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A329086 Decimal expansion of Sum_{k>=0} 1/(k^2+5).

%I A329086 #14 Jun 15 2023 02:29:20
%S A329086 8,0,2,4,8,2,5,8,4,8,0,6,7,8,6,8,8,6,8,3,5,8,4,4,9,5,4,4,8,6,5,5,7,7,
%T A329086 0,9,4,0,7,1,6,0,7,2,9,7,9,0,5,7,0,1,3,6,4,1,9,8,5,9,5,9,3,9,6,0,9,4,
%U A329086 0,1,4,9,5,4,0,5,3,4,0,8,0,4,5,5,2,9,1,0,9,3,9
%N A329086 Decimal expansion of Sum_{k>=0} 1/(k^2+5).
%C A329086 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 A329086 F(z) = (1 + sqrt(z)*Pi*coth(sqrt(z)*Pi))/(2z), z != 0, -1, -4, -9, -16, ...;
%C A329086 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 A329086 This sequence gives F(5).
%C A329086 This and A329093 are essentially the same, but both sequences are added because some people may search for this, and some people may search for A329093.
%F A329086 Sum_{k>=0} 1/(k^2+5) = (1 + (sqrt(5)*Pi)*coth(sqrt(5)*Pi))/10 = (1 + (sqrt(-5)*Pi)*cot(sqrt(-5)*Pi))/10.
%e A329086 Sum_{k>=0} 1/(k^2+5) = 0.80248258480678688683...
%t A329086 RealDigits[(1 + Sqrt[5]*Pi*Coth[Sqrt[5]*Pi])/10, 10, 120][[1]] (* _Amiram Eldar_, Jun 15 2023 *)
%o A329086 (PARI) default(realprecision, 100); my(F(x) = (1 + (sqrt(x)*Pi)/tanh(sqrt(x)*Pi))/(2*x)); F(5)
%o A329086 (PARI) sumnumrat(1/(x^2+5), 0) \\ _Charles R Greathouse IV_, Jan 20 2022
%Y A329086 Cf. A329080 (F(-5)), A329081 (F(-3)), A329082 (F(-2)), A113319 (F(1)), A329083 (F(2)), A329084 (F(3)), A329085 (F(4)), this sequence (F(5)).
%Y A329086 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 A329086 nonn,cons
%O A329086 0,1
%A A329086 _Jianing Song_, Nov 04 2019