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A255698
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Decimal expansion of the Plouffe sum S(3,1) = Sum_{n >= 1} 1/(n^3*(exp(Pi*n)-1)).
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8
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4, 5, 4, 0, 2, 6, 1, 7, 2, 7, 7, 2, 8, 1, 1, 1, 6, 7, 3, 1, 5, 8, 3, 1, 1, 5, 8, 4, 6, 8, 9, 6, 1, 8, 9, 6, 9, 6, 1, 0, 6, 0, 6, 9, 0, 4, 0, 5, 8, 4, 9, 0, 1, 9, 9, 4, 3, 5, 7, 1, 2, 8, 8, 5, 9, 3, 7, 5, 9, 0, 7, 7, 7, 7, 9, 9, 8, 0, 0, 0, 5, 6, 4, 3, 4, 1, 4, 8, 9, 0, 5, 2, 9, 0, 4, 4, 5, 4, 5, 5, 6, 5, 4
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OFFSET
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-1,1
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LINKS
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FORMULA
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This is the case k=3, m=1 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(Pi*k)). - Amiram Eldar, Jun 05 2023
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EXAMPLE
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0.0454026172772811167315831158468961896961060690405849...
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MATHEMATICA
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digits = 103; S[3, 1] = NSum[1/(n^3*(Exp[ Pi*n] - 1)), {n, 1, Infinity}, WorkingPrecision -> digits+10, NSumTerms -> digits]; RealDigits[S[3, 1], 10, digits] // First
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CROSSREFS
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Cf. A255695 (S(1,1)), A084254 (S(1,2)), A255697 (S(1,4)), A255699 (S(3,2)), A255700 (S(3,4)), A255701 (S(5,1)), A255702 (S(5,2)), A255703 (S(5,4)).
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KEYWORD
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AUTHOR
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STATUS
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approved
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