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A329084 Decimal expansion of Sum_{k>=0} 1/(k^2+3).

%I A329084 #18 Jun 17 2023 03:14:33
%S A329084 1,0,7,3,6,0,0,4,0,9,9,1,5,1,8,4,1,1,5,9,1,3,9,3,6,2,9,8,1,5,8,1,4,5,
%T A329084 3,1,1,2,7,6,4,4,2,6,3,5,7,1,8,7,8,4,5,7,8,9,6,0,3,6,8,7,5,1,9,5,8,6,
%U A329084 6,7,5,2,3,1,8,4,5,6,3,4,5,9,8,8,5,8,4,8,2,4,9
%N A329084 Decimal expansion of Sum_{k>=0} 1/(k^2+3).
%C A329084 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 A329084 F(z) = (1 + sqrt(z)*Pi*coth(sqrt(z)*Pi))/(2z), z != 0, -1, -4, -9, -16, ...;
%C A329084 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 A329084 This sequence gives F(3).
%F A329084 Equals (1 + (sqrt(3)*Pi)*coth(sqrt(3)*Pi))/6 = (1 + (sqrt(-3)*Pi)*cot(sqrt(-3)*Pi))/6.
%e A329084 1.07360040991518411591...
%t A329084 RealDigits[Sum[1/(k^2+3),{k,0,\[Infinity]}],10,120][[1]] (* _Harvey P. Dale_, Jul 05 2021 *)
%t A329084 RealDigits[(1 + Sqrt[3]*Pi*Coth[Sqrt[3]*Pi])/6, 10, 120][[1]] (* _Amiram Eldar_, Jun 17 2023 *)
%o A329084 (PARI) default(realprecision, 100); my(F(x) = (1 + (sqrt(x)*Pi)/tanh(sqrt(x)*Pi))/(2*x)); F(3)
%o A329084 (PARI) sumnumrat(1/(x^2+3), 0) \\ _Charles R Greathouse IV_, Jan 20 2022
%Y A329084 Cf. A329080 (F(-5)), A329081 (F(-3)), A329082 (F(-2)), A113319 (F(1)), A329083 (F(2)), this sequence (F(3)), A329085 (F(4)), A329086 (F(5)).
%Y A329084 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 A329084 nonn,cons
%O A329084 1,3
%A A329084 _Jianing Song_, Nov 04 2019