A247277 Decimal expansion of gamma_3, a lattice sum constant, analog of Euler's constant for 3-dimensional lattices.
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
Offset: 0
Examples
0.58174804565972267655489926584685317714602246563144492431364...
References
- Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 80.
Links
- Eric Weisstein's MathWorld, Lattice Sum
Programs
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Mathematica
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
Formula
gamma_3 = (1/8)*(delta_3 + 3*(- Pi/6 + log((sqrt(3) + 1)/(sqrt(3) - 1))) - 12*gamma_2 - 6*EulerGamma).