A329080 Decimal expansion of Sum_{k>=0} 1/(k^2-5), negated.

8, 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 A329087 are essentially the same, but both sequences are added because some people may search for this, and some people may search for A329087.

Links

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

Formula

Equals (1 + (sqrt(-5)*Pi)*coth(sqrt(-5)*Pi))/(-10).

Equals (1 + (sqrt(5)*Pi)*cot(sqrt(5)*Pi))/(-10).

Example

-0.86683259566274485298...

Mathematica

RealDigits[(1 + Sqrt[5]*Pi*Cot[Sqrt[5]*Pi])/10, 10, 120][[1]] (* Amiram Eldar, Jun 17 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. this sequence (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)), A329091 (f(3)), A329092 (f(4)), A329093 (f(5)).

Sequence in context: A241994 A275828 A154499 * A092750 A218197 A181048

Adjacent sequences: A329077 A329078 A329079 * A329081 A329082 A329083

Keyword

nonn,cons

Author

Jianing Song, Nov 04 2019

Status

approved