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 A338612 Decimal expansion of Sum_{k>=1} (-1)^(k+1)/L(k) where L(k) is the k-th Lucas number (A000032). 0
 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 (list; constant; graph; refs; listen; history; text; internal format)
 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 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 A200300 Adjacent sequences:  A338609 A338610 A338611 * A338613 A338614 A338615 KEYWORD nonn,cons AUTHOR Amiram Eldar, Nov 03 2020 STATUS approved

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Last modified August 5 03:00 EDT 2021. Contains 346457 sequences. (Running on oeis4.)