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A192043 Decimal approximation of x such that f(x)=r, where f is the Fibonacci function described in Comments and r=(golden ratio).

%I A192043 #7 Feb 13 2014 13:23:56
%S A192043 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 A192043 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 A192043 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 A192043 Decimal approximation of x such that f(x)=r, where f is the Fibonacci function described in Comments and r=(golden ratio).
%C A192043 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 A192043 2.6141654966507095222450798053609573198964859263002877
%t A192043 r = GoldenRatio; s = 1/Sqrt[5];
%t A192043 f[x_] := s (r^x - r^-x Cos[Pi x]);
%t A192043 x /. FindRoot[Fibonacci[x] == r, {x, 5}, WorkingPrecision -> 100]
%t A192043 RealDigits[%, 10]
%t A192043 (Show[Plot[#1, #2], ListPlot[Table[{x, #1}, #2]]] &)[
%t A192043 Fibonacci[x], {x, -7, 7}]
%t A192043 (* _Peter J. C. Moses_, Jun 21 2011 *)
%Y A192043 Cf. A192038.
%K A192043 nonn,cons
%O A192043 1,1
%A A192043 _Clark Kimberling_, Jun 21 2011