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A329088 Decimal expansion of Sum_{k>=1} 1/(k^2-3). 13
9, 7, 6, 6, 5, 0, 1, 8, 9, 9, 8, 6, 0, 9, 3, 6, 1, 7, 1, 0, 5, 8, 4, 9, 0, 5, 5, 1, 4, 1, 7, 1, 6, 2, 6, 2, 4, 4, 3, 0, 5, 9, 4, 1, 1, 4, 4, 5, 5, 1, 6, 9, 1, 9, 3, 8, 6, 9, 6, 6, 1, 7, 6, 6, 3, 5, 2, 1, 6, 5, 1, 8, 2, 9, 1, 7, 2, 9, 3, 7, 0, 2, 5, 9, 4, 8, 0, 4, 5, 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(-3).

LINKS

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

FORMULA

Sum_{k>=1} 1/(k^2-3) = (-1 + (sqrt(-3)*Pi)*coth(sqrt(-3)*Pi))/(-6) = (-1 + (sqrt(3)*Pi)*cot(sqrt(3)*Pi))/(-6).

EXAMPLE

Sum_{k>=1} 1/(k^2-3) = 0.97665018998609361710...

PROG

(PARI) default(realprecision, 100); my(f(x) = (-1 + (sqrt(x)*Pi)/tanh(sqrt(x)*Pi))/(2*x)); f(-3)

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)), this sequence (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: A020840 A069181 A214552 * A154678 A021915 A163828

Adjacent sequences:  A329085 A329086 A329087 * A329089 A329090 A329091

KEYWORD

nonn,cons

AUTHOR

Jianing Song, Nov 04 2019

STATUS

approved

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Last modified April 6 08:27 EDT 2020. Contains 333268 sequences. (Running on oeis4.)