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%I #26 Jan 15 2021 21:21:29
%S 0,9,1,8,0,7,2,6,2,5,5,2,1,0,9,0,7,5,6,4,3,2,7,6,3,6,6,6,3,0,3,8,0,0,
%T 9,5,2,2,2,5,2,5,9,0,2,5,9,9,2,3,3,4,3,3,1,3,6,2,3,7,3,0,9,8,1,3,6,2,
%U 2,8,2,5,1,1,1,4,0,9,9,3,2,3,8,0,4,2,1,9,9,3,8,7,4,8,0,6,5,8,1,5,0,3,1,1,4,8
%N Decimal expansion of Sum_{k>=1} (zeta(3*k+1) - 1).
%F Equals Sum_{k>=2} (k^5 - 3*k^4 + k^3 - k^2 + k - 1)/(k*(k^6 - 1)).
%F Equals 1/3 - 2*gamma/3 - (2/3)*Re(Psi(1/2 + i*sqrt(3)/2)), where Psi is the digamma function, gamma is the Euler-Mascheroni constant (see A001620), and i=sqrt(-1).
%F Equals 1/3 - 2*gamma/3 + 2*A339135/3 - 4*log(2)/9.
%F Equals 1 - A339604 - A339606.
%F Equals Sum_{k>=2} 1/(k*(k^3 - 1)). - _Vaclav Kotesovec_, Dec 24 2020
%e 0.09180726255210907564327636663...
%t Join[{0}, RealDigits[Chop[N[Sum[Zeta[3 n + 1] - 1, {n, 1, Infinity}], 105]]][[1]]]
%o (PARI) suminf(k=1, zeta(3*k+1)-1) \\ _Michel Marcus_, Dec 09 2020
%Y Cf. A256919, A338815, A339135, A339529, A339530, A339604, A339606.
%K nonn,cons
%O 0,2
%A _Artur Jasinski_, Dec 09 2020
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