A329087 Decimal expansion of Sum_{k>=1} 1/(k^2-5), negated.

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

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) (negated).

This and A329080 are essentially the same, but both sequences are added because some people may search for this, and some people may search for A329080.

Links

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

Formula

Sum_{k>=1} 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>=1} 1/(k^2-5) = -0.66683259566274485298...

Mathematica

RealDigits[(1 - Sqrt[5]*Pi*Cot[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), 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. this sequence (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: A276861 A131703 A352764 * A135357 A322346 A332559

Adjacent sequences: A329084 A329085 A329086 * A329088 A329089 A329090

Keyword

nonn,cons

Author

Jianing Song, Nov 04 2019

Status

approved