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A329083 Decimal expansion of Sum_{k>=0} 1/(k^2+2).

%I A329083 #16 Jun 17 2023 03:15:35
%S A329083 1,3,6,1,0,2,8,1,0,0,5,7,3,7,2,7,9,2,2,8,2,1,3,3,2,1,5,8,5,1,8,2,3,4,
%T A329083 6,3,6,8,7,2,8,5,3,5,6,0,7,0,6,9,3,0,7,2,3,3,4,9,4,7,8,9,0,0,1,6,0,7,
%U A329083 8,2,1,1,4,6,3,6,5,5,4,4,4,5,7,3,7,6,1,5,1,4,7
%N A329083 Decimal expansion of Sum_{k>=0} 1/(k^2+2).
%C A329083 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 A329083 F(z) = (1 + sqrt(z)*Pi*coth(sqrt(z)*Pi))/(2z), z != 0, -1, -4, -9, -16, ...;
%C A329083 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 A329083 This sequence gives F(2).
%C A329083 This and A329090 are essentially the same, but both sequences are added because some people may search for this, and some people may search for A329090.
%F A329083 Equals (1 + (sqrt(2)*Pi)*coth(sqrt(2)*Pi))/4 = (1 + (sqrt(-2)*Pi)*cot(sqrt(-2)*Pi))/4.
%e A329083 1.36102810057372792282...
%t A329083 RealDigits[(1 + Sqrt[2]*Pi*Coth[Sqrt[2]*Pi])/4, 10, 120][[1]] (* _Amiram Eldar_, Jun 17 2023 *)
%o A329083 (PARI) default(realprecision, 100); my(F(x) = (1 + (sqrt(x)*Pi)/tanh(sqrt(x)*Pi))/(2*x)); F(2)
%o A329083 (PARI) sumnumrat(1/(x^2+2),0) \\ _Charles R Greathouse IV_, Jan 20 2022
%Y A329083 Cf. A329080 (F(-5)), A329081 (F(-3)), A329082 (F(-2)), A113319 (F(1)), this sequence (F(2)), A329084 (F(3)), A329085 (F(4)), A329086 (F(5)).
%Y A329083 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 A329083 nonn,cons
%O A329083 1,2
%A A329083 _Jianing Song_, Nov 04 2019