OFFSET
1,2
COMMENTS
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:
F(z) = (1 + sqrt(z)*Pi*coth(sqrt(z)*Pi))/(2z), z != 0, -1, -4, -9, -16, ...;
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.
This sequence gives F(2).
FORMULA
Equals (1 + (sqrt(2)*Pi)*coth(sqrt(2)*Pi))/4 = (1 + (sqrt(-2)*Pi)*cot(sqrt(-2)*Pi))/4.
EXAMPLE
1.36102810057372792282...
MATHEMATICA
RealDigits[(1 + Sqrt[2]*Pi*Coth[Sqrt[2]*Pi])/4, 10, 120][[1]] (* Amiram Eldar, Jun 17 2023 *)
PROG
(PARI) default(realprecision, 100); my(F(x) = (1 + (sqrt(x)*Pi)/tanh(sqrt(x)*Pi))/(2*x)); F(2)
(PARI) sumnumrat(1/(x^2+2), 0) \\ Charles R Greathouse IV, Jan 20 2022
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jianing Song, Nov 04 2019
STATUS
approved