A387638 Decimal expansion of Sum_{n>=0} (-1)^n/((4*n+1)^2*(4*n+3)^2).

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

Offset

0,4

Comments

Sum_{n>=0} 1/((4*n+1)^2*(4*n+3)^2) = Pi*(Pi-2)/32. - Vaclav Kotesovec, Oct 07 2025

References

A. P. Prudnikov, Yu. A. Brychkov, and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", New York, Gordon and Breach Science Publishers, 1986-1992 p. 678 eq. 54.

Links

Table of n, a(n) for n=0..104.

Formula

Equals (zeta(2,1/8)+zeta(2,3/8)-zeta(2,5/8)-zeta(2,7/8))/256-log(1+sqrt(2))/(4*sqrt(2)), where zeta(s,a) is the Hurwitz zeta function.

Example

0.1103772327258182142497018428...

Mathematica

RealDigits[-Log[1 + Sqrt[2]]/(4 Sqrt[2]) + (PolyGamma[1, 1/8] + PolyGamma[1, 3/8] - PolyGamma[1, 5/8] - PolyGamma[1, 7/8])/256, 10, 105][[1]]

Prog

(PARI) sumalt(k=0, (-1)^k/((4*k+1)^2*(4*k+3)^2))

Crossrefs

Cf. A309710, A328895.

Sequence in context: A131707 A348722 A388320 * A388373 A349614 A349613

Adjacent sequences: A387635 A387636 A387637 * A387639 A387640 A387641

Keyword

nonn,cons

Author

Artur Jasinski, Oct 06 2025

Status

approved