A391782 Decimal expansion of Sum_{k>=1} eta(2*k) * Lucas(2*k) / 5^k, where eta is the Dirichlet eta function.

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, 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, 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

Offset

1,3

Links

Table of n, a(n) for n=1..105.

Khristo Boyadzhiev and Robert Frontczak, Series Involving Euler's Eta (or Dirichlet Eta) Function, Journal of Integer Sequences, Vol. 24 (2021), Article 21.9.1. See p. 8, eq. (12).

Robert Frontczak, proposer, Problem H-872, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 59, No. 1 (2021), p. 90; Fibonacci numbers and the alternating Riemann zeta function, Solution to Problem H-872 by the proposer, ibid., Vol. 60, No. 4 (2022), pp. 374-375.

Eric Weisstein's World of Mathematics, Dirichlet Eta Function.

Wikipedia, Dirichlet eta function.

Formula

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

Equals 5 * A391781 - 1.

Example

1.05806011897843731375271608862367495615612193924278...

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A153420 A193505 A141848 * A349398 A349397 A140249

Adjacent sequences: A391779 A391780 A391781 * A391783 A391784 A391785

Keyword

nonn,cons

Author

Amiram Eldar, Dec 20 2025

Status

approved