A247046 Decimal expansion of delta_3, a constant associated with a certain 3-dimensional lattice sum.

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, 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, 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

Offset: 1

Author

Jean-François Alcover, Sep 10 2014

Keywords

• nonn
• cons
• easy

Examples

			-2.313698703882320603588809874061155008356357135595965962...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 79.

Links

• Eric Weisstein's World of Mathematics, Lattice Sum.

Crossrefs

Cf. A088537, A185576, A247042.

Programs

Mathematica

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

Formula

Pi_0 + Pi/6, where Pi_0 is A185576.