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

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.

Links

Table of n, a(n) for n=1..91.

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

Adjacent sequences: A329080 A329081 A329082 * A329084 A329085 A329086

Keyword

nonn,cons

Author

Jianing Song, Nov 04 2019

Status

approved