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A329091
Decimal expansion of Sum_{k>=1} 1/(k^2+3).
13
7, 4, 0, 2, 6, 7, 0, 7, 6, 5, 8, 1, 8, 5, 0, 7, 8, 2, 5, 8, 0, 6, 0, 2, 9, 6, 4, 8, 2, 4, 8, 1, 1, 9, 7, 7, 9, 4, 3, 1, 0, 9, 3, 0, 2, 3, 8, 5, 4, 5, 1, 2, 4, 5, 6, 2, 7, 0, 3, 5, 4, 1, 8, 6, 2, 5, 3, 3, 4, 1, 8, 9, 8, 5, 1, 2, 3, 0, 1, 2, 6, 5, 5, 2, 5, 1, 4, 9, 1, 6, 1
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).
FORMULA
Equals (-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) = 0.74026707658185078258...
MATHEMATICA
RealDigits[(-1 + Sqrt[3]*Pi*Coth[Sqrt[3]*Pi])/6, 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(3)
(PARI) sumnumrat(1/(x^2+3), 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)), this sequence (f(3)), A329092 (f(4)), A329093 (f(5)).
Sequence in context: A221388 A303982 A175998 * A306398 A093825 A229784
KEYWORD
nonn,cons
AUTHOR
Jianing Song, Nov 04 2019
STATUS
approved