Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #19 Nov 06 2016 01:48:48
%S 1,6,5,3,2,3,1,5,9,5,9,7,6,1,6,6,9,6,4,3,8,9,2,7,0,4,5,9,2,8,8,7,8,5,
%T 1,7,4,3,8,3,4,1,2,9,0,7,0,2,5,5,1,8,6,8,8,6,1,1,7,7,9,3,6,5,9,5,6,0,
%U 2,7,0,3,0,9,4,9,5,1,0,8,5,2,3,0,7,7,8
%N Decimal expansion of Sum'_{(x,y,z)=-infinity..infinity} 1/(x^2+y^2+z^2)^2, where the 'prime' indicates that the term x=y=z=0 is to be left out.
%H Jon E. Schoenfield, <a href="/A259281/b259281.txt">Table of n, a(n) for n = 2..401</a>
%H M. Kontsevich and D. Zagier, <a href="http://www.ihes.fr/~maxim/TEXTS/Periods.pdf">Periods</a>, Institut des Hautes Etudes Scientifiques 2001 IHES/M/01/22 p. 25.
%H Jon E. Schoenfield, <a href="/A259281/a259281_1.txt">Magma program</a>
%H I. J. Zucker, <a href="http://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 (1975) 1734, Table 4, s=2, a(2s).
%e 16.532315959761669643892704592887851743834129...
%t (* This script gives only 10 correct digits *) s1 = NSum[(-2 + Pi*x*(Coth[Pi*x] + Pi*x*Csch[Pi*x]^2))/(4*x^4), {x, 1, Infinity} ];s2 = NSum[-((Csch[Pi*x]^2*(2 + 2*Pi^2*x^2 - 2*Cosh[2*Pi*x] + Pi*x*Sinh[2*Pi*x]))/(16*x^4)), {x, 1, Infinity} ]; f[y_?NumericQ] := NSum[(Pi*Coth[Pi*Sqrt[x^2 + y^2]])/(4*(x^2 + y^2)^(3/2)), {x, 1, Infinity} ]; s3 = NSum[f[y], {y, 1, Infinity} ]; g[y_?NumericQ] := NSum[2*((Pi^2*x^2*Csch[Pi*Sqrt[x^2 + y^2]]^2)/(4*(x^2 + y^2)^2)), {x, 1, Infinity} ]; s4 = NSum[g[y], {y, 1, Infinity} ]; s = Pi^4/15 + 12*s1 + 8*(s2 + s3 + s4); RealDigits[s, 10, 10] // First
%K nonn,cons
%O 2,2
%A _Jean-François Alcover_, Jun 23 2015