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A329082
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Decimal expansion of Sum_{k>=0} 1/(k^2-2), negated.
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13
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5, 5, 6, 8, 1, 0, 4, 0, 7, 7, 0, 0, 6, 2, 0, 0, 8, 2, 5, 5, 2, 9, 8, 1, 6, 0, 9, 1, 1, 2, 5, 9, 7, 3, 4, 7, 0, 9, 8, 7, 0, 9, 2, 7, 0, 2, 5, 7, 0, 4, 0, 8, 7, 8, 5, 5, 1, 0, 0, 1, 9, 8, 3, 4, 8, 6, 3, 2, 8, 1, 0, 3, 7, 4, 4, 1, 5, 7, 0, 0, 2, 4, 6, 1, 7, 4, 5, 6, 5, 7, 7
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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(-2) (negated).
This and A329089 are essentially the same, but both sequences are added because some people may search for this, and some people may search for A329089.
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LINKS
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FORMULA
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Equals (1 + (sqrt(-2)*Pi)*coth(sqrt(-2)*Pi))/(-4) = (1 + (sqrt(2)*Pi)*cot(sqrt(2)*Pi))/(-4).
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EXAMPLE
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-0.55681040770062008255...
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MATHEMATICA
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RealDigits[(1 + Sqrt[2]*Pi*Cot[Sqrt[2]*Pi])/4, 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(-2)
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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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