A338612 Decimal expansion of Sum_{k>=1} (-1)^(k+1)/L(k) where L(k) is the k-th Lucas number (A000032).

8, 3, 0, 5, 0, 2, 8, 2, 1, 5, 8, 6, 8, 7, 6, 6, 8, 2, 3, 1, 6, 9, 3, 6, 4, 8, 6, 2, 5, 1, 0, 5, 9, 5, 1, 9, 1, 7, 7, 3, 0, 4, 6, 2, 1, 4, 3, 0, 4, 0, 8, 2, 8, 0, 1, 4, 6, 0, 2, 6, 4, 1, 3, 9, 0, 7, 9, 1, 0, 4, 9, 8, 4, 8, 6, 0, 4, 3, 0, 0, 6, 7, 4, 9, 3, 3, 0

Offset: 0

Author

Amiram Eldar, Nov 03 2020

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)).

Examples

			0.83050282158687668231693648625105951917730462143040...
		

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.
• Yohei Tachiya, Irrationality of certain Lambert series, Tokyo Journal of Mathematics, Vol. 27, No. 1 (2004), pp. 75-85.
• Eric Weisstein's World of Mathematics, Reciprocal Lucas Constant.

Crossrefs

Cf. A000032, A000045, A001622, A079586, A093540, A153415, A153416, A158933.

Programs

Mathematica

RealDigits[Sum[(-1)^(n+1)/LucasL[n], {n, 1, 1000}], 10, 120][[1]]

Formula

Equals A153416 - A153415.

Equals Sum_{k>=1} (-1)^(k+1) * Fibonacci(k)/Fibonacci(2*k).

Equals Sum_{k>=1} (-1)^(k+1)/(phi^k + (1-phi)^k), where phi is the golden ratio (A001622).

Equals Sum_{k>=0} 1/(phi^(2*k+1) + (-1)^k).