%I #15 Jun 17 2023 03:15:49
%S 6,4,3,3,1,6,8,5,6,6,5,2,7,6,0,2,8,3,7,7,2,5,1,5,7,2,1,8,0,8,3,8,2,9,
%T 2,9,1,0,9,7,2,6,0,7,8,1,1,2,1,8,3,5,8,6,0,5,3,6,3,2,8,4,3,3,0,1,8,8,
%U 3,1,8,4,9,5,8,3,9,6,0,3,6,9,2,6,1,4,7,1,1,8,8
%N Decimal expansion of Sum_{k>=0} 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 0.64331685665276028377...
%t RealDigits[(1 + Sqrt[3]*Pi*Cot[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), 0) \\ _Charles R Greathouse IV_, Jan 20 2022
%Y Cf. A329080 (F(-5)), this sequence (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)), A329091 (f(3)), A329092 (f(4)), A329093 (f(5)).
%K nonn,cons
%O 0,1
%A _Jianing Song_, Nov 04 2019