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A093540 Decimal expansion of Sum_{n >= 1} 1/L(n), where L(n) is the n-th Lucas number. 14
1, 9, 6, 2, 8, 5, 8, 1, 7, 3, 2, 0, 9, 6, 4, 5, 7, 8, 2, 8, 6, 8, 7, 9, 5, 1, 2, 8, 6, 7, 5, 1, 8, 3, 5, 2, 6, 6, 4, 9, 5, 9, 3, 0, 1, 7, 1, 6, 2, 2, 1, 9, 4, 2, 1, 1, 3, 0, 7, 1, 5, 2, 4, 0, 4, 1, 7, 0, 6, 1, 6, 0, 7, 5, 4, 6, 4, 6, 0, 3, 7, 7, 9, 7, 9, 0, 4, 1, 8, 9, 9, 0, 8, 4, 0, 3, 4, 6, 9, 6, 2, 2 (list; constant; graph; refs; listen; history; text; internal format)
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
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 A153415 + A153416. (End)
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
Sequence in context: A019961 A327996 A357762 * A252837 A198573 A068925
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Jan 04 2004
STATUS
approved

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