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A329085
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Decimal expansion of Sum_{k>=0} 1/(k^2+4).
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13
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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
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OFFSET
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0,1
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COMMENTS
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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.
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LINKS
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FORMULA
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Equals (1 + (2*Pi)*coth(2*Pi))/8 = (1 + (2*Pi*i)*cot(2*Pi*i))/8, i = sqrt(-1).
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EXAMPLE
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0.91040364132111511419...
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MATHEMATICA
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RealDigits[(1 + 2*Pi*Coth[2*Pi])/8, 10, 120][[1]] (* Amiram Eldar, Jun 17 2023 *)
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PROG
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(PARI) default(realprecision, 100); my(F(x) = (1 + (sqrt(x)*Pi)/tanh(sqrt(x)*Pi))/(2*x)); F(4)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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