A192039 Decimal approximation of x such that f(x)=6, where f is the Fibonacci function described in Comments.

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

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

5.391849606901775521280408442083479799478829143140

Mathematica

r = GoldenRatio; s = 1/Sqrt[5];
f[x_] := s (r^x - r^-x Cos[Pi x]);
x /. FindRoot[Fibonacci[x] == 6, {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: A078302 A199438 A220129 * A305327 A112812 A241624

Adjacent sequences: A192036 A192037 A192038 * A192040 A192041 A192042

Keyword

nonn,cons

Author

Clark Kimberling, Jun 21 2011

Status

approved