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A329082 Decimal expansion of Sum_{k>=0} 1/(k^2-2), negated.

%I A329082 #16 Jun 17 2023 03:16:01
%S A329082 5,5,6,8,1,0,4,0,7,7,0,0,6,2,0,0,8,2,5,5,2,9,8,1,6,0,9,1,1,2,5,9,7,3,
%T A329082 4,7,0,9,8,7,0,9,2,7,0,2,5,7,0,4,0,8,7,8,5,5,1,0,0,1,9,8,3,4,8,6,3,2,
%U A329082 8,1,0,3,7,4,4,1,5,7,0,0,2,4,6,1,7,4,5,6,5,7,7
%N A329082 Decimal expansion of Sum_{k>=0} 1/(k^2-2), negated.
%C A329082 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 A329082 F(z) = (1 + sqrt(z)*Pi*coth(sqrt(z)*Pi))/(2z), z != 0, -1, -4, -9, -16, ...;
%C A329082 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 A329082 This sequence gives F(-2) (negated).
%C A329082 This and A329089 are essentially the same, but both sequences are added because some people may search for this, and some people may search for A329089.
%F A329082 Equals (1 + (sqrt(-2)*Pi)*coth(sqrt(-2)*Pi))/(-4) = (1 + (sqrt(2)*Pi)*cot(sqrt(2)*Pi))/(-4).
%e A329082 -0.55681040770062008255...
%t A329082 RealDigits[(1 + Sqrt[2]*Pi*Cot[Sqrt[2]*Pi])/4, 10, 120][[1]] (* _Amiram Eldar_, Jun 17 2023 *)
%o A329082 (PARI) default(realprecision, 100); my(F(x) = (1 + (sqrt(x)*Pi)/tanh(sqrt(x)*Pi))/(2*x)); F(-2)
%o A329082 (PARI) sumnumrat(1/(x^2-2), 0) \\ _Charles R Greathouse IV_, Jan 20 2022
%Y A329082 Cf. A329080 (F(-5)), A329081 (F(-3)), this sequence (F(-2)), A113319 (F(1)), A329083 (F(2)), A329084 (F(3)), A329085 (F(4)), A329086 (F(5)).
%Y A329082 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 A329082 nonn,cons
%O A329082 0,1
%A A329082 _Jianing Song_, Nov 04 2019