OFFSET
1,2
COMMENTS
André-Jeannin (1989) proved that this constant is irrational, and Tachiya (2004) proved that it does not belong to the quadratic number field Q(sqrt(5)). - Amiram Eldar, Oct 30 2020
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..10000
Richard André-Jeannin, Irrationalité de la somme des inverses de certaines suites récurrentes, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, Vol. 308, No. 19 (1989), pp. 539-541.
Paul S. Bruckman, Problem B-603, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 25, No. 3 (1987), p. 280; Lucas Analogue, Solution to Problem B-603 by C. Georghiou, ibid., Vol. 26, No. 3 (1988), p. 282.
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 (May-1988), pp. 98-114.
Yohei Tachiya, Irrationality of certain Lambert series, Tokyo Journal of Mathematics, Vol. 27, No. 1 (2004), pp. 75-85.
FORMULA
From Amiram Eldar, Oct 04 2020: (Start)
Equals Sum_{k>=0} 1/(phi^(2*k+1) - (-1)^k), where phi is the golden ratio (A001622).
Equals 7/3 - 10 * Sum_{k>=1} 1/(L(2*k-1)*L(2*k+1)*L(2*k+2)) (Bruckman, 1987). - Amiram Eldar, Jan 27 2022
EXAMPLE
1.96285817320964578286879512867518352664959301716221...
MATHEMATICA
RealDigits[Sum[1/LucasL[n], {n, 2000}], 10, 120][[1]] (* Harvey P. Dale, Jan 15 2012 *)
PROG
(PARI) suminf(n=1, 1/(fibonacci(n-1)+fibonacci(n+1))) \\ Charles R Greathouse IV, Jan 15 2012
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Jan 04 2004
STATUS
approved