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 Fibonacci

 http://functions.wolfram.com/04.11.23.0013.01

 Input Form

 Sum[Binomial[n, k] Fibonacci[k] 2^k, {k, 0, n}] == Fibonacci[3 n] /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["Fibonacci", "[", "k", "]"]], SuperscriptBox["2", "k"]]]]], "\[Equal]", RowBox[List["Fibonacci", "[", RowBox[List["3", "n"]], "]"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", " ", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 k = 0 n ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["k", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] F TagBox["F", Fibonacci] k 2 k F TagBox["F", Fibonacci] 3 n /; n Condition k 0 n Binomial n k Fibonacci k 2 k Fibonacci 3 n n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "0"]], "n_"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n_", ",", "k_"]], "]"]], " ", RowBox[List["Fibonacci", "[", "k_", "]"]], " ", SuperscriptBox["2", "k_"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["3", " ", "n"]], "]"]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02