A093540 Decimal expansion of Sum_{n >= 1} 1/L(n), where L(n) is the n-th Lucas number.

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

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 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

Cf. A001622, A000204, A079586, A153415, A153416.

Sequence in context: A019961 A327996 A357762 * A252837 A198573 A068925

Adjacent sequences: A093537 A093538 A093539 * A093541 A093542 A093543

Keyword

nonn,cons

Author

Eric W. Weisstein, Jan 04 2004

Status

approved