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A192041 Decimal approximation of x such that f(x)=1/2, where f is the Fibonacci function described in Comments. 1
4, 5, 0, 7, 0, 6, 6, 6, 6, 5, 7, 4, 5, 4, 4, 6, 0, 0, 2, 3, 0, 6, 0, 5, 0, 6, 3, 1, 4, 0, 3, 2, 8, 5, 7, 1, 5, 1, 8, 1, 4, 4, 0, 2, 4, 0, 2, 0, 3, 6, 2, 2, 4, 6, 1, 8, 7, 8, 4, 7, 5, 3, 5, 5, 7, 7, 8, 1, 6, 3, 5, 8, 9, 8, 9, 0, 4, 0, 4, 7, 9, 9, 3, 5, 5, 7, 5, 9, 8, 7, 3, 2, 9, 4, 1, 0, 4, 3, 4, 3 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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=0..99.

EXAMPLE

0.450706666574544600230605063140328571518144024020

MATHEMATICA

r = GoldenRatio; s = 1/Sqrt[5];

f[x_] := s (r^x - r^-x Cos[Pi x]);

x /. FindRoot[Fibonacci[x] == 1/2, {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 *)

CROSSREFS

Cf. A192038.

Sequence in context: A164357 A092487 A322505 * A132022 A319459 A318740

Adjacent sequences:  A192038 A192039 A192040 * A192042 A192043 A192044

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Jun 21 2011

STATUS

approved

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Last modified October 23 21:27 EDT 2021. Contains 348217 sequences. (Running on oeis4.)