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
Examples
4.549112556507743239203225039690296777977751571212553...
Links
- Eric W. Weisstein, MathWorld: Fibonacci Number
Crossrefs
Programs
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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 *) -
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
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]