A192039 - OEIS

%I #7 Feb 13 2014 13:24:52

%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^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 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