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http://functions.wolfram.com/04.11.16.0001.01
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Fibonacci[m + n] == Fibonacci[m - 1] Fibonacci[n] +
Fibonacci[m] Fibonacci[n + 1] /; Element[m, Integers] &&
Element[n, Integers]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["m", "+", "n"]], "]"]], "\[Equal]", " ", RowBox[List[RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["m", "-", "1"]], "]"]], " ", RowBox[List["Fibonacci", "[", "n", "]"]]]], " ", "+", RowBox[List[RowBox[List["Fibonacci", "[", "m", "]"]], " ", RowBox[List["Fibonacci", "[", RowBox[List["n", "+", "1"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[Element]", "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["F", Fibonacci] </annotation> </semantics> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> </msub> <mo> ⩵ </mo> <mrow> <mrow> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox["F", Fibonacci] </annotation> </semantics> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ⁢ </mo> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox["F", Fibonacci] </annotation> </semantics> <mi> m </mi> </msub> </mrow> <mo> + </mo> <mrow> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox["F", Fibonacci] </annotation> </semantics> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ⁢ </mo> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox["F", Fibonacci] </annotation> </semantics> <mi> n </mi> </msub> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> m </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Fibonacci </ci> <apply> <plus /> <ci> m </ci> <ci> n </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Fibonacci </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Fibonacci </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <apply> <ci> Fibonacci </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Fibonacci </ci> <ci> n </ci> </apply> </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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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Fibonacci", "[", RowBox[List["m_", "+", "n_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["m", "-", "1"]], "]"]], " ", RowBox[List["Fibonacci", "[", "n", "]"]]]], "+", RowBox[List[RowBox[List["Fibonacci", "[", "m", "]"]], " ", RowBox[List["Fibonacci", "[", RowBox[List["n", "+", "1"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[Element]", "Integers"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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