This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A339606 #27 Jan 07 2024 01:49:56
%S A339606 6,8,6,5,0,3,3,4,2,3,3,8,6,2,3,8,8,5,9,6,4,6,0,5,2,1,2,1,8,6,8,5,4,7,
%T A339606 5,2,1,8,2,2,3,2,6,9,9,2,1,9,6,3,6,1,8,8,4,5,8,6,3,4,4,1,4,9,2,8,8,5,
%U A339606 6,1,4,9,9,4,5,9,7,4,1,3,1,9,4,2,1,8,2,5,6,1,1,8,2,1,2,0,7,1,4,0,3,6,3,9,9
%N A339606 Decimal expansion of Sum_{k>=0} (zeta(3*k+2)-1).
%F A339606 Equals Sum_{k>=1} 1/(k^3 + 1).
%F A339606 Equals -1/3 + gamma/3 + (1/3)*Re(Psi(1/2 + i*sqrt(3)/2)) + sqrt(3)*Pi*tanh(sqrt(3)*Pi/2)/6, where Psi is digamma function, gamma is Euler-Mascheroni constant (see A001620), and i=sqrt(-1).
%F A339606 Equals -1/3 + gamma/3 - (1/3)*A339135 + 2*log(2)/9 + sqrt(3)*Pi*tanh(sqrt(3)*Pi/2)/6.
%F A339606 Equals 1 - A339605 - A339604.
%F A339606 Equals 1/2 + Sum_{k>=1} (-1)^(k+1) * (zeta(3*k)-1). - _Amiram Eldar_, Jan 07 2024
%e A339606 0.6865033423386238859646...
%p A339606 evalf(Re(sum(1/(k^3+1), k=1..infinity)), 120); # _Alois P. Heinz_, Dec 12 2020
%t A339606 RealDigits[Chop[N[Sum[Zeta[3 n + 2] - 1, {n, 0, Infinity}], 105]]][[1]]
%o A339606 (PARI) suminf(k=0, zeta(3*k+2)-1) \\ _Michel Marcus_, Dec 09 2020
%Y A339606 Cf. A024006, A256919, A338815, A339135, A339529, A339530, A339604, A339605.
%K A339606 nonn,cons
%O A339606 0,1
%A A339606 _Artur Jasinski_, Dec 09 2020