A185579 Decimal expansion of Sum_{m,n,p = -infinity..infinity} (-1)^m/sqrt(m^2 + (n-1/2)^2 + (p-1/2)^2).
1, 5, 4, 0, 1, 7, 0, 9, 0, 1, 8, 5, 5, 5, 4, 3, 6, 1, 7, 4, 3, 6, 4, 6, 6, 6, 6, 3, 8, 6, 4, 8, 0, 3, 9, 7, 8, 4, 2, 9, 6, 2, 7, 5, 6, 4, 1, 5, 6, 1, 4, 5, 9, 4, 8, 4, 2, 1, 8, 9, 5, 5, 2, 9, 4, 6, 0, 3, 7, 9, 1, 5, 8, 7, 6, 0, 1, 2, 7, 6, 9, 7, 9, 2, 0, 7, 4, 3, 0, 7, 6, 9, 2, 2, 7, 8, 9, 1, 3, 0, 2, 5, 3, 8, 5
Offset: 1
Examples
1.5401709018555436174364666638648...
Links
- 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 e(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 e(1).
Programs
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Mathematica
digits = 105; Clear[f]; f[n_, p_] := f[n, p] = (d = Sqrt[n^2 + (p - 1/2)^2]; (-1)^n*(Csch[d*Pi]/d) // N[#, digits+10]&); f[m_] := f[m] = 2*Log[1 + Sqrt[2]] + 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 *)
Formula
Equals 2*log(1+sqrt(2)) + 4*Sum_{n>=1, p>=1} (-1)^n*cosech(d*Pi)/d where d = sqrt(n^2 + (p-1/2)^2).
Extensions
More terms from Jean-François Alcover, Feb 21 2013