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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). 0
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 (list; constant; graph; refs; listen; history; text; internal format)
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

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Last modified April 20 04:42 EDT 2019. Contains 322294 sequences. (Running on oeis4.)