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A329088 Decimal expansion of Sum_{k>=1} 1/(k^2-3).

%I A329088 #13 Jun 15 2023 02:30:30
%S A329088 9,7,6,6,5,0,1,8,9,9,8,6,0,9,3,6,1,7,1,0,5,8,4,9,0,5,5,1,4,1,7,1,6,2,
%T A329088 6,2,4,4,3,0,5,9,4,1,1,4,4,5,5,1,6,9,1,9,3,8,6,9,6,6,1,7,6,6,3,5,2,1,
%U A329088 6,5,1,8,2,9,1,7,2,9,3,7,0,2,5,9,4,8,0,4,5,2,1
%N A329088 Decimal expansion of Sum_{k>=1} 1/(k^2-3).
%C A329088 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 A329088 F(z) = (1 + sqrt(z)*Pi*coth(sqrt(z)*Pi))/(2z), z != 0, -1, -4, -9, -16, ...;
%C A329088 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 A329088 This sequence gives f(-3).
%F A329088 Sum_{k>=1} 1/(k^2-3) = (-1 + (sqrt(-3)*Pi)*coth(sqrt(-3)*Pi))/(-6) = (-1 + (sqrt(3)*Pi)*cot(sqrt(3)*Pi))/(-6).
%e A329088 Sum_{k>=1} 1/(k^2-3) = 0.97665018998609361710...
%t A329088 RealDigits[(1 - Sqrt[3]*Pi*Cot[Sqrt[3]*Pi])/6, 10, 120][[1]] (* _Amiram Eldar_, Jun 15 2023 *)
%o A329088 (PARI) default(realprecision, 100); my(f(x) = (-1 + (sqrt(x)*Pi)/tanh(sqrt(x)*Pi))/(2*x)); f(-3)
%o A329088 (PARI) sumnumrat(1/(x^2-3), 1) \\ _Charles R Greathouse IV_, Jan 20 2022
%Y A329088 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 A329088 Cf. A329087 (f(-5)), this sequence (f(-3)), A329089 (f(-2)), A013661 (f(0)), A259171 (f(1)), A329090 (f(2)), A329091 (f(3)), A329092 (f(4)), A329093 (f(5)).
%K A329088 nonn,cons
%O A329088 0,1
%A A329088 _Jianing Song_, Nov 04 2019