A387206 Decimal expansion of Sum_{k>=1} (-1)^(k+1)/(6*k-5)^6 + (-1)^(k+1)/(6*k-1)^6.

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

Offset: 1

Author

Jason Bard, Aug 21 2025

Keywords

• nonn
• cons
• new

Examples

			1.000055160794869463473756618844925625878197693652065...
		

Links

• Jason Bard, Table of n, a(n) for n = 1..1000
• Michael I. Shamos, A catalog of the real numbers, (2007). See p. 21. (Misprint contains erroneous minus sign)

Programs

Maple

c:= Re(sum((-1)^(k+1)/(6*k-5)^6+(-1)^(k+1)/(6*k-1)^6, k=1..infinity)):
evalf(c, 140);  # Alois P. Heinz, Aug 21 2025

Mathematica

RealDigits[(1/358318080)*(PolyGamma[5, 1/12] + PolyGamma[5, 5/12] - PolyGamma[5, 7/12] - PolyGamma[5, 11/12]), 10, 100][[1]]

PARI

(365/1492992) * (zetahurwitz(6, 1/4) - zetahurwitz(6, 3/4))

Formula

Equals (1/358318080) * (PolyGamma(5, 1/12) + PolyGamma(5, 5/12) - PolyGamma(5, 7/12) - PolyGamma(5, 11/12)).

Equals (73/35831808) * (PolyGamma(5, 1/4) - PolyGamma(5, 3/4)). - Amiram Eldar, Aug 22 2025