A329092 Decimal expansion of Sum_{k>=1} 1/(k^2+4).

6, 6, 0, 4, 0, 3, 6, 4, 1, 3, 2, 1, 1, 1, 5, 1, 1, 4, 1, 9, 3, 0, 4, 3, 8, 2, 4, 9, 2, 6, 4, 4, 3, 6, 0, 9, 6, 1, 1, 6, 9, 5, 0, 6, 5, 7, 9, 4, 6, 5, 0, 4, 4, 8, 9, 0, 2, 5, 8, 5, 8, 8, 0, 4, 5, 3, 5, 8, 0, 8, 3, 1, 1, 4, 9, 4, 5, 5, 2, 0, 6, 2, 5, 2, 8, 4, 5, 3, 1, 7, 8

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(4).

This and A329085 are essentially the same, but both sequences are added because some people may search for this, and some people may search for A329085.

Links

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

Formula

Equals (-1 + (2*Pi)*coth(2*Pi))/8 = (-1 + (2*Pi*i)*cot(2*Pi*i))/8, i = sqrt(-1).

Equals Integral_{x=0..oo} sin(x)*cos(x)/(exp(x) - 1) dx. - Amiram Eldar, Aug 16 2020

Example

Sum_{k>=1} 1/(k^2+4) = 0.66040364132111511419...

Mathematica

RealDigits[(-1 + 2*Pi*Coth[2*Pi])/8, 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(4)
(PARI) sumnumrat(1/(x^2+4), 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)), A329090 (f(2)), A329091 (f(3)), this sequence (f(4)), A329093 (f(5)).

Sequence in context: A281056 A273989 A197013 * A081825 A272648 A212708

Adjacent sequences: A329089 A329090 A329091 * A329093 A329094 A329095

Keyword

nonn,cons

Author

Jianing Song, Nov 04 2019

Status

approved