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

Author

Eric W. Weisstein, Jan 04 2004

Keywords

• nonn
• cons

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

Examples

			1.96285817320964578286879512867518352664959301716221...
		

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.

Crossrefs

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

Programs

Mathematica

RealDigits[Sum[1/LucasL[n],{n,2000}],10,120][[1]] (* Harvey P. Dale, Jan 15 2012 *)

PARI

suminf(n=1,1/(fibonacci(n-1)+fibonacci(n+1))) \\ Charles R Greathouse IV, Jan 15 2012

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