A329090 Decimal expansion of Sum_{k>=1} 1/(k^2+2).

8, 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, 1

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(2).

This and A329083 are essentially the same, but both sequences are added because some people may search for this, and some people may search for A329083.

Links

Table of n, a(n) for n=0..90.

Formula

Equals (-1 + (sqrt(2)*Pi)*coth(sqrt(2)*Pi))/4.

Equals (-1 + (sqrt(-2)*Pi)*cot(sqrt(-2)*Pi))/4.

Example

Sum_{k>=1} 1/(k^2+2) = 0.86102810057372792282...

Mathematica

RealDigits[(-1 + Sqrt[2]*Pi*Coth[Sqrt[2]*Pi])/4, 10, 120][[1]] (* Amiram Eldar, Jun 18 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), 1) \\ Charles R Greathouse IV, Jan 20 2022

Crossrefs

Cf. A329080 (F(-5)), A329081 (F(-3)), A329082 (F(-2)), A113319 (F(1)), A329083 (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)), this sequence (f(2)), A329091 (f(3)), A329092 (f(4)), A329093 (f(5)).

Sequence in context: A240805 A010115 A268438 * A197759 A199070 A372268

Adjacent sequences: A329087 A329088 A329089 * A329091 A329092 A329093

Keyword

nonn,cons

Author

Jianing Song, Nov 04 2019

Status

approved