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 DigitCount

 http://functions.wolfram.com/13.10.23.0001.01

 Input Form

 Sum[DigitCount[k, 2, 1], {k, 0, n - 1}] == (n/2) Log[2, n] + n F[Log[2, n]] /; (F[n] == F[n + 1] && F[n] == 2^(Floor[Log[2, n]] - 1) (2 f[n/2^Floor[Log[2, n]] - 1] + (n/2^Floor[Log[2, n]]) Log[2, n/2^Floor[Log[2, n]]] - 2 (n/2^Floor[Log[2, n]] - 1)) && (f[x] == Sum[g[2^k x]/2^k, {k, 0, Infinity}] /; g[x] == (1/2) Mod[x, 1] UnitStep[1/2 - Mod[x, 1]] + (1/2) (1 - Mod[x, 1]) UnitStep[Mod[x, 1] - 1/2])) || (F[x] == Sum[Subscript[c, k] E^(2 Pi I k x), {k, 0, Infinity}] /; Subscript[c, 0] == Log[2, Pi]/2 - 1/(2 Log[2]) - 1/4 && Subscript[c, k] == (-(Log[2]/(2 I k Pi Log[2] - 4 k^2 Pi^2))) Zeta[(2 I k Pi)/Log[2]])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List["DigitCount", "[", RowBox[List["k", ",", "2", ",", "1"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["n", "2"], " ", RowBox[List["Log", "[", RowBox[List["2", ",", "n"]], "]"]]]], "+", RowBox[List["n", " ", RowBox[List["F", "[", RowBox[List["Log", "[", RowBox[List["2", ",", "n"]], "]"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[RowBox[List["F", "[", "n", "]"]], "\[Equal]", RowBox[List["F", "[", RowBox[List["n", "+", "1"]], "]"]]]], "\[And]", RowBox[List[RowBox[List["F", "[", "n", "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["Floor", "[", RowBox[List["Log", "[", RowBox[List["2", ",", "n"]], "]"]], "]"]], "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["f", "[", RowBox[List[FractionBox["n", SuperscriptBox["2", RowBox[List["Floor", "[", RowBox[List["Log", "[", RowBox[List["2", ",", "n"]], "]"]], "]"]]]], "-", "1"]], "]"]]]], "+", RowBox[List[FractionBox["n", SuperscriptBox["2", RowBox[List["Floor", "[", RowBox[List["Log", "[", RowBox[List["2", ",", "n"]], "]"]], "]"]]]], RowBox[List["Log", "[", RowBox[List["2", ",", FractionBox["n", SuperscriptBox["2", RowBox[List["Floor", "[", RowBox[List["Log", "[", RowBox[List["2", ",", "n"]], "]"]], "]"]]]]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List[FractionBox["n", SuperscriptBox["2", RowBox[List["Floor", "[", RowBox[List["Log", "[", RowBox[List["2", ",", "n"]], "]"]], "]"]]]], "-", "1"]], ")"]]]]]], ")"]]]]]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["f", "[", "x", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SuperscriptBox["2", RowBox[List["-", "k"]]], " ", RowBox[List["g", "[", RowBox[List[SuperscriptBox["2", "k"], " ", "x"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["g", "[", "x", "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Mod", "[", RowBox[List["x", ",", "1"]], "]"]], " ", RowBox[List["UnitStep", "[", RowBox[List[FractionBox["1", "2"], "-", RowBox[List["Mod", "[", RowBox[List["x", ",", "1"]], "]"]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["x", ",", "1"]], "]"]]]], ")"]], " ", RowBox[List["UnitStep", "[", RowBox[List[RowBox[List["Mod", "[", RowBox[List["x", ",", "1"]], "]"]], "-", FractionBox["1", "2"]]], "]"]]]]]]]]]], ")"]]]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["F", "[", "x", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SubscriptBox["c", "k"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "k", " ", "x"]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["c", "0"], "\[Equal]", RowBox[List[FractionBox[RowBox[List["Log", "[", RowBox[List["2", ",", "\[Pi]"]], "]"]], "2"], "-", FractionBox["1", RowBox[List["2", " ", RowBox[List["Log", "[", "2", "]"]]]]], "-", FractionBox["1", "4"]]]]], "\[And]", RowBox[List[SubscriptBox["c", "k"], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["Log", "[", "2", "]"]], RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "k", " ", "\[Pi]", " ", RowBox[List["Log", "[", "2", "]"]]]], "-", RowBox[List["4", " ", SuperscriptBox["k", "2"], " ", SuperscriptBox["\[Pi]", "2"]]]]]]]], RowBox[List["Zeta", "[", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "k", " ", "\[Pi]"]], RowBox[List["Log", "[", "2", "]"]]], "]"]]]]]]]]]], ")"]]]]]]]]

 MathML Form

 k = 0 n - 1 s 2 ( 1 ) ( k ) n F ( log 2 ( n ) ) + 1 2 n log 2 ( n ) /; F ( n ) F ( n + 1 ) F ( n ) 2 log 2 ( n ) - 1 ( - 2 ( n 2 log 2 ( n ) - 1 ) + 2 f ( n 2 log 2 ( n ) - 1 ) + n 2 log 2 ( n ) log 2 ( n 2 log 2 ( n ) ) ) ( f ( x ) k = 0 2 - k g ( 2 k x ) /; g ( x ) 1 2 ( 1 - x mod 1 \$CellContext`x 1 ) θ UnitStep ( x mod 1 \$CellContext`x 1 - 1 2 ) + 1 2 ( x mod 1 \$CellContext`x 1 ) θ UnitStep ( 1 2 - x mod 1 \$CellContext`x 1 ) ) ( F ( x ) k = 0 c k 2 π k x /; c 0 log 2 ( π ) 2 - 1 4 - 1 2 log ( 2 ) c k - log ( 2 ) 2 k π log ( 2 ) - 4 k 2 π 2 ζ ( 2 k π log ( 2 ) ) TagBox[RowBox[List["\[Zeta]", "(", TagBox[FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "k", " ", "\[Pi]"]], RowBox[List["log", "(", "2", ")"]]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[\$CellContext`e, Zeta[\$CellContext`e]]]] ) Condition k 0 n -1 Subscript s 2 1 k n EllipticF 2 n 1 2 n 2 n EllipticF n EllipticF n 1 EllipticF n 2 2 n -1 -2 n 2 2 n -1 -1 2 f n 2 2 n -1 -1 n 2 2 n -1 2 n 2 2 n -1 Condition f x k 0 2 -1 k g 2 k x g x 1 2 1 -1 \$CellContext`x 1 UnitStep \$CellContext`x 1 -1 1 2 1 2 \$CellContext`x 1 UnitStep 1 2 -1 \$CellContext`x 1 Condition EllipticF x k 0 Subscript c k 2 k x Subscript c 0 2 2 -1 -1 1 4 -1 1 2 2 -1 Subscript c k -1 2 2 k 2 -1 4 k 2 2 -1 Zeta 2 k 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n_", "-", "1"]]], RowBox[List["DigitCount", "[", RowBox[List["k", ",", "2", ",", "1"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "n", " ", RowBox[List["Log", "[", RowBox[List["2", ",", "n"]], "]"]]]], "+", RowBox[List["n", " ", RowBox[List["F", "[", RowBox[List["Log", "[", RowBox[List["2", ",", "n"]], "]"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["F", "[", "n", "]"]], "\[Equal]", RowBox[List["F", "[", RowBox[List["n", "+", "1"]], "]"]]]], "&&", RowBox[List[RowBox[List["F", "[", "n", "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["Floor", "[", RowBox[List["Log", "[", RowBox[List["2", ",", "n"]], "]"]], "]"]], "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["f", "[", RowBox[List[FractionBox["n", SuperscriptBox["2", RowBox[List["Floor", "[", RowBox[List["Log", "[", RowBox[List["2", ",", "n"]], "]"]], "]"]]]], "-", "1"]], "]"]]]], "+", FractionBox[RowBox[List["n", " ", RowBox[List["Log", "[", RowBox[List["2", ",", FractionBox["n", SuperscriptBox["2", RowBox[List["Floor", "[", RowBox[List["Log", "[", RowBox[List["2", ",", "n"]], "]"]], "]"]]]]]], "]"]]]], SuperscriptBox["2", RowBox[List["Floor", "[", RowBox[List["Log", "[", RowBox[List["2", ",", "n"]], "]"]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List[FractionBox["n", SuperscriptBox["2", RowBox[List["Floor", "[", RowBox[List["Log", "[", RowBox[List["2", ",", "n"]], "]"]], "]"]]]], "-", "1"]], ")"]]]]]], ")"]]]]]], "&&", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["f", "[", "x", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SuperscriptBox["2", RowBox[List["-", "k"]]], " ", RowBox[List["g", "[", RowBox[List[SuperscriptBox["2", "k"], " ", "x"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["g", "[", "x", "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Mod", "[", RowBox[List["x", ",", "1"]], "]"]], " ", RowBox[List["UnitStep", "[", RowBox[List[FractionBox["1", "2"], "-", RowBox[List["Mod", "[", RowBox[List["x", ",", "1"]], "]"]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["x", ",", "1"]], "]"]]]], ")"]], " ", RowBox[List["UnitStep", "[", RowBox[List[RowBox[List["Mod", "[", RowBox[List["x", ",", "1"]], "]"]], "-", FractionBox["1", "2"]]], "]"]]]]]]]]]], ")"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["F", "[", "x", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SubscriptBox["c", "k"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "k", " ", "x"]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["c", "0"], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Log", "[", RowBox[List["2", ",", "\[Pi]"]], "]"]]]], "-", FractionBox["1", RowBox[List["2", " ", RowBox[List["Log", "[", "2", "]"]]]]], "-", FractionBox["1", "4"]]]]], "&&", RowBox[List[SubscriptBox["c", "k"], "\[Equal]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["Log", "[", "2", "]"]], " ", RowBox[List["Zeta", "[", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "k", " ", "\[Pi]"]], RowBox[List["Log", "[", "2", "]"]]], "]"]]]], RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "k", " ", "\[Pi]", " ", RowBox[List["Log", "[", "2", "]"]]]], "-", RowBox[List["4", " ", SuperscriptBox["k", "2"], " ", SuperscriptBox["\[Pi]", "2"]]]]]]]]]]]]]], ")"]]]]]]]]]]

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

 2001-10-29