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A329080 Decimal expansion of Sum_{k>=0} 1/(k^2-5), negated.

%I A329080 #21 Jun 17 2023 03:24:43
%S A329080 8,6,6,8,3,2,5,9,5,6,6,2,7,4,4,8,5,2,9,8,2,9,6,3,3,3,9,7,6,6,9,6,8,1,
%T A329080 5,7,5,4,3,4,3,2,5,6,6,2,3,8,0,3,9,6,4,0,4,0,5,8,3,3,4,5,8,2,7,1,4,8,
%U A329080 6,8,3,3,7,2,8,9,9,0,6,0,3,4,3,6,8,6,0,4,9,2,1
%N A329080 Decimal expansion of Sum_{k>=0} 1/(k^2-5), negated.
%C A329080 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 A329080 F(z) = (1 + sqrt(z)*Pi*coth(sqrt(z)*Pi))/(2z), z != 0, -1, -4, -9, -16, ...;
%C A329080 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 A329080 This sequence gives F(-5) (negated).
%C A329080 This and A329087 are essentially the same, but both sequences are added because some people may search for this, and some people may search for A329087.
%F A329080 Equals (1 + (sqrt(-5)*Pi)*coth(sqrt(-5)*Pi))/(-10).
%F A329080 Equals (1 + (sqrt(5)*Pi)*cot(sqrt(5)*Pi))/(-10).
%e A329080 -0.86683259566274485298...
%t A329080 RealDigits[(1 + Sqrt[5]*Pi*Cot[Sqrt[5]*Pi])/10, 10, 120][[1]] (* _Amiram Eldar_, Jun 17 2023 *)
%o A329080 (PARI) default(realprecision, 100); my(F(x) = (1 + (sqrt(x)*Pi)/tanh(sqrt(x)*Pi))/(2*x)); F(-5)
%o A329080 (PARI) sumnumrat(1/(x^2-5), 0) \\ _Charles R Greathouse IV_, Jan 20 2022
%Y A329080 Cf. this sequence (F(-5)), A329081 (F(-3)), A329082 (F(-2)), A113319 (F(1)), A329083 (F(2)), A329084 (F(3)), A329085 (F(4)), A329086 (F(5)).
%Y A329080 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 A329080 nonn,cons
%O A329080 0,1
%A A329080 _Jianing Song_, Nov 04 2019