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%I #22 Mar 11 2018 05:58:32
%S 1,4,8,0,3,8,9,8,0,6,5,1,2,2,2,2,5,9,7,9,0,7,7,6,1,7,0,6,3,5,2,8,1,7,
%T 5,5,5,7,0,7,6,6,0,5,0,8,5,1,3,6,8,8,5,5,3,6,4,5,5,3,6,2,5,7,0,0,8,7,
%U 5,7,3,1,7,4,3,5,0,4,6,1,2,7,3,9,8,8,9,1,0,7,8,8,9,0,2,0,4,5,9,0,1,8,6,7,9
%N Decimal expansion of Sum'_{m,n,p = -infinity .. infinity} (-1)^(m + n)/sqrt(m^2 + n^2 + p^2), negated.
%C The prime at the sum symbol means the term at m=n=p=0 is omitted.
%H Y. Sakamoto, <a href="https://doi.org/10.1063/1.1744060">Madelung constants of simple crystals expressed in terms of Born's basic potentials of 15 figures</a>, J. Chem. Phys. 28 (1958) 164, variable Pi_3.
%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 c(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 c(1).
%F Equals Pi/2 - 9*log(2)/2 + 4*Sum_{p>=1, n>=1} (1+(-1)^n+(-1)^(n+p))*cosech(d*Pi)/d where d = sqrt(n^2 + p^2).
%e 1.48038980651222259790776170...
%t digits = 105; Clear[f]; f[n_, p_] := f[n, p] = (s = Sqrt[n^2 + p^2]; ((1 + (-1)^n + (-1)^(n + p))*Csch[s*Pi])/s // N[#, digits+10]&); f[m_] := f[m] = Pi/2 - 9*Log[2]/2 + 4*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 20 2013 *)
%Y Cf. A185576, A185577, A185579, A185580, A185581, A185582, A185583.
%K cons,nonn
%O 1,2
%A _R. J. Mathar_, Jan 31 2011
%E More terms from _Jean-François Alcover_, Feb 20 2013
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