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%I #14 Jun 17 2023 03:13:57
%S 7,4,0,2,6,7,0,7,6,5,8,1,8,5,0,7,8,2,5,8,0,6,0,2,9,6,4,8,2,4,8,1,1,9,
%T 7,7,9,4,3,1,0,9,3,0,2,3,8,5,4,5,1,2,4,5,6,2,7,0,3,5,4,1,8,6,2,5,3,3,
%U 4,1,8,9,8,5,1,2,3,0,1,2,6,5,5,2,5,1,4,9,1,6,1
%N Decimal expansion of Sum_{k>=1} 1/(k^2+3).
%C In general, for complex numbers z, if we define F(z) = Sum_{k>=0} 1/(k^2+z), f(z) = Sum_{k>=1} 1/(k^2+z), then we have:
%C F(z) = (1 + sqrt(z)*Pi*coth(sqrt(z)*Pi))/(2z), z != 0, -1, -4, -9, -16, ...;
%C f(z) = (-1 + sqrt(z)*Pi*coth(sqrt(z)*Pi))/(2z), z != 0, -1, -4, -9, -16, ...; Pi^2/6, z = 0. Note that f(z) is continuous at z = 0.
%C This sequence gives f(3).
%F Equals (-1 + (sqrt(3)*Pi)*coth(sqrt(3)*Pi))/6 = (-1 + (sqrt(-3)*Pi)*cot(sqrt(-3)*Pi))/6.
%e Sum_{k>=1} 1/(k^2+3) = 0.74026707658185078258...
%t RealDigits[(-1 + Sqrt[3]*Pi*Coth[Sqrt[3]*Pi])/6, 10, 120][[1]] (* _Amiram Eldar_, Jun 17 2023 *)
%o (PARI) default(realprecision, 100); my(f(x) = (-1 + (sqrt(x)*Pi)/tanh(sqrt(x)*Pi))/(2*x)); f(3)
%o (PARI) sumnumrat(1/(x^2+3), 1) \\ _Charles R Greathouse IV_, Jan 20 2022
%Y Cf. A329080 (F(-5)), A329081 (F(-3)), A329082 (F(-2)), A113319 (F(1)), A329083 (F(2)), A329084 (F(3)), A329085 (F(4)), A329086 (F(5)).
%Y Cf. A329087 (f(-5)), A329088 (f(-3)), A329089 (f(-2)), A013661 (f(0)), A259171 (f(1)), A329090 (f(2)), this sequence (f(3)), A329092 (f(4)), A329093 (f(5)).
%K nonn,cons
%O 0,1
%A _Jianing Song_, Nov 04 2019
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