A185577 Decimal expansion of Sum'_{m,n,p = -infinity..infinity} (-1)^m/sqrt(m^2 + n^2 + p^2), negated.

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

Offset

0,1

Comments

The prime at the sum symbol means that the term at m=n=p=0 is omitted.

Links

Table of n, a(n) for n=0..104.

Y. Sakamoto, Madelung constants of simple crystals expressed in terms of Born's basic potentials of 15 figures, J. Chem. Phys. 28 (1958) 164, variable Pi_2.

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 b(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 b(1).

Formula

sqrt(3)*(3*(this value) + A085469)/4 = A181152.

Equals Pi/2 - 7*log(2)/2 + 4*Sum_{n>=1, p>=1} (2+(-1)^n) *cosech(d*Pi)/d with d = sqrt(n^2 + p^2).

Example

0.77438614142400281521275138640678...

Mathematica

digits = 105; Clear[f]; f[n_, p_] := f[n, p] =(s = Sqrt[n^2 + p^2]; ((2 + (-1)^n)*Csch[s*Pi])/s // N[#, digits+10]&); f[m_] := f[m] = Pi/2 - (7*Log[2])/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 20 2013 *)

Crossrefs

Cf. A185576, A185578, A185579, A185580, A185581, A185582, A185583.

Sequence in context: A353127 A247274 A086315 * A201517 A010513 A225402

Adjacent sequences: A185574 A185575 A185576 * A185578 A185579 A185580

Keyword

cons,nonn

Author

R. J. Mathar, Jan 31 2011

Extensions

More terms from Jean-François Alcover, Feb 20 2013

Status

approved