OFFSET
0,1
FORMULA
Equals (1/2) * lim_{n -> infinity} n*sqrt(5) - Sum_{k=1..n} F(2*k)/F(k)^2.
From Amiram Eldar, Oct 05 2020: (Start)
Equals Sum_{k>=1} (-1)^(k+1)/(phi^k * F(k)).
Equals sqrt(5) * Sum_{k>=1} (-1)^(k+1)/(phi^(2*k) - (-1)^k). (End)
EXAMPLE
0.31845296407450108129217572132624763993618782273...
MATHEMATICA
f[n_] := f[n] = n *GoldenRatio - Sum[Fibonacci[k + 1]/Fibonacci[k], {k, 1, n}] // RealDigits[#, 10, 104]& // First; f[n=100]; While[f[n] != f[n-100], n = n+100]; f[n] (* Jean-François Alcover, Feb 13 2013 *)
CROSSREFS
KEYWORD
AUTHOR
Benoit Cloitre, Sep 01 2002
STATUS
approved