A329091 Decimal expansion of Sum_{k>=1} 1/(k^2+3).

7, 4, 0, 2, 6, 7, 0, 7, 6, 5, 8, 1, 8, 5, 0, 7, 8, 2, 5, 8, 0, 6, 0, 2, 9, 6, 4, 8, 2, 4, 8, 1, 1, 9, 7, 7, 9, 4, 3, 1, 0, 9, 3, 0, 2, 3, 8, 5, 4, 5, 1, 2, 4, 5, 6, 2, 7, 0, 3, 5, 4, 1, 8, 6, 2, 5, 3, 3, 4, 1, 8, 9, 8, 5, 1, 2, 3, 0, 1, 2, 6, 5, 5, 2, 5, 1, 4, 9, 1, 6, 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) = 0.74026707658185078258...
		

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)), this sequence (f(3)), A329092 (f(4)), A329093 (f(5)).

Programs

Mathematica

RealDigits[(-1 + Sqrt[3]*Pi*Coth[Sqrt[3]*Pi])/6, 10, 120][[1]] (* Amiram Eldar, Jun 17 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

Equals (-1 + (sqrt(3)*Pi)*coth(sqrt(3)*Pi))/6 = (-1 + (sqrt(-3)*Pi)*cot(sqrt(-3)*Pi))/6.