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 Fibonacci

 http://functions.wolfram.com/04.11.17.0013.01

 Input Form

 Fibonacci[n + 1] == Floor[GoldenRatio Fibonacci[n] + 1/2] /; Element[n - 1, Integers] && n - 1 > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["n", "+", "1"]], "]"]], "\[Equal]", RowBox[List["Floor", "[", RowBox[List[RowBox[List["GoldenRatio", " ", RowBox[List["Fibonacci", "[", "n", "]"]]]], "+", FractionBox["1", "2"]]], "]"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["n", "-", "1"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["n", "-", "1"]], ">", "0"]]]]]]]]

 MathML Form

 F TagBox["F", Fibonacci] n + 1 ϕ TagBox["\[Phi]", Function[GoldenRatio]] F TagBox["F", Fibonacci] n + 1 2 /; n - 1 + TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]] Condition Fibonacci n 1 GoldenRatio Fibonacci n 1 2 n -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Fibonacci", "[", RowBox[List["n_", "+", "1"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Floor", "[", RowBox[List[RowBox[List["GoldenRatio", " ", RowBox[List["Fibonacci", "[", "n", "]"]]]], "+", FractionBox["1", "2"]]], "]"]], "/;", RowBox[List[RowBox[List[RowBox[List["n", "-", "1"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["n", "-", "1"]], ">", "0"]]]]]]]]]]

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

 2001-10-29