%I #36 Jan 14 2023 13:23:50

%S 1,8,2,4,5,1,5,1,5,7,4,0,6,9,2,4,5,6,8,1,4,2,1,5,8,4,0,6,2,6,7,3,2,8,

%T 1,7,3,3,2,1,8,9,3,5,4,2,6,6,0,8,2,9,9,2,3,2,6,0,2,9,0,1,5,0,1,9,4,0,

%U 8,3,0,4,0,3,6,7,7,7,3,9,6,7,5,9,8,9,1,3,8,9,9,8,1,9,8,2,0,7,5,0,7,6,4,2,4

%N Decimal expansion of Sum_{n>=1} 1/Fibonacci(2*n-1).

%C Borwein et al. express the sum in terms of theta functions. - _N. J. A. Sloane_, May 16 2011

%C Duverney et al. (1997) proved that this constant is transcendental. - _Amiram Eldar_, Oct 30 2020

%D 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.

%H Joerg Arndt, <a href="/A153387/b153387.txt">Table of n, a(n) for n = 1..1000</a>

%H Joerg Arndt, <a href="http://arxiv.org/abs/1202.6525">On computing the generalized Lambert series</a>, arXiv:1202.6525v3 [math.CA], (2012).

%H L. Carlitz, <a href="https://fq.math.ca/Scanned/5-1/elementary5-1.pdf">Problem B-110</a>, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 5, No. 1 (1967), p. 108; <a href="https://fq.math.ca/Scanned/5-5/elementary5-5.pdf">An Infinite Series Equality</a>, Solution to Problem B-110 by the proposer, ibid., Vol. 5, No. 5 (1967), pp. 469-470.

%H Daniel Duverney, Keiji Nishioka, Kumiko Nishioka and Iekata Shiokawa, <a href="http://doi.org/10.3792/pjaa.73.140">Transcendence of Rogers-Ramanujan continued fraction and reciprocal sums of Fibonacci numbers</a>, Proceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 73, No. 7 (1997), pp. 140-142.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%F 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

%F Equals sqrt(5) * Sum_{n>=0} (-1)^n/Lucas(2*n+1) (Carlitz, 1967). - _Amiram Eldar_, Feb 05 2022

%e 1.8245151574069245681...

%o (PARI) sumpos(n=1, 1/fibonacci(2*n-1)) \\ _Michel Marcus_, Feb 05 2022

%Y Cf. A000032, A002878, A079586, A153386, A190649.

%K nonn,cons

%O 1,2

%A _Eric W. Weisstein_, Dec 25 2008

%E Definition reconciled to sequence and example by _Clark Kimberling_, Aug 06 2013