OFFSET
0,1
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(-5) (negated).
FORMULA
Equals (1 + (sqrt(-5)*Pi)*coth(sqrt(-5)*Pi))/(-10).
Equals (1 + (sqrt(5)*Pi)*cot(sqrt(5)*Pi))/(-10).
EXAMPLE
-0.86683259566274485298...
MATHEMATICA
RealDigits[(1 + Sqrt[5]*Pi*Cot[Sqrt[5]*Pi])/10, 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(-5)
(PARI) sumnumrat(1/(x^2-5), 0) \\ Charles R Greathouse IV, Jan 20 2022
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jianing Song, Nov 04 2019
STATUS
approved