A391782 - OEIS

%I #8 Jan 17 2026 17:22:47

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

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

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

%N Decimal expansion of Sum_{k>=1} eta(2*k) * Lucas(2*k) / 5^k, where eta is the Dirichlet eta function.

%H Khristo Boyadzhiev and Robert Frontczak, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Frontczak/front22.html">Series Involving Euler's Eta (or Dirichlet Eta) Function</a>, Journal of Integer Sequences, Vol. 24 (2021), Article 21.9.1. See p. 8, eq. (12).

%H Robert Frontczak, proposer, <a href="https://www.fq.math.ca/Problems/Feb2021AdvProb.pdf">Problem H-872</a>, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 59, No. 1 (2021), p. 90; <a href="https://fq.math.ca/Problems/AdvProbNov2022.pdf">Fibonacci numbers and the alternating Riemann zeta function</a>, Solution to Problem H-872 by the proposer, ibid., Vol. 60, No. 4 (2022), pp. 374-375.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DirichletEtaFunction.html">Dirichlet Eta Function</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dirichlet_eta_function">Dirichlet eta function</a>.

%F Equals Pi/(2*cos(Pi/(2*sqrt(5)))) - 1.

%F Equals 5 * A391781 - 1.

%e 1.05806011897843731375271608862367495615612193924278...

%t RealDigits[Pi/(2*Cos[Pi/(2*Sqrt[5])]) - 1, 10, 120][[1]]

%o (PARI) Pi/(2*cos(Pi/(2*sqrt(5)))) - 1

%Y Cf. A000032, A005248, A350760 (analogous sum with zeta), A391781.

%K nonn,cons

%O 1,3

%A _Amiram Eldar_, Dec 20 2025