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A185577 Decimal expansion of Sum'_{m,n,p = -infinity..infinity} (-1)^m/sqrt(m^2 + n^2 + p^2), negated. 9

%I #21 Mar 11 2018 05:58:11

%S 7,7,4,3,8,6,1,4,1,4,2,4,0,0,2,8,1,5,2,1,2,7,5,1,3,8,6,4,0,6,7,8,8,7,

%T 9,8,8,5,3,1,7,1,0,4,8,1,0,3,2,1,4,4,5,9,3,0,7,2,4,0,9,6,6,4,0,2,1,4,

%U 3,5,1,9,2,1,6,3,0,6,7,8,8,7,7,8,2,3,0,9,9,7,6,7,0,9,7,0,4,8,1,6,2,9,6,6,9

%N Decimal expansion of Sum'_{m,n,p = -infinity..infinity} (-1)^m/sqrt(m^2 + n^2 + p^2), negated.

%C The prime at the sum symbol means that 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_2.

%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 b(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 b(1).

%F sqrt(3)*(3*(this value) + A085469)/4 = A181152.

%F Equals Pi/2 - 7*log(2)/2 + 4*Sum_{n>=1, p>=1} (2+(-1)^n) *cosech(d*Pi)/d with d = sqrt(n^2 + p^2).

%e 0.77438614142400281521275138640678...

%t digits = 105; Clear[f]; f[n_, p_] := f[n, p] =(s = Sqrt[n^2 + p^2]; ((2 + (-1)^n)*Csch[s*Pi])/s // N[#, digits+10]&); f[m_] := f[m] = Pi/2 - (7*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, A185578, A185579, A185580, A185581, A185582, A185583.

%K cons,nonn

%O 0,1

%A _R. J. Mathar_, Jan 31 2011

%E More terms from _Jean-François Alcover_, Feb 20 2013

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