%I
%S 5,3,9,1,8,4,9,6,0,6,9,0,1,7,7,5,5,2,1,2,8,0,4,0,8,4,4,2,0,8,3,4,7,9,
%T 7,9,9,4,7,8,8,2,9,1,4,3,1,4,0,1,3,1,5,4,6,1,7,4,8,8,4,9,8,6,2,7,3,6,
%U 3,1,8,8,4,9,3,1,9,9,0,9,7,2,6,0,8,6,8,1,5,8,8,5,9,1,4,0,4,1,1,9
%N Decimal approximation of x such that f(x)=6, where f is the Fibonacci function described in Comments.
%C f(x)=(r^xr^(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 5.391849606901775521280408442083479799478829143140
%t r = GoldenRatio; s = 1/Sqrt[5];
%t f[x_] := s (r^x  r^x Cos[Pi x]);
%t x /. FindRoot[Fibonacci[x] == 6, {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
