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A247277
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Decimal expansion of gamma_3, a lattice sum constant, analog of Euler's constant for 3-dimensional lattices.
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0
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5, 8, 1, 7, 4, 8, 0, 4, 5, 6, 5, 9, 7, 2, 2, 6, 7, 6, 5, 5, 4, 8, 9, 9, 2, 6, 5, 8, 4, 6, 8, 5, 3, 1, 7, 7, 1, 4, 6, 0, 2, 2, 4, 6, 5, 6, 3, 1, 4, 4, 4, 9, 2, 4, 3, 1, 3, 6, 4, 0, 0, 8, 7, 5, 4, 3, 8, 9, 5, 6, 2, 1, 9, 4, 8, 9, 2, 7, 8, 6, 3, 8, 0, 3, 4, 3, 4, 7, 4, 4, 7, 9, 9, 5, 9, 0, 4, 4, 5, 3, 2, 4
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OFFSET
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0,1
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REFERENCES
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Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 80.
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LINKS
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FORMULA
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gamma_3 = (1/8)*(delta_3 + 3*(- Pi/6 + log((sqrt(3) + 1)/(sqrt(3) - 1))) - 12*gamma_2 - 6*EulerGamma).
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EXAMPLE
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0.58174804565972267655489926584685317714602246563144492431364...
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MATHEMATICA
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digits = 100; k0 = 10; dk = 10; Clear[s]; s[k_] := s[k] = 7*(Pi/6) - 19/2*Log[2] + 4*Sum[(3 + 3*(-1)^m + (-1)^(m + n))*Csch[Pi*Sqrt[m^2 + n^2]]/Sqrt[m^2 + n^2], {m, 1, k}, {n, 1, k}] // N[#, digits + 10] &; s[k0]; s[k = k0 + dk]; While[RealDigits[s[k], 10, digits + 5][[1]] != RealDigits[s[k - dk], 10, digits + 5][[1]], k = k + dk]; Pi0 = s[k]; delta2 = 2*Zeta[1/2]*(Zeta[1/2, 1/4] - Zeta[1/2, 3/4]); delta3 = Pi0 + Pi/6; gamma2 = (1/4)*(delta2 + 2*Log[(Sqrt[2] + 1)/(Sqrt[2] - 1)] - 4*EulerGamma); gamma3 = (1/8)*(delta3 + 3*(- Pi/6 + Log[(Sqrt[3] + 1)/(Sqrt[3] - 1)]) - 12*gamma2 - 6*EulerGamma); RealDigits[gamma3, 10, 102] // First
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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