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%I #18 Apr 24 2016 19:51:39
%S 2,5,1,9,3,5,6,1,5,2,0,8,9,4,4,5,3,1,3,3,4,2,7,1,1,7,2,7,3,2,9,4,3,7,
%T 9,1,2,1,1,6,4,9,9,1,3,6,7,5,1,7,3,2,5,7,7,5,0,0,6,6,0,7,8,5,6,7,7,4,
%U 3,9,0,1,2,6,9,1,8,7,2,7,7,4,0,9,6,4,2,8,0,2,1,0,1,6,2,3,7,3,0,3,1
%N Decimal expansion of the doubly infinite sum N_3 = Sum_{i,j,k = -inf..inf} (-1)^(i+j+k)/(i^2+j^2+k^2), a lattice constant analog of Madelung's constant (negated).
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.10 Madelung's constant, p. 77.
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/MadelungConstants.html">Madelung Constants</a>
%F N_3 = Pi^2/3-Pi*log(2)-(Pi/sqrt(2))*log(2(sqrt(2)+1))+8 Pi*Sum_{m,n >= 1} (-1)^n csch(Pi*sqrt(m^2+2n^2))/sqrt(m^2+2n^2).
%e -2.51935615208944531334271172732943791211649913675173257750066...
%t digits = 101; Clear[s]; s[max_] := s[max] = NSum[(-1)^n Csch[Pi *Sqrt[m^2 + 2 n^2]]/Sqrt[m^2 + 2 n^2], {m, 1, max}, {n, 1, max}, Method -> "AlternatingSigns", WorkingPrecision -> digits + 10]; s[10]; s[max = 20]; Print[max]; While[RealDigits[s[max], 10, digits + 5][[1]] != RealDigits[s[max/2], 10, digits + 5][[1]], max = max*2; Print[max]]; N3 = Pi^2/3 - Pi*Log[2] - Pi/Sqrt[2] Log[2 (Sqrt[2] + 1)] + 8 Pi*s[max]; RealDigits[N3, 10, digits][[1]]
%Y Cf. A088537 (M_2), A085469 (M_3), A090734 (M_4), A086054 (N_2).
%K nonn,cons
%O 1,1
%A _Jean-François Alcover_, Apr 24 2016
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