A329086 Decimal expansion of Sum_{k>=0} 1/(k^2+5).

8, 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 A329093 are essentially the same, but both sequences are added because some people may search for this, and some people may search for A329093.

Links

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

Formula

Sum_{k>=0} 1/(k^2+5) = (1 + (sqrt(5)*Pi)*coth(sqrt(5)*Pi))/10 = (1 + (sqrt(-5)*Pi)*cot(sqrt(-5)*Pi))/10.

Example

Sum_{k>=0} 1/(k^2+5) = 0.80248258480678688683...

Mathematica

RealDigits[(1 + Sqrt[5]*Pi*Coth[Sqrt[5]*Pi])/10, 10, 120][[1]] (* Amiram Eldar, Jun 15 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), 0) \\ 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)), this sequence (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: A232227 A322231 A261168 * A342200 A265294 A062522

Adjacent sequences: A329083 A329084 A329085 * A329087 A329088 A329089

Keyword

nonn,cons

Author

Jianing Song, Nov 04 2019

Status

approved