A185580 Decimal expansion of Sum_{m,n,p = -infinity..infinity} (-1)^(m+n)/sqrt( m^2 + n^2 + (p-1/2)^2 ).

1, 3, 1, 9, 6, 7, 0, 5, 8, 6, 9, 6, 7, 4, 3, 6, 2, 0, 0, 9, 2, 9, 5, 2, 3, 3, 7, 8, 2, 8, 6, 1, 0, 1, 7, 5, 3, 6, 7, 6, 2, 7, 3, 4, 6, 3, 7, 2, 1, 9, 1, 4, 8, 3, 8, 7, 1, 6, 8, 1, 5, 4, 3, 5, 0, 4, 3, 9, 7, 9, 1, 6, 8, 4, 9, 8, 9, 4, 7, 5, 2, 5, 6, 6, 1, 3, 3, 8, 1, 1, 9, 3, 2, 2, 7, 0, 0, 6, 9, 4, 2, 2, 0, 2, 0

Offset

1,2

Links

Table of n, a(n) for n=1..105.

I. J. Zucker, Madelung constants and lattice sums for invariant cubic lattice complexes and certain tetragonal structures, J. Phys. A: Math. Gen. 8 (11) (1975) 1734, variable f(1).

I. J. Zucker, Functional equations for poly-dimensional zeta functions and the evaluation of Madelung constants, J. Phys. A: Math. Gen. 9 (4) (1976) 499, variable f(1).

Formula

Equals 4*Sum_{n>=1, p>=1} cosech(d*Pi)/d where d = sqrt((n-1/2)^2 + (p-1/2)^2).

Mathematica

digits = 105; Clear[f]; f[n_, p_] := f[n, p] = (d = Sqrt[(n - 1/2)^2 + (p - 1/2)^2]; (Csch[d*Pi]/d) // N[#, digits + 10] &); f[m_] := f[m] = 4*Sum[f[n, p], {n, 1, m}, {p, 1, m}] // RealDigits[#, 10, digits + 10] & // First; f[0]; f[m = 10]; While[f[m] != f[m - 10], Print[m]; m = m + 10]; f[m][[1 ;; digits]] (* Jean-François Alcover, Feb 21 2013 *)

Crossrefs

Cf. A185576, A185577, A185578, A185579, A185581, A185582, A185583.

Sequence in context: A217629 A127552 A229759 * A052931 A368379 A006803

Adjacent sequences: A185577 A185578 A185579 * A185581 A185582 A185583

Keyword

cons,nonn

Author

R. J. Mathar, Jan 31 2011

Extensions

More terms from Jean-François Alcover, Feb 21 2013

Status

approved