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%I #20 Mar 11 2018 05:57:48
%S 2,5,3,3,5,5,7,4,0,4,4,3,3,1,2,1,0,2,5,2,9,4,8,6,2,7,9,5,7,1,8,9,2,9,
%T 1,1,1,1,2,9,7,9,6,9,6,3,9,8,2,7,4,9,9,5,8,9,7,0,3,6,9,7,0,6,5,3,4,5,
%U 3,6,3,0,6,1,2,0,3,5,5,6,9,7,0,8,0,1,6,4,9,3,0,6,1,0,8,8,8,1,1,3,7,1,0,4,2
%N Decimal expansion of 8*Sum_{m,n,p = -infinity..infinity} (-1)^(m + n + p)/ sqrt( (2*m-1/2)^2+(2*n-1/2)^2+(2*p-1/2)^2 ).
%C The formula for g(1) in the 1976 paper on page 503 is a factor 2 too large.
%H I. J. Zucker, <a href="https://dx.doi.org/10.1088/0305-4470/8/11/008">Madelung constants and lattice sums for invariant cubic lattice complexes and certain tetragonal structures</a>, J. Phys. A: Math. Gen. 8 (11) (1975) 1734, variable g(1).
%H I. J. Zucker, <a href="http://dx.doi.org/10.1088/0305-4470/9/4/006">Functional equations for poly-dimensional zeta functions and the evaluation of Madelung constants</a>, J. Phys. A: Math. Gen. 9 (4) (1976) 499, variable g(1).
%F Equals 2*sqrt(2)*Sum_{n,p = -infinity..infinity} (-1)^n*cosech(d*Pi)/d where d = sqrt( (2*n-1/2)^2/2 + (p-1/2)^2 ).
%e 2.533557404433121025294862795718...
%t digits = 105; Clear[f]; f[n_, p_] := f[n, p] = (d = Sqrt[(2 n - 1/2)^2/2 + (p - 1/2)^2]; (-1)^n*(Csch[d*Pi]/d) // N[#, digits + 10] &); f[m_] := f[m] = 2 Sqrt[2]*Sum[f[n, p], {n, -m, m}, {p, -m, m}] // RealDigits[#, 10, digits + 10] & // First; f[0]; f[m = 10]; While[ f[m] != f[m - 10], Print[m]; m = m + 10]; f[m][[1 ;; digits]] (* _Jean-François Alcover_, Feb 21 2013 *)
%Y Cf. A185576, A185577, A185578, A185579, A185580, A185582, A185583.
%K cons,nonn
%O 1,1
%A _R. J. Mathar_, Jan 31 2011
%E More terms from _Jean-François Alcover_, Feb 21 2013
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