login
A329083
Decimal expansion of Sum_{k>=0} 1/(k^2+2).
14
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, 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, 8, 2, 1, 1, 4, 6, 3, 6, 5, 5, 4, 4, 4, 5, 7, 3, 7, 6, 1, 5, 1, 4, 7
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).
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.
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
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)).
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)).
Sequence in context: A009036 A021281 A034004 * A188882 A016660 A142464
KEYWORD
nonn,cons
AUTHOR
Jianing Song, Nov 04 2019
STATUS
approved