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

1,2

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

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.

Example

-1.9655570390090782813123135557351853678689767284464645117...

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]]

Prog

(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

Crossrefs

Cf. A059750, A088537, A085469, A090734.

Sequence in context: A347330 A343947 A367709 * A019942 A102047 A144665

Adjacent sequences: A247037 A247038 A247039 * A247041 A247042 A247043

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Sep 10 2014

Status

approved