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A329092
Decimal expansion of Sum_{k>=1} 1/(k^2+4).
13
6, 6, 0, 4, 0, 3, 6, 4, 1, 3, 2, 1, 1, 1, 5, 1, 1, 4, 1, 9, 3, 0, 4, 3, 8, 2, 4, 9, 2, 6, 4, 4, 3, 6, 0, 9, 6, 1, 1, 6, 9, 5, 0, 6, 5, 7, 9, 4, 6, 5, 0, 4, 4, 8, 9, 0, 2, 5, 8, 5, 8, 8, 0, 4, 5, 3, 5, 8, 0, 8, 3, 1, 1, 4, 9, 4, 5, 5, 2, 0, 6, 2, 5, 2, 8, 4, 5, 3, 1, 7, 8
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(4).
This and A329085 are essentially the same, but both sequences are added because some people may search for this, and some people may search for A329085.
FORMULA
Equals (-1 + (2*Pi)*coth(2*Pi))/8 = (-1 + (2*Pi*i)*cot(2*Pi*i))/8, i = sqrt(-1).
Equals Integral_{x=0..oo} sin(x)*cos(x)/(exp(x) - 1) dx. - Amiram Eldar, Aug 16 2020
EXAMPLE
Sum_{k>=1} 1/(k^2+4) = 0.66040364132111511419...
MATHEMATICA
RealDigits[(-1 + 2*Pi*Coth[2*Pi])/8, 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(4)
(PARI) sumnumrat(1/(x^2+4), 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)), A329089 (f(-2)), A013661 (f(0)), A259171 (f(1)), A329090 (f(2)), A329091 (f(3)), this sequence (f(4)), A329093 (f(5)).
Sequence in context: A281056 A273989 A197013 * A081825 A272648 A212708
KEYWORD
nonn,cons
AUTHOR
Jianing Song, Nov 04 2019
STATUS
approved