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Decimal expansion of Sum_{k>=0} (zeta(3*k+2)-1).
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%I #27 Jan 07 2024 01:49:56

%S 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 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 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 Decimal expansion of Sum_{k>=0} (zeta(3*k+2)-1).

%F Equals Sum_{k>=1} 1/(k^3 + 1).

%F 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 Equals -1/3 + gamma/3 - (1/3)*A339135 + 2*log(2)/9 + sqrt(3)*Pi*tanh(sqrt(3)*Pi/2)/6.

%F Equals 1 - A339605 - A339604.

%F Equals 1/2 + Sum_{k>=1} (-1)^(k+1) * (zeta(3*k)-1). - _Amiram Eldar_, Jan 07 2024

%e 0.6865033423386238859646...

%p evalf(Re(sum(1/(k^3+1), k=1..infinity)), 120); # _Alois P. Heinz_, Dec 12 2020

%t RealDigits[Chop[N[Sum[Zeta[3 n + 2] - 1, {n, 0, Infinity}], 105]]][[1]]

%o (PARI) suminf(k=0, zeta(3*k+2)-1) \\ _Michel Marcus_, Dec 09 2020

%Y Cf. A024006, A256919, A338815, A339135, A339529, A339530, A339604, A339605.

%K nonn,cons

%O 0,1

%A _Artur Jasinski_, Dec 09 2020