A329089 Decimal expansion of Sum_{k>=1} 1/(k^2-2), negated.

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

Offset

-1,1

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(-2) (negated).

This and A329082 are essentially the same, but both sequences are added because some people may search for this, and some people may search for A329082.

Links

Table of n, a(n) for n=-1..89.

Formula

Equals (-1 + (sqrt(-2)*Pi)*coth(sqrt(-2)*Pi))/(-4).

Equals (-1 + (sqrt(2)*Pi)*cot(sqrt(2)*Pi))/(-4).

Example

Sum_{k>=1} 1/(k^2-2) = -0.05681040770062008255...

Mathematica

RealDigits[(-1 + Sqrt[2]*Pi*Cot[Sqrt[2]*Pi])/4, 10, 120][[1]] (* Amiram Eldar, Jun 18 2023 *)

Prog

(PARI) default(realprecision, 100); my(f(x) = (-1 + (sqrt(x)*Pi)/tanh(sqrt(x)*Pi))/(2*x)); f(-2)
(PARI) sumnumrat(1/(x^2-2), 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. A329087 (f(-5)), A329088 (f(-3)), this sequence (f(-2)), A013661 (f(0)), A259171 (f(1)), A329090 (f(2)), A329091 (f(3)), A329092 (f(4)), A329093 (f(5)).

Sequence in context: A294071 A240661 A153430 * A100379 A351885 A021180

Adjacent sequences: A329086 A329087 A329088 * A329090 A329091 A329092

Keyword

nonn,cons

Author

Jianing Song, Nov 04 2019

Status

approved