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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 (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(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. LINKS FORMULA Sum_{k>=1} 1/(k^2+4) = (-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... PROG (PARI) default(realprecision, 100); my(f(x) = (-1 + (sqrt(x)*Pi)/tanh(sqrt(x)*Pi))/(2*x)); f(4) 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 Adjacent sequences:  A329089 A329090 A329091 * A329093 A329094 A329095 KEYWORD nonn,cons AUTHOR Jianing Song, Nov 04 2019 STATUS approved

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Last modified November 26 15:51 EST 2020. Contains 338640 sequences. (Running on oeis4.)