A329088 Decimal expansion of Sum_{k>=1} 1/(k^2-3).

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

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(-3).

Examples

			Sum_{k>=1} 1/(k^2-3) = 0.97665018998609361710...
		

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)), this sequence (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[3]*Pi*Cot[Sqrt[3]*Pi])/6, 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(-3)

PARI

sumnumrat(1/(x^2-3), 1) \\ Charles R Greathouse IV, Jan 20 2022

Formula

Sum_{k>=1} 1/(k^2-3) = (-1 + (sqrt(-3)*Pi)*coth(sqrt(-3)*Pi))/(-6) = (-1 + (sqrt(3)*Pi)*cot(sqrt(3)*Pi))/(-6).