A264157 Decimal expansion of M_7, the 7-dimensional analog of Madelung's constant (negated).
2, 0, 1, 2, 4, 0, 5, 9, 8, 9, 7, 9, 7, 9, 8, 6, 0, 6, 4, 3, 9, 5, 0, 3, 0, 6, 3, 5, 8, 0, 4, 3, 0, 0, 4, 4, 1, 6, 5, 6, 7, 8, 0, 6, 5, 8, 1, 2, 1, 9, 2, 9, 3, 2, 8, 7, 8, 4, 9, 0, 4, 6, 9, 1, 1, 7, 3
Offset: 1
Examples
-2.01240598979798606439503063580430044165678065812192932878490469117330...
References
- Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 77.
Links
- Eric Weisstein's World of Mathematics, Madelung Constants.
Programs
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Mathematica
digits = 32; f[n_, x_] := 1/Sqrt[Pi*x]*(EllipticTheta[4, 0, Exp[-x]]^n - 1); M[7] = NIntegrate[f[7, x], {x, 0, Infinity}, WorkingPrecision -> digits + 5]; RealDigits[M[7], 10, digits] // First -
PARI
th4(x)=1+2*sumalt(n=1,(-1)^n*x^n^2) intnum(x=0,[oo,1], (th4(exp(-x))^7-1)/sqrt(Pi*x)) \\ Charles R Greathouse IV, Jun 06 2016
Formula
Equals (1/sqrt(Pi))*Integral_{t=0..oo} ((Sum_{k=-oo..oo} (-1)^k*exp(-k^2*t))^7-1)/sqrt(t) dt.
Extensions
More terms from Charles R Greathouse IV, Jun 06 2016