A329093 Decimal expansion of Sum_{k>=1} 1/(k^2+5).

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

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).

This and A329086 are essentially the same, but both sequences are added because some people may search for this, and some people may search for A329086.

Links

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

Formula

Equals (-1 + (sqrt(5)*Pi)*coth(sqrt(5)*Pi))/10.

Equals (-1 + (sqrt(-5)*Pi)*cot(sqrt(-5)*Pi))/10.

Example

Sum_{k>=1} 1/(k^2+5) = 0.60248258480678688683...

Mathematica

RealDigits[(-1 + Sqrt[5]*Pi*Coth[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), 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)), A329092 (f(4)), this sequence (f(5)).

Sequence in context: A327837 A261166 A021170 * A195406 A021628 A371923

Adjacent sequences: A329090 A329091 A329092 * A329094 A329095 A329096

Keyword

nonn,cons

Author

Jianing Song, Nov 04 2019

Status

approved