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A192041
Decimal approximation of x such that f(x)=1/2, where f is the Fibonacci function described in Comments.
1
4, 5, 0, 7, 0, 6, 6, 6, 6, 5, 7, 4, 5, 4, 4, 6, 0, 0, 2, 3, 0, 6, 0, 5, 0, 6, 3, 1, 4, 0, 3, 2, 8, 5, 7, 1, 5, 1, 8, 1, 4, 4, 0, 2, 4, 0, 2, 0, 3, 6, 2, 2, 4, 6, 1, 8, 7, 8, 4, 7, 5, 3, 5, 5, 7, 7, 8, 1, 6, 3, 5, 8, 9, 8, 9, 0, 4, 0, 4, 7, 9, 9, 3, 5, 5, 7, 5, 9, 8, 7, 3, 2, 9, 4, 1, 0, 4, 3, 4, 3
OFFSET
0,1
COMMENTS
f(x)=(r^x-r^(-x*cos[pi*x]))/sqrt(5), where r=(golden ratio)=(1+sqrt(5))/2. This function, a variant of the Binet formula, gives Fibonacci numbers for integer values of x; e.g., f(3)=2, f(4)=3, f(5)=5.
EXAMPLE
0.450706666574544600230605063140328571518144024020
MATHEMATICA
r = GoldenRatio; s = 1/Sqrt[5];
f[x_] := s (r^x - r^-x Cos[Pi x]);
x /. FindRoot[Fibonacci[x] == 1/2, {x, 5}, WorkingPrecision -> 100]
RealDigits[%, 10]
(Show[Plot[#1, #2], ListPlot[Table[{x, #1}, #2]]] &)[
Fibonacci[x], {x, -7, 7}]
(* Peter J. C. Moses, Jun 21 2011 *)
CROSSREFS
Cf. A192038.
Sequence in context: A164357 A092487 A322505 * A132022 A319459 A318740
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Jun 21 2011
STATUS
approved