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http://functions.wolfram.com/04.11.06.0015.01
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Fibonacci[\[Nu]] \[Proportional] Fibonacci[Subscript[\[Nu], 0]] +
((1/Sqrt[5]) ((1 + Sqrt[5])^(2 Subscript[\[Nu], 0]) ArcCsch[2] +
4^Subscript[\[Nu], 0] (ArcCsch[2] Cos[Pi Subscript[\[Nu], 0]] +
Pi Sin[Pi Subscript[\[Nu], 0]])) (\[Nu] - Subscript[\[Nu], 0]))/
(2 (1 + Sqrt[5]))^Subscript[\[Nu], 0] +
(1/10) (5 ArcCsch[2]^2 Fibonacci[Subscript[\[Nu], 0]] +
Sqrt[5] ((1/2) (-1 + Sqrt[5]))^Subscript[\[Nu], 0] Pi
(Pi Cos[Pi Subscript[\[Nu], 0]] - 2 ArcCsch[2]
Sin[Pi Subscript[\[Nu], 0]])) (\[Nu] - Subscript[\[Nu], 0])^2 +
\[Ellipsis] /; (\[Nu] -> Subscript[\[Nu], 0])
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Fibonacci", "[", "\[Nu]", "]"]], "\[Proportional]", RowBox[List[RowBox[List["Fibonacci", "[", SubscriptBox["\[Nu]", "0"], "]"]], "+", RowBox[List[FractionBox["1", SqrtBox["5"]], SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]]]], ")"]], RowBox[List["-", SubscriptBox["\[Nu]", "0"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]], RowBox[List["2", " ", SubscriptBox["\[Nu]", "0"]]]], " ", RowBox[List["ArcCsch", "[", "2", "]"]]]], "+", RowBox[List[SuperscriptBox["4", SubscriptBox["\[Nu]", "0"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["ArcCsch", "[", "2", "]"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", SubscriptBox["\[Nu]", "0"]]], "]"]]]], "+", RowBox[List["\[Pi]", " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", SubscriptBox["\[Nu]", "0"]]], "]"]]]]]], ")"]]]]]], ")"]], RowBox[List["(", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]", "0"]]], ")"]]]], "+", RowBox[List[FractionBox["1", "10"], " ", RowBox[List["(", RowBox[List[RowBox[List["5", " ", SuperscriptBox[RowBox[List["ArcCsch", "[", "2", "]"]], "2"], " ", RowBox[List["Fibonacci", "[", SubscriptBox["\[Nu]", "0"], "]"]]]], "+", RowBox[List[SqrtBox["5"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["5"]]], ")"]]]], ")"]], SubscriptBox["\[Nu]", "0"]], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", SubscriptBox["\[Nu]", "0"]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["ArcCsch", "[", "2", "]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", SubscriptBox["\[Nu]", "0"]]], "]"]]]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]", "0"]]], ")"]], "2"]]], "+", "\[Ellipsis]"]]]], "/;", RowBox[List["(", RowBox[List["\[Nu]", "\[Rule]", SubscriptBox["\[Nu]", "0"]]], ")"]]]]]]
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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["F", Fibonacci] </annotation> </semantics> <mi> ν </mi> </msub> <mo> ∝ </mo> <mrow> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox["F", Fibonacci] </annotation> </semantics> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </msub> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mn> 5 </mn> </msqrt> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mn> 4 </mn> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 10 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mn> 5 </mn> </msqrt> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> + </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </msup> </mrow> <mo> + </mo> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <msup> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox["F", Fibonacci] </annotation> </semantics> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mo> … </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> Fibonacci </ci> <ci> ν </ci> </apply> <apply> <plus /> <apply> <ci> Fibonacci </ci> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 4 </cn> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> <apply> <cos /> <apply> <times /> <pi /> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <pi /> <apply> <sin /> <apply> <times /> <pi /> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 10 </cn> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <pi /> <apply> <plus /> <apply> <times /> <pi /> <apply> <cos /> <apply> <times /> <pi /> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> <apply> <sin /> <apply> <times /> <pi /> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <power /> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Fibonacci </ci> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> … </ci> </apply> </apply> <apply> <ci> Rule </ci> <ci> ν </ci> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Fibonacci", "[", "\[Nu]_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Fibonacci", "[", SubscriptBox["\[Nu]\[Nu]", "0"], "]"]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]]]], ")"]], RowBox[List["-", SubscriptBox["\[Nu]\[Nu]", "0"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]], RowBox[List["2", " ", SubscriptBox["\[Nu]\[Nu]", "0"]]]], " ", RowBox[List["ArcCsch", "[", "2", "]"]]]], "+", RowBox[List[SuperscriptBox["4", SubscriptBox["\[Nu]\[Nu]", "0"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["ArcCsch", "[", "2", "]"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", SubscriptBox["\[Nu]\[Nu]", "0"]]], "]"]]]], "+", RowBox[List["\[Pi]", " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", SubscriptBox["\[Nu]\[Nu]", "0"]]], "]"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], ")"]]]], SqrtBox["5"]], "+", RowBox[List[FractionBox["1", "10"], " ", RowBox[List["(", RowBox[List[RowBox[List["5", " ", SuperscriptBox[RowBox[List["ArcCsch", "[", "2", "]"]], "2"], " ", RowBox[List["Fibonacci", "[", SubscriptBox["\[Nu]\[Nu]", "0"], "]"]]]], "+", RowBox[List[SqrtBox["5"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["5"]]], ")"]]]], ")"]], SubscriptBox["\[Nu]\[Nu]", "0"]], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", SubscriptBox["\[Nu]\[Nu]", "0"]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["ArcCsch", "[", "2", "]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", SubscriptBox["\[Nu]\[Nu]", "0"]]], "]"]]]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], ")"]], "2"]]], "+", "\[Ellipsis]"]], "/;", RowBox[List["(", RowBox[List["\[Nu]", "\[Rule]", SubscriptBox["\[Nu]\[Nu]", "0"]]], ")"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
