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%I #15 Jan 28 2017 14:43:22
%S 1,9,6,5,5,5,7,0,3,9,0,0,9,0,7,8,2,8,1,3,1,2,3,1,3,5,5,5,7,3,5,1,8,5,
%T 3,6,7,8,6,8,9,7,6,7,2,8,4,4,6,4,6,4,5,1,1,7,0,8,5,6,5,2,8,8,7,8,1,7,
%U 9,6,4,0,1,4,3,2,5,3,5,4,5,7,6,4,9,3,1,3,4,2,6,6,6,3,6,7,2,6,7,6,4,2,9,8
%N Decimal expansion of M_6, the 6th Madelung constant.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 77.
%H G. C. Greubel, <a href="/A247040/b247040.txt">Table of n, a(n) for n = 1..5000</a>
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/DirichletBetaFunction.html"> Dirichlet Beta Function</a>
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/MadelungConstants.html"> Madelung Constants</a>
%F M6 = (3/Pi^2)*(4*(sqrt(2)-1)*zeta(1/2)*beta(5/2) - (4*sqrt(2)-1)*zeta(5/2)*beta(1/2)), where beta is Dirichlet's "beta" function.
%e -1.9655570390090782813123135557351853678689767284464645117...
%t beta[x_] := (Zeta[x, 1/4] - Zeta[x, 3/4])/4^x; M6 = (3/Pi^2)*(4*(Sqrt[2]-1)*Zeta[1/2]*beta[5/2] - (4*Sqrt[2]-1)*Zeta[5/2]*beta[1/2]); RealDigits[M6, 10, 104][[1]]
%o (PARI) th4(x)=1+2*sumalt(n=1, (-1)^n*x^n^2)
%o intnum(x=0, [oo, 1], (th4(exp(-x))^6-1)/sqrt(Pi*x)) \\ _Charles R Greathouse IV_, Jun 07 2016
%o (PARI) th4(x)=1+2*sumalt(n=1, (-1)^n*x^n^2)
%o intnum(x=0, [oo, 1], (th4(exp(-x))^6-1)/sqrt(Pi*x)) \\ _Charles R Greathouse IV_, Jun 06 2016
%Y Cf. A059750, A088537, A085469, A090734.
%K nonn,cons,easy
%O 1,2
%A _Jean-François Alcover_, Sep 10 2014
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