This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A084254 #21 Feb 17 2025 06:07:38
%S A084254 0,0,1,8,7,2,6,8,2,4,4,9,7,6,8,5,4,6,1,1,5,6,3,8,5,7,9,4,7,9,9,6,1,3,
%T A084254 9,8,8,6,9,1,6,2,8,9,5,6,5,2,6,1,9,5,6,3,8,4,1,3,3,1,5,7,4,5,3,7,8,8,
%U A084254 4,3,1,9,5,1,7,0,9,8,0,2,2,6,7,5,1,7,0,7,2,7,8,4,0,2,4,5,6,7,9,7,9,9,8,7
%N A084254 Decimal expansion of Sum_{k>=1} 1/(k*(exp(2*Pi*k)-1)).
%D A084254 Bruce C. Berndt, Ramanujan Notebook part II, Infinite series, Springer Verlag, 1989, pp. 280-281.
%H A084254 Steven R. Finch, <a href="https://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, arXiv:2001.00578 [math.HO], 2020-2022, p. 6.
%H A084254 Simon Plouffe, <a href="http://www.plouffe.fr/simon/inspired2.pdf">Identities inspired by Ramanujan Notebooks (part 2)</a>, April 2006.
%H A084254 Linas Vepštas, <a href="https://doi.org/10.1007/s11139-011-9335-9">On Plouffe's Ramanujan identities</a>, The Ramanujan Journal, Vol. 27 (2012), pp. 387-408; <a href="https://arxiv.org/abs/math/0609775">arXiv preprint</a>, arXiv:math/0609775 [math.NT], 2006-2010.
%F A084254 Equals log(4/Pi)/4 - Pi/12 + log(Gamma(3/4)).
%F A084254 From _Jean-François Alcover_, Mar 02 2015: (Start)
%F A084254 This is the case k=1, m=2 of the Plouffe sum S(k,m) = Sum_{n >= 1} 1/(n^k*(exp(m*Pi*n)-1)).
%F A084254 Pi = 72*S(1,1) - 96*S(1,2) + 24*S(1,4). (End)
%F A084254 Equals Sum_{k>=1} sigma(k)/(k*exp(2*Pi*k)). - _Amiram Eldar_, Jun 05 2023
%e A084254 0.00187268244976854611563857947996139886916289565261...
%t A084254 digits = 104; S[1, 2] = NSum[1/(n*(Exp[2*Pi*n] - 1)), {n, 1, Infinity}, WorkingPrecision -> digits+10, NSumTerms -> digits]; RealDigits[S[1, 2], 10, digits] // First (* _Jean-François Alcover_, Mar 02 2015 *)
%t A084254 Join[{0,0},RealDigits[Log[4/Pi]/4 - Pi/12 + Log[Gamma[3/4]], 10, 100][[1]]] (* _Amiram Eldar_, May 21 2022 *)
%o A084254 (PARI) 1/4*log(4/Pi)-Pi/12+log(gamma(3/4))
%Y A084254 Cf. A000203, A003881, A068465, A094640.
%Y A084254 Cf. A255695 (S(1,1)), A255697 (S(1,4)), A255698 (S(3,1)), A255699 (S(3,2)), A255700 (S(3,4)), A255701 (S(5,1)), A255702 (S(5,2)), A255703 (S(5,4)).
%K A084254 cons,nonn,easy
%O A084254 0,4
%A A084254 _Benoit Cloitre_, Jun 21 2003