A086723 -id:A086723 - cp's OEIS frontend

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

N. J. A. Sloane, Jul 31 2003

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

			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 ++--....

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

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.

Cf. A086723-A086731, A214552.

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 *)

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

A086729 Decimal expansion of Pi^2/72.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jul 31 2003

The original name was: Decimal expansion of Sum_{m=0..infinity} 1/(6*m+3)^2.

			0.1370778389040188697...

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

Index entries for transcendental numbers

Cf. A086722, A086723, A086724, A086725, A086726, A086727, A086728, A086730, A086731, A111003, A353908.

evalf(Pi^2/72) ;  # R. J. Mathar, Dec 18 2010

PARI
```
Pi^2/72 \\ Omar E. Pol, May 12 2022
```

Equals A111003/9. - R. J. Mathar, Dec 18 2010
From Amiram Eldar, Jul 19 2020: (Start)
Sum_{k>=0} (1/(12*k+3)^2 + 1/(12*k+9)^2).
Equals Integral_{x=1..oo} log(1 + 1/x^6)/x dx. (End)
Equals A353908/2. - Omar E. Pol, May 12 2022

New name after R. J. Mathar's Maple program. - Omar E. Pol, May 12 2022

cp's OEIS Frontend

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

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