A271872 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).

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, 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, 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

Offset

1,1

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.10 Madelung's constant, p. 77.

Links

Table of n, a(n) for n=1..101.

Eric Weisstein's MathWorld, Madelung Constants

Formula

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).

Example

-2.51935615208944531334271172732943791211649913675173257750066...

Mathematica

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]]

Crossrefs

Cf. A088537 (M_2), A085469 (M_3), A090734 (M_4), A086054 (N_2).

Sequence in context: A174815 A021401 A280637 * A286677 A199868 A010588

Adjacent sequences: A271869 A271870 A271871 * A271873 A271874 A271875

Keyword

nonn,cons

Author

Jean-François Alcover, Apr 24 2016

Status

approved