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A255699 Decimal expansion of the Plouffe sum S(3,2) = Sum_{n >= 1} 1/(n^3*(exp(2*Pi*n)-1)). 8
1, 8, 7, 1, 3, 7, 2, 7, 5, 9, 3, 6, 6, 0, 2, 7, 3, 7, 8, 8, 3, 7, 0, 4, 5, 5, 4, 9, 0, 2, 4, 3, 5, 5, 7, 7, 1, 8, 8, 7, 4, 6, 4, 8, 3, 3, 0, 7, 2, 0, 9, 7, 6, 0, 1, 1, 1, 9, 1, 2, 9, 9, 0, 3, 0, 9, 4, 1, 3, 1, 6, 1, 3, 1, 2, 1, 0, 7, 2, 3, 0, 3, 9, 0, 2, 3, 6, 1, 0, 5, 1, 4, 9, 9, 6, 0, 2, 4, 5, 8, 0, 4 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
-2,2
LINKS
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2022, p. 6.
Linas Vepštas, On Plouffe's Ramanujan identities, The Ramanujan Journal, Vol. 27 (2012), pp. 387-408; alternative link; arXiv preprint, arXiv:math/0609775 [math.NT], 2006-2010.
FORMULA
This is the case k=3, m=2 of S(k,m) = Sum_{n >= 1} 1/(n^k*(exp(m*Pi*n)-1)).
Pi^3 = 720*S(3,1) - 900*S(3,2) + 180*S(3,4).
zeta(3) = 28*S(3,1) - 37*S(3,2) + 7*S(3,4).
Equals Sum_{k>=1} sigma_3(k)/(k^3*exp(2*Pi*k)). - Amiram Eldar, Jun 05 2023
EXAMPLE
0.0018713727593660273788370455490243557718874648330720976...
MATHEMATICA
digits = 102; S[3, 2] = NSum[1/(n^3*(Exp[2* Pi*n] - 1)), {n, 1, Infinity}, WorkingPrecision -> digits+10, NSumTerms -> digits]; RealDigits[S[3, 2], 10, digits] // First
CROSSREFS
Cf. A255695 (S(1,1)), A084254 (S(1,2)), A255697 (S(1,4)), A255698 (S(3,1)), A255700 (S(3,4)), A255701 (S(5,1)), A255702 (S(5,2)), A255703 (S(5,4)).
Cf. A001158 (sigma_3), A002117 (zeta(3)).
Sequence in context: A248582 A362120 A362122 * A343965 A201768 A019814
KEYWORD
nonn,cons,easy
AUTHOR
STATUS
approved

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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)