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

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, 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, 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

Offset: 1

Author

Clark Kimberling, Jun 21 2011

Keywords

• nonn
• cons

Comments

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.

Examples

			2.6141654966507095222450798053609573198964859263002877
		

Crossrefs

Cf. A192038.

Programs

Mathematica

r = GoldenRatio; s = 1/Sqrt[5];
f[x_] := s (r^x - r^-x Cos[Pi x]);
x /. FindRoot[Fibonacci[x] == r, {x, 5}, WorkingPrecision -> 100]
RealDigits[%, 10]
(Show[Plot[#1, #2], ListPlot[Table[{x, #1}, #2]]] &)[
Fibonacci[x], {x, -7, 7}]
(* Peter J. C. Moses, Jun 21 2011 *)