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A192044
Decimal approximation of x such that f(x)=r+1, where f is the Fibonacci function described in Comments and r=(golden ratio).
1
3, 7, 0, 8, 2, 2, 8, 3, 1, 9, 6, 1, 1, 8, 1, 5, 4, 4, 6, 2, 2, 7, 9, 5, 6, 9, 7, 6, 0, 4, 7, 6, 2, 9, 0, 3, 1, 4, 1, 4, 4, 4, 7, 8, 0, 1, 5, 1, 4, 7, 0, 4, 6, 7, 1, 2, 4, 7, 2, 4, 0, 2, 3, 9, 9, 5, 4, 0, 8, 0, 1, 9, 6, 5, 8, 7, 3, 7, 9, 3, 6, 4, 3, 9, 8, 5, 9, 4, 2, 2, 6, 1, 1, 6, 1, 6, 0, 6, 3, 3
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.
EXAMPLE
3.70822831961181544622795697604762903141444780151470467124724
MATHEMATICA
r = GoldenRatio; s = 1/Sqrt[5];
f[x_] := s (r^x - r^-x Cos[Pi x]);
x /. FindRoot[Fibonacci[x] == r+1, {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
Sequence in context: A332527 A175576 A134976 * A010622 A103844 A199068
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Jun 21 2011
STATUS
approved