cp's OEIS Frontend

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

%I A338612 #5 Nov 04 2020 17:11:59
%S A338612 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,
%T A338612 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,
%U A338612 0,4,9,8,4,8,6,0,4,3,0,0,6,7,4,9,3,3,0
%N A338612 Decimal expansion of Sum_{k>=1} (-1)^(k+1)/L(k) where L(k) is the k-th Lucas number (A000032).
%C A338612 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)).
%H A338612 Richard André-Jeannin, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k5686125p/f9.image">Irrationalité de la somme des inverses de certaines suites récurrentes</a>, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, Vol. 308, No. 19 (1989), pp. 539-541.
%H A338612 Yohei Tachiya, <a href="https://projecteuclid.org/euclid.tjm/1244208475">Irrationality of certain Lambert series</a>, Tokyo Journal of Mathematics, Vol. 27, No. 1 (2004), pp. 75-85.
%H A338612 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ReciprocalLucasConstant.html">Reciprocal Lucas Constant</a>.
%F A338612 Equals A153416 - A153415.
%F A338612 Equals Sum_{k>=1} (-1)^(k+1) * Fibonacci(k)/Fibonacci(2*k).
%F A338612 Equals Sum_{k>=1} (-1)^(k+1)/(phi^k + (1-phi)^k), where phi is the golden ratio (A001622).
%F A338612 Equals Sum_{k>=0} 1/(phi^(2*k+1) + (-1)^k).
%e A338612 0.83050282158687668231693648625105951917730462143040...
%t A338612 RealDigits[Sum[(-1)^(n+1)/LucasL[n], {n, 1, 1000}], 10, 120][[1]]
%Y A338612 Cf. A000032, A000045, A001622, A079586, A093540, A153415, A153416, A158933.
%K A338612 nonn,cons
%O A338612 0,1
%A A338612 _Amiram Eldar_, Nov 03 2020