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%I #22 Aug 14 2017 02:38:59
%S 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,
%T 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,
%U 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
%N Decimal expansion of Madelung's constant M2.
%D S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 76-81.
%H I. J. Zucker, <a href="https://doi.org/10.1088/0305-4470/7/13/011">Exact results for some lattice sums in 2, 4, 6 and 8 dimensions</a>, J. Phys. A: Math., Gen. vol. 7 (1974) no. 13, pp. 1568-1575.
%F M2 = Sum_{ -oo < i < oo, -oo < j < oo, (i,j) != (0,0) } (-1)^(i + j)/sqrt(i^2 + j^2)).
%F M2 = 4*(sqrt(2) - 1)*zeta(1/2)*beta(1/2) (beta=Dirichlet beta function).
%e M2 = -1.6155426267....
%p 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
%t (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 *)
%o (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
%Y Cf. A059750.
%K nonn,cons
%O 1,2
%A _Benoit Cloitre_, Nov 16 2003
%E More terms from _R. J. Mathar_, Jul 20 2007
%E More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 10 2009
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