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URL: https://oeis.org/A329085

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A329085
Decimal expansion of Sum_{k>=0} 1/(k^2+4).
14
9, 1, 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 A329092 are essentially the same, but both sequences are added because some people may search for this, and some people may search for A329092.
FORMULA
Equals (1 + (2*Pi)*coth(2*Pi))/8 = (1 + (2*Pi*i)*cot(2*Pi*i))/8, i = sqrt(-1).
EXAMPLE
0.91040364132111511419...
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), 0) \\ 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)), this sequence (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: A269948 A121935 A070060 * A273636 A248414 A176980
KEYWORD
nonn,cons
AUTHOR
Jianing Song, Nov 04 2019
STATUS
approved