A247040 Decimal expansion of M_6, the 6th Madelung constant.

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, 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, 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

Offset: 1

Author

Jean-François Alcover, Sep 10 2014

Keywords

• nonn
• cons
• easy

Examples

			-1.9655570390090782813123135557351853678689767284464645117...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 77.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• Eric Weisstein's MathWorld, Dirichlet Beta Function
• Eric Weisstein's MathWorld, Madelung Constants

Crossrefs

Cf. A059750, A088537, A085469, A090734.

Programs

Mathematica

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]]

PARI

th4(x)=1+2*sumalt(n=1, (-1)^n*x^n^2)
intnum(x=0, [oo, 1], (th4(exp(-x))^6-1)/sqrt(Pi*x)) \\ Charles R Greathouse IV, Jun 07 2016

PARI

th4(x)=1+2*sumalt(n=1, (-1)^n*x^n^2)
intnum(x=0, [oo, 1], (th4(exp(-x))^6-1)/sqrt(Pi*x)) \\ Charles R Greathouse IV, Jun 06 2016

Formula

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.