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%I #20 Mar 11 2018 05:57:31
%S 3,6,3,4,8,9,9,0,1,1,0,4,9,1,4,8,7,1,1,3,6,8,0,4,2,9,9,2,0,0,7,8,1,5,
%T 3,2,7,9,9,0,1,4,4,3,0,9,3,5,5,3,4,0,1,8,0,6,2,1,3,0,9,1,5,2,6,9,1,2,
%U 1,5,4,8,4,1,7,8,4,5,8,8,7,2,9,1,0,0,9,3,7,4,5,0,9,4,7,6,5,1,2,5,8,0,8,9,1
%N Decimal expansion of Sum_{m,n,p = -infinity..infinity} 4*(-1)^(m+p)/sqrt(m^2 + (2n-1/2)^2 + (2p-1/2)^2).
%C The defining equation (3.10i) on page 1738 of the 1975 paper has a typo (m for n).
%C The value in Table 4 (page 1742, column h(2s)) seems to have two digits swapped.
%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 h(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 h(1).
%F Equals 4*log(1+sqrt(2)) + 8*Sum_{n>=1, p>=1} cosech(d*Pi)/d where d = sqrt(2*n^2 + (p-1/2)^2).
%e 3.634899011049148711368042992007815327990...
%t digits = 105; Clear[f]; f[n_, p_] := f[n, p] = (d = Sqrt[2 n^2 + (p - 1/2)^2]; (Csch[d*Pi]/d) // N[#, digits + 10] &); f[m_] := f[m] = 4 Log[1 + Sqrt[2]] + 8*Sum[f[n, p], {n, 1, m}, {p, 1, 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, A185581, 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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