A192038 Decimal approximation of x such that f(x)=4, where f is the Fibonacci function.

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

Offset

1,1

Links

Table of n, a(n) for n=1..100.

Eric W. Weisstein, MathWorld: Fibonacci Number

Formula

f(x) = (phi^x - cos(Pi*x) * phi^(-x))/sqrt(5), where phi = (1+sqrt(5))/2 (the golden ratio). The function f, a generalization over the reals of the Binet formula, gives Fibonacci numbers for integer values of x; e.g., f(3) = 2, f(4) = 3, f(5) = 5. [Corrected by Daniel Forgues, Oct 05 2016]

Example

4.549112556507743239203225039690296777977751571212553...

Mathematica

r = GoldenRatio; s = 1/Sqrt[5];
f[x_] := s*(r^x - Cos[Pi*x] * r^(-x));
x /. FindRoot[Fibonacci[x] == 4, {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 *)

Prog

(PARI) phi = (1+sqrt(5))/2; solve(x=4, 5, (phi^x - cos(Pi*x) * phi^(-x))/sqrt(5) - 4) \\ Michel Marcus, Oct 05 2016

Crossrefs

Cf. A192039, A192040, A192041, A192042, A192043, A192044 (these correspond to f(x) = 6, 7, 1/2, 3/2, phi, phi^2 respectively); A171909, A172081.

Sequence in context: A246954 A045834 A106148 * A046577 A176016 A184833

Adjacent sequences: A192035 A192036 A192037 * A192039 A192040 A192041

Keyword

nonn,cons

Author

Clark Kimberling, Jun 21 2011

Status

approved