OFFSET
1,2
COMMENTS
Borwein et al. express the sum in terms of theta functions. - N. J. A. Sloane, May 16 2011
Duverney et al. (1997) proved that this constant is transcendental. - Amiram Eldar, Oct 30 2020
REFERENCES
J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See pp. 202-203.
LINKS
Joerg Arndt, Table of n, a(n) for n = 1..1000
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
L. Carlitz, Problem B-110, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 5, No. 1 (1967), p. 108; An Infinite Series Equality, Solution to Problem B-110 by the proposer, ibid., Vol. 5, No. 5 (1967), pp. 469-470.
Daniel Duverney, Keiji Nishioka, Kumiko Nishioka and Iekata Shiokawa, Transcendence of Rogers-Ramanujan continued fraction and reciprocal sums of Fibonacci numbers, Proceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 73, No. 7 (1997), pp. 140-142.
FORMULA
Equals sqrt(5)/4 * (T(b^2)^2 - T(b)^2) where T(q) = 1 + 2*Sum_{n>=1} q^(n^2) and b = 1/2*(1-sqrt(5)); see the Arndt reference and the references cited there. - Joerg Arndt, Feb 01 2014
Equals sqrt(5) * Sum_{n>=0} (-1)^n/Lucas(2*n+1) (Carlitz, 1967). - Amiram Eldar, Feb 05 2022
EXAMPLE
1.8245151574069245681...
PROG
(PARI) sumpos(n=1, 1/fibonacci(2*n-1)) \\ Michel Marcus, Feb 05 2022
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Dec 25 2008
EXTENSIONS
Definition reconciled to sequence and example by Clark Kimberling, Aug 06 2013
STATUS
approved