A259281 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.

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, 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, 2, 7, 0, 3, 0, 9, 4, 9, 5, 1, 0, 8, 5, 2, 3, 0, 7, 7, 8

Offset

2,2

Links

Jon E. Schoenfield, Table of n, a(n) for n = 2..401

M. Kontsevich and D. Zagier, Periods, Institut des Hautes Etudes Scientifiques 2001 IHES/M/01/22 p. 25.

Jon E. Schoenfield, Magma program

I. J. Zucker, Madelung constants and lattice sums for invariant cubic lattice complexes and certain tetragonal structures, J. Phys. A: Math. Gen. 8 (1975) 1734, Table 4, s=2, a(2s).

Example

16.532315959761669643892704592887851743834129...

Mathematica

(* 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

Crossrefs

Sequence in context: A245632 A356983 A158038 * A153330 A225661 A225662

Adjacent sequences: A259278 A259279 A259280 * A259282 A259283 A259284

Keyword

nonn,cons

Author

Jean-François Alcover, Jun 23 2015

Status

approved