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%I #12 May 20 2022 06:34:23
%S 1,5,1,7,1,0,0,7,3,4,0,3,3,2,1,6,4,2,6,1,5,2,9,0,7,6,4,4,9,0,2,4,1,3,
%T 8,5,8,0,6,2,2,1,1,3,2,2,5,2,9,8,4,4,6,7,2,8,4,7,6,3,4,8,9,9,0,3,7,9,
%U 0,1,3,5,0,5,3,5,7,9,8,7,2,0,0,7,8,4,3,6,9,3,6,9,3,3,0,0,6,4,3,7,0,6,6,6,4
%N Decimal expansion of Sum_{k>=0} 1 / (k^6 + 1).
%F Equals 1/2 + (coth(Pi) + (sinh(Pi) + sqrt(3)*sin(sqrt(3)*Pi)) / (cosh(Pi) - cos(sqrt(3)*Pi)))*Pi/6.
%F Equal 3/2 + Sum_{k>=1} (-1)^(k+1) * (zeta(6*k)-1). - _Amiram Eldar_, May 20 2022
%e 1.517100734033216426152907644902413858062211322529844672847634899037901...
%p evalf(1/2 + (coth(Pi) + (sinh(Pi) + sqrt(3)*sin(sqrt(3)*Pi)) / (cosh(Pi) - cos(sqrt(3)*Pi)))*Pi/6, 100);
%t RealDigits[Chop[N[Sum[1/(k^6 + 1), {k, 0, Infinity}], 105]]][[1]]
%o (PARI) sumpos(k=0, 1/(k^6 + 1))
%Y Cf. A002604, A113319, A354051, A354053.
%K nonn,cons
%O 1,2
%A _Vaclav Kotesovec_, May 16 2022
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