A192042 Decimal approximation of x such that f(x)=3/2, where f is the Fibonacci function described in Comments.

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

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

			2.50939491635468709205638984467935130148690741498451
		

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] == 3/2, {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 *)