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A329084
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Decimal expansion of Sum_{k>=0} 1/(k^2+3).
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13
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1, 0, 7, 3, 6, 0, 0, 4, 0, 9, 9, 1, 5, 1, 8, 4, 1, 1, 5, 9, 1, 3, 9, 3, 6, 2, 9, 8, 1, 5, 8, 1, 4, 5, 3, 1, 1, 2, 7, 6, 4, 4, 2, 6, 3, 5, 7, 1, 8, 7, 8, 4, 5, 7, 8, 9, 6, 0, 3, 6, 8, 7, 5, 1, 9, 5, 8, 6, 6, 7, 5, 2, 3, 1, 8, 4, 5, 6, 3, 4, 5, 9, 8, 8, 5, 8, 4, 8, 2, 4, 9
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OFFSET
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1,3
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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(3).
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LINKS
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FORMULA
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Equals (1 + (sqrt(3)*Pi)*coth(sqrt(3)*Pi))/6 = (1 + (sqrt(-3)*Pi)*cot(sqrt(-3)*Pi))/6.
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EXAMPLE
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1.07360040991518411591...
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MATHEMATICA
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RealDigits[Sum[1/(k^2+3), {k, 0, \[Infinity]}], 10, 120][[1]] (* Harvey P. Dale, Jul 05 2021 *)
RealDigits[(1 + Sqrt[3]*Pi*Coth[Sqrt[3]*Pi])/6, 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(3)
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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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