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A192039 Decimal approximation of x such that f(x)=6, where f is the Fibonacci function described in Comments.

%I A192039 #7 Feb 13 2014 13:24:52
%S A192039 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 A192039 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 A192039 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 A192039 Decimal approximation of x such that f(x)=6, where f is the Fibonacci function described in Comments.
%C A192039 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 A192039 5.391849606901775521280408442083479799478829143140
%t A192039 r = GoldenRatio; s = 1/Sqrt[5];
%t A192039 f[x_] := s (r^x - r^-x Cos[Pi x]);
%t A192039 x /. FindRoot[Fibonacci[x] == 6, {x, 5}, WorkingPrecision -> 100]
%t A192039 RealDigits[%, 10]
%t A192039 (Show[Plot[#1, #2], ListPlot[Table[{x, #1}, #2]]] &)[
%t A192039 Fibonacci[x], {x, -7, 7}]
%t A192039 (* _Peter J. C. Moses_, Jun 21 2011 *)
%Y A192039 Cf. A192038.
%K A192039 nonn,cons
%O A192039 1,1
%A A192039 _Clark Kimberling_, Jun 21 2011