A086722 Decimal expansion of g(1)+g(2)-g(4)-g(5), where g(k) = Sum_{m>=0} (1/(6*m+k)^2).

1, 1, 7, 1, 9, 5, 3, 6, 1, 9, 3, 4, 4, 7, 2, 9, 4, 4, 5, 3, 0, 0, 7, 8, 1, 1, 4, 4, 4, 3, 6, 1, 3, 8, 5, 3, 4, 5, 4, 7, 7, 0, 1, 5, 0, 4, 8, 1, 7, 9, 2, 8, 1, 3, 0, 3, 3, 3, 1, 5, 0, 0, 9, 4, 4, 5, 0, 3, 3, 0, 4, 7, 6, 9, 7, 7, 4, 2, 7, 3, 2, 7, 1, 3, 9, 3, 0, 4, 3, 5, 6, 2, 4, 8, 3, 1, 4, 7, 9

Offset: 1

Author

N. J. A. Sloane, Jul 31 2003

Keywords

• nonn
• cons

Comments

By summing over g(1)-g(5) and g(2)-g(4) separately we obtain A214552 for the first difference and a quarter of A086724 for the second difference. - R. J. Mathar, Jul 20 2012

2/3 times this constant equals A086724 [Bailey, Borwein and Crandall, 2006] - R. J. Mathar, Jul 20 2012

Examples

			1.1719536193447294453... = A214552 + A086724/4 = 1/1^2 +1/2^2 -1/4^2 -1/5^2 +1/7^2 +1/8^2 -1/10^2 -1/11^2 ++--....
		

References

• L. Fejes Tóth, Lagerungen in der Ebene, auf der Kugel und im Raum, 2nd. ed., Springer-Verlag, Berlin, Heidelberg 1972; see p. 213.

Links

• David H. Bailey, Jonathan M. Borwein, and Richard E. Crandall, Integrals of the Ising class, Journal of Physics A: Mathematical and General, Vol. 39, No. 40 (2006), 12271; alternative link.

Crossrefs

Cf. A086723-A086731, A214552.

Programs

Mathematica

g[k_] := PolyGamma[1, k/6]/36; RealDigits[g[1] + g[2] - g[4] - g[5], 10, 99] // First (* Jean-François Alcover, Feb 12 2013 *)

Formula

Equals -Integral_{x=0..1} log(x)/(x^2-x+1) dx. - Jean-François Alcover, Aug 29 2014

Equals Integral_{x>=0} x/(exp(x) + exp(-x) - 1) dx. - Amiram Eldar, May 22 2023