A088537 Decimal expansion of Madelung's constant M2.

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

Offset: 1

Author

Benoit Cloitre, Nov 16 2003

Keywords

• nonn
• cons

Examples

			M2 = -1.61554262671282472386792333275861809...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 76-81.

Links

• I. J. Zucker, Exact results for some lattice sums in 2, 4, 6 and 8 dimensions, J. Phys. A: Math., Gen. vol. 7 (1974) no. 13, pp. 1568-1575.

Crossrefs

Cf. A059750.

Programs

Maple

M2:=evalf(4*(sqrt(2)-1)*Zeta(1/2)*sum('(-1)^n/sqrt(2*n+1)','n'=0..infinity),120); # Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 10 2009

Mathematica

(2-2*I)*(Sqrt[2]-1)*Zeta[1/2]*(PolyLog[1/2, -I]-Zeta[1/2, 1/4]) // Re // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Feb 15 2013 *)

PARI

DirBet=sumalt(n=0, (-1)^n/sqrt(2*n+1)); print(4.0*(sqrt(2)-1)*zeta(0.5)*DirBet) ; \\ R. J. Mathar, Jul 20 2007

Formula

M2 = Sum_{ -oo < i < oo, -oo < j < oo, (i,j) != (0,0) } (-1)^(i + j)/sqrt(i^2 + j^2).

M2 = 4*(sqrt(2) - 1)*zeta(1/2)*beta(1/2) (beta=Dirichlet beta function).

Extensions

More terms from R. J. Mathar, Jul 20 2007

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 10 2009