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

0,1

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)).

Links

Table of n, a(n) for n=0..86.

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.

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).

Example

0.83050282158687668231693648625105951917730462143040...

Mathematica

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

Crossrefs

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

Sequence in context: A256783 A154538 A154166 * A010521 A200025 A372585

Adjacent sequences: A338609 A338610 A338611 * A338613 A338614 A338615

Keyword

nonn,cons

Author

Amiram Eldar, Nov 03 2020

Status

approved