A294967 - OEIS

%I #29 Feb 16 2025 08:33:51

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

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

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

%N Decimal expansion of (1/9)*Hurwitz Zeta(2, 2/3) = (1/9)*Psi(1, 2/3), with the Polygamma function Psi.

%C This is the value of the series Sum_{n=0..infinity} 1/(3*n+2)^2. For (3*n+2)^2 see A016790.

%C For the partial sums see A173986(n+1)/A173987(n+1), n >= 0.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HurwitzZetaFunction.html">Hurwitz Zeta Function</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolygammaFunction.html">Polygamma Function</a>.

%F From _Amiram Eldar_, Oct 02 2020: (Start)

%F Equals Integral_{1..oo} log(x)/(x^3-1) dx.

%F Equals 4*Pi^2/27 - A214550. (End)

%e 0.340430601039857489998590803697298350359188343374823262215864737125448716...

%t RealDigits[4*Pi^2/27 - PolyGamma[1, 1/3]/9, 10, 126][[1]] (* _Amiram Eldar_, Oct 02 2020 *)

%t RealDigits[HurwitzZeta[2, 2/3]/9, 10, 150][[1]] (* _Vaclav Kotesovec_, Oct 02 2020 *)

%o (PARI) zetahurwitz(2,2/3)/9 \\ _Charles R Greathouse IV_, Jan 30 2018

%Y Cf. A016790, A173986/A173987, A214550.

%K nonn,cons,changed

%O 0,1

%A _Wolfdieter Lang_, Nov 12 2017

%E Data corrected by _Amiram Eldar_, Oct 02 2020