cp's OEIS Frontend

A192044 Decimal approximation of x such that f(x)=r+1, where f is the Fibonacci function described in Comments and r=(golden ratio).

%I A192044 #10 Feb 13 2014 13:23:59
%S A192044 3,7,0,8,2,2,8,3,1,9,6,1,1,8,1,5,4,4,6,2,2,7,9,5,6,9,7,6,0,4,7,6,2,9,
%T A192044 0,3,1,4,1,4,4,4,7,8,0,1,5,1,4,7,0,4,6,7,1,2,4,7,2,4,0,2,3,9,9,5,4,0,
%U A192044 8,0,1,9,6,5,8,7,3,7,9,3,6,4,3,9,8,5,9,4,2,2,6,1,1,6,1,6,0,6,3,3
%N A192044 Decimal approximation of x such that f(x)=r+1, where f is the Fibonacci function described in Comments and r=(golden ratio).
%C A192044 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 A192044 3.70822831961181544622795697604762903141444780151470467124724
%t A192044 r = GoldenRatio; s = 1/Sqrt[5];
%t A192044 f[x_] := s (r^x - r^-x Cos[Pi x]);
%t A192044 x /. FindRoot[Fibonacci[x] == r+1, {x, 5}, WorkingPrecision -> 100]
%t A192044 RealDigits[%, 10]
%t A192044 (Show[Plot[#1, #2], ListPlot[Table[{x, #1}, #2]]] &)[
%t A192044 Fibonacci[x], {x, -7, 7}]
%t A192044 (* _Peter J. C. Moses_, Jun 21 2011 *)
%Y A192044 Cf. A192038, A104457.
%K A192044 nonn,cons
%O A192044 1,1
%A A192044 _Clark Kimberling_, Jun 21 2011