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%I #7 Feb 13 2014 13:23:56
%S 2,6,1,4,1,6,5,4,9,6,6,5,0,7,0,9,5,2,2,2,4,5,0,7,9,8,0,5,3,6,0,9,5,7,
%T 3,1,9,8,9,6,4,8,5,9,2,6,3,0,0,2,8,7,7,3,7,8,8,3,4,0,7,2,9,6,4,4,1,5,
%U 4,2,7,4,4,2,5,6,6,8,5,7,3,0,9,6,1,1,6,1,3,2,6,8,1,3,1,7,6,7,3,6
%N Decimal approximation of x such that f(x)=r, where f is the Fibonacci function described in Comments and r=(golden ratio).
%C 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.
%e 2.6141654966507095222450798053609573198964859263002877
%t r = GoldenRatio; s = 1/Sqrt[5];
%t f[x_] := s (r^x - r^-x Cos[Pi x]);
%t x /. FindRoot[Fibonacci[x] == r, {x, 5}, WorkingPrecision -> 100]
%t RealDigits[%, 10]
%t (Show[Plot[#1, #2], ListPlot[Table[{x, #1}, #2]]] &)[
%t Fibonacci[x], {x, -7, 7}]
%t (* _Peter J. C. Moses_, Jun 21 2011 *)
%Y Cf. A192038.
%K nonn,cons
%O 1,1
%A _Clark Kimberling_, Jun 21 2011
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