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 Fibonacci

 http://functions.wolfram.com/04.11.06.0008.01

 Input Form

 Fibonacci[2 n + 1] == Sum[Binomial[n + k, 2 k], {k, 0, n}] /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List[RowBox[List["2", "n"]], "+", "1"]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "+", "k"]], ",", RowBox[List["2", "k"]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 F TagBox["F", Fibonacci] 2 n + 1 k = 0 n ( k + n 2 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["k", "+", "n"]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List["2", " ", "k"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] /; n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition Fibonacci 2 n 1 k 0 n Binomial k n 2 k n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Fibonacci", "[", RowBox[List[RowBox[List["2", " ", "n_"]], "+", "1"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "+", "k"]], ",", RowBox[List["2", " ", "k"]]]], "]"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2001-10-29