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

1,1

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.

Links

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

Example

2.6141654966507095222450798053609573198964859263002877

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 *)

Crossrefs

Cf. A192038.

Sequence in context: A360857 A124320 A156146 * A154584 A129677 A324033

Adjacent sequences: A192040 A192041 A192042 * A192044 A192045 A192046

Keyword

nonn,cons

Author

Clark Kimberling, Jun 21 2011

Status

approved