OFFSET
1,2
LINKS
Richard André-Jeannin, Problem B-745, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 31, No. 3 (1993), p. 277; Fun with Unit Fractions, Solution to Problem B-745 by Paul S. Bruckman, ibid., Vol. 33, No. 1 (1995), p. 86.
A. F. Horadam, Elliptic Functions and Lambert Series in the Summation of Reciprocals in Certain Recurrence-Generated Sequences, The Fibonacci Quarterly, Vol. 26, No. 2 (1988), pp. 98-114.
FORMULA
Equals sqrt(5) * (L((3-sqrt(5))/2) - L((7-3*sqrt(5))/2)), where L(x) = Sum_{k>=1} x^k/(1-x^k) (Horadam, 1988, equation (4.6)). - Amiram Eldar, Oct 04 2020
From Gleb Koloskov, Sep 04 2021: (Start)
Equals 1/2 + (sqrt(5)/log(phi))*(log(5)/8 + 3*Integral_{x=0..infinity} sin(x)/((4*sin(x)^2+5)*(exp(Pi*x/log(phi))-1)) dx), where phi = (1+sqrt(5))/2 = A001622.
Equals 1/2 + (A002163/A002390)*(A016628/8 + 3*Integral_{x=0..infinity} sin(x)/((4*sin(x)^2+5)*(A001113^(A000796*x/A002390)-1)) dx). (End)
Equals 1 + Sum_{n>=1} 1/A065563(2*n-1) (André-Jeannin, 1993). - Amiram Eldar, Jan 15 2022
From Peter Bala, Aug 17 2022: (Start)
Equals 5/3 - 3*Sum_{n >= 1} 1/(F(2*n)*F(2*n+2)*F(2*n+4)), where F(n) = Fibonacci(n).
Conjecture: Equals 151/96 - 6*Sum_{n >= 1} 1/(F(2*n)*F(2*n+4)*F(2*n+6)). (End)
Equals A360928 * sqrt(5). - Kevin Ryde, Feb 27 2023
EXAMPLE
1.5353705088362529850...
MATHEMATICA
rd[k_] := rd[k] = RealDigits[ N[ Sum[ 1/Fibonacci[2*n], {n, 1, 2^k}], 105]][[1]]; rd[k = 4]; While[ rd[k] != rd[k - 1], k++]; rd[k] (* Jean-François Alcover, Oct 29 2012 *)
RealDigits[Sqrt[5] * (Log[5] + 2*QPolyGamma[0, 1, 1/GoldenRatio^4] - 4*QPolyGamma[0, 1, 1/GoldenRatio^2]) / (8*ArcCsch[2]), 10, 105][[1]] (* Vaclav Kotesovec, Feb 26 2023 *)
PROG
(PARI) sumpos(n=1, 1/fibonacci(2*n)) \\ Michel Marcus, Sep 04 2021
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Dec 25 2008
STATUS
approved