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%I #8 Mar 07 2023 10:31:21
%S 2,3,1,3,6,9,8,7,0,3,8,8,2,3,2,0,6,0,3,5,8,8,8,0,9,8,7,4,0,6,1,1,5,5,
%T 0,0,8,3,5,6,3,5,7,1,3,5,5,9,5,9,6,5,9,6,2,1,7,4,5,6,1,5,7,4,9,4,7,9,
%U 4,4,9,7,6,7,8,1,2,3,8,4,7,6,3,6,9,3,6,9,0,5,9,9,0,2,3,5,8,1,9,0
%N Decimal expansion of delta_3, a constant associated with a certain 3-dimensional lattice sum.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 79.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LatticeSum.html">Lattice Sum</a>.
%F Pi_0 + Pi/6, where Pi_0 is A185576.
%e -2.313698703882320603588809874061155008356357135595965962...
%t digits = 100; k0 = 10; dk = 10; Clear[s]; s[k_] := s[k] = 7*(Pi/6) - 19/2*Log[2] + 4*Sum[(3 + 3*(-1)^m + (-1)^(m + n))*Csch[Pi*Sqrt[m^2 + n^2]]/Sqrt[m^2 + n^2], {m, 1, k}, {n, 1, k}] // N[#, digits + 10] &; s[k0]; s[k = k0 + dk]; While[RealDigits[s[k], 10, digits + 5][[1]] != RealDigits[s[k - dk], 10, digits + 5][[1]], Print["s(", k, ") = ", s[k]]; k = k + dk]; Pi0 = s[k]; delta3 = Pi0 + Pi/6; RealDigits[delta3, 10, digits] // First
%Y Cf. A088537, A185576, A247042.
%K nonn,cons,easy
%O 1,1
%A _Jean-François Alcover_, Sep 10 2014
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