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A153387 Decimal expansion of Sum_{n>=1} 1/Fibonacci(2*n-1). 15
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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
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

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

Sequence in context: A019775 A274211 A138499 * A342261 A010520 A169847

Adjacent sequences:  A153384 A153385 A153386 * A153388 A153389 A153390

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

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Last modified September 30 21:21 EDT 2022. Contains 357106 sequences. (Running on oeis4.)