A390263 Decimal expansion of Sum_{k>=1} (-1)^(k+1) * (1/(4*k-3)^6 + 1/(4*k-1)^6).

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

Offset

1,5

Comments

Shamos shows a minus instead of a plus.

Sum_{k>=1} 1/(4*k-3)^6 + 1/(4*k-1)^6 = Pi^6/960 = A300709. - Vaclav Kotesovec, Oct 30 2025

Links

Table of n, a(n) for n=1..100.

Michael I. Shamos, Shamos's Catalog of the Real Numbers, 2011, p. 29.

Formula

Equals (1/31457280) * (PolyGamma(5, 1/8) + PolyGamma(5, 3/8) - PolyGamma(5, 5/8) - PolyGamma(5, 7/8)).

Equals (1/262144) * (zeta(6, 1/8) + zeta(6, 3/8) - zeta(6, 5/8) - zeta(6, 7/8)).

Equals Sum_{k>=1} (-1)^(k+1) * (1/A016813(k)^6 + 1/A004767(k)^6)

Example

1.00130144245443407196300635551705830033223670716771...

Mathematica

First[RealDigits[(1/31457280)*(PolyGamma[5, 1/8] + PolyGamma[5, 3/8] - PolyGamma[5, 5/8] - PolyGamma[5, 7/8]), 10, 100]]

Prog

(PARI) (zetahurwitz(6, 1/8) + zetahurwitz(6, 3/8) - zetahurwitz(6, 5/8) - zetahurwitz(6, 7/8)) / 262144 \\ Amiram Eldar, Oct 30 2025

Crossrefs

Cf. A004767, A016813, A300709.

Sequence in context: A091613 A039727 A137176 * A143949 A124323 A250104

Adjacent sequences: A390260 A390261 A390262 * A390264 A390265 A390266

Keyword

nonn,cons

Author

Jason Bard, Oct 30 2025

Status

approved