A185576 - OEIS

%I #25 Mar 11 2018 05:58:21

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

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

%U 4,7,7,1,1,1,6,2,2,7,3,3,0,9,4,7,4,0,4,1,5,3,0,7,9,2,7,1,1,0,3,5

%N Decimal expansion of Born's basic potential Pi_0.

%C Decimal expansion of Sum'_{m,n,p = -infinity..infinity} 1/(m^2 + n^2 + p^2)^s, analytic continuation to s=1/2. The prime at the sum symbol means the term at m=n=p=0 is omitted.

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

%H Y. Sakamoto, <a href="https://doi.org/10.1063/1.1744060">Madelung constants of simple crystals expressed in terms of Born's basic potentials of 15 figures</a>, J. Chem. Phys. 28 (1958) 164, variable Pi_0.

%H I. J. Zucker, <a href="https://dx.doi.org/10.1088/0305-4470/8/11/008">Madelung constants and lattice sums for invariant cubic lattice complexes and certain tetragonal structures</a>, J. Phys. A: Math. Gen. 8 (11) (1975) 1734, variable a(1).

%H I. J. Zucker, <a href="http://dx.doi.org/10.1088/0305-4470/9/4/006">Functional equations for poly-dimensional zeta functions and the evaluation of Madelung constants</a>, J. Phys. A: Math. Gen. 9 (4) (1976) 499, variable a(1).

%F Equals A085469/3 + A185577 + A185578.

%e 2.8372974794806194766659171046...

%t 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]; RealDigits[s[k], 10, digits] // First (* _Jean-François Alcover_, Sep 10 2014 *)

%Y Cf. A185577, A185578, A185579, A185580, A185581, A185582, A185583.

%K cons,nonn

%O 1,1

%A _R. J. Mathar_, Jan 31 2011

%E More terms from _Jean-François Alcover_, Sep 10 2014