A192041 Decimal approximation of x such that f(x)=1/2, where f is the Fibonacci function described in Comments.

4, 5, 0, 7, 0, 6, 6, 6, 6, 5, 7, 4, 5, 4, 4, 6, 0, 0, 2, 3, 0, 6, 0, 5, 0, 6, 3, 1, 4, 0, 3, 2, 8, 5, 7, 1, 5, 1, 8, 1, 4, 4, 0, 2, 4, 0, 2, 0, 3, 6, 2, 2, 4, 6, 1, 8, 7, 8, 4, 7, 5, 3, 5, 5, 7, 7, 8, 1, 6, 3, 5, 8, 9, 8, 9, 0, 4, 0, 4, 7, 9, 9, 3, 5, 5, 7, 5, 9, 8, 7, 3, 2, 9, 4, 1, 0, 4, 3, 4, 3

Offset: 0

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

			0.450706666574544600230605063140328571518144024020
		

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] == 1/2, {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 *)