25) (6 + 4 (-1)^n Fibonacci[2 n] - Fibonacci[4 n] - 8 (-1)^n Fibonacci[1 + 2 n] + 2 Fibonacci[1 + 4 n]) /; Element[n, Integers]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Fibonacci", "[", "n", "]"]], "4"], "\[Equal]", RowBox[List[FractionBox["1", "25"], " ", RowBox[List["(", RowBox[List["6", "+", RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["Fibonacci", "[", RowBox[List["2", " ", "n"]], "]"]]]], "-", RowBox[List["Fibonacci", "[", RowBox[List["4", " ", "n"]], "]"]], "-", RowBox[List["8", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["Fibonacci", "[", RowBox[List["1", "+", RowBox[List["2", " ", "n"]]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["Fibonacci", "[", RowBox[List["1", "+", RowBox[List["4", " ", "n"]]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msubsup> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mi> n </mi> <mn> 4 </mn> </msubsup> <mo> &#63449; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 25 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; </mo> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </msub> </mrow> <mo> - </mo> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </msub> <mo> - </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; </mo> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> + </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </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> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <power /> <apply> <ci> Fibonacci </ci> <ci> n </ci> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 25 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <ci> Fibonacci </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Fibonacci </ci> <apply> <times /> <cn type='integer'> 4 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <ci> Fibonacci </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Fibonacci </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["Fibonacci", "[", "n_", "]"]], "4"], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "25"], " ", RowBox[List["(", RowBox[List["6", "+", RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["Fibonacci", "[", RowBox[List["2", " ", "n"]], "]"]]]], "-", RowBox[List["Fibonacci", "[", RowBox[List["4", " ", "n"]], "]"]], "-", RowBox[List["8", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["Fibonacci", "[", RowBox[List["1", "+", RowBox[List["2", " ", "n"]]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["Fibonacci", "[", RowBox[List["1", "+", RowBox[List["4", " ", "n"]]]], "]"]]]]]], ")"]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]]]]


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





2007-05-02