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A329087 Decimal expansion of Sum_{k>=1} 1/(k^2-5), negated. 13
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 (list; constant; graph; refs; listen; history; text; internal format)
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...

PROG

(PARI) default(realprecision, 100); my(f(x) = (-1 + (sqrt(x)*Pi)/tanh(sqrt(x)*Pi))/(2*x)); f(-5)

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: A098537 A276861 A131703 * A135357 A322346 A322345

Adjacent sequences:  A329084 A329085 A329086 * A329088 A329089 A329090

KEYWORD

nonn,cons

AUTHOR

Jianing Song, Nov 04 2019

STATUS

approved

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Last modified January 17 20:36 EST 2020. Contains 330987 sequences. (Running on oeis4.)