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

Author

Jianing Song, Nov 04 2019

Keywords

• nonn
• cons

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.

Examples

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

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)).

Programs

Mathematica

RealDigits[(1 + Sqrt[5]*Pi*Coth[Sqrt[5]*Pi])/10, 10, 120][[1]] (* Amiram Eldar, Jun 15 2023 *)

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

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.