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A129658
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Numerators of the convergents of the continued fraction for L(3, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3.
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15
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0, 1, 0, 1, 7, 8, 15, 23, 38, 61, 343, 404, 747, 7127, 29255, 387442, 1579023, 1966465, 5511953, 150789196, 156301149, 4527221368, 4683522517, 13894266402, 32472055321, 111310432365, 255092920051, 1896960872722, 2152053792773
(list; graph; refs; listen; history; internal format)
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OFFSET
| -2,5
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REFERENCES
| Leonhard Euler, ``Introductio in Analysin Infinitorum'', First Part, Articles 176 and 292
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FORMULA
| chi3(k) = Kronecker(-3, k); chi3(k) is 0, 1, -1 when k reduced modulo 3 is 0, 1, 2, respectively; chi3 is A049347 shifted.
Series: L(3, chi3) = sum_{k=1..infinity} chi3(k) k^{-3} = 1 - 1/2^3 + 1/4^3 - 1/5^3 + 1/7^3 - 1/8^3 + 1/10^3 - 1/11^3 + ...
Closed form: L(3, chi3) = 4 pi^3/(81 sqrt(3))
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EXAMPLE
| L(3, chi3) = 0.8840238117500798567430579168710118077... = [0; 1, 7, 1, 1, 1, 1, 1, 5, 1, 1, 9, 4, 13, 4, ...], the convergents of which are 0/1, 1/0, [0/1], 1/1, 7/8, 8/9, 15/17, 23/26, 38/43, 61/69, 343/388, 404/457, 747/845, 7127/8062, 29255/33093, 387442/438271, 1579023/1786177, ..., with brackets marking index 0. Those prior to index 0 are for initializing the recurrence.
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MATHEMATICA
| nmax = 100; cfrac = ContinuedFraction[4 Pi^3/(81 Sqrt[3]), nmax + 1]; Join[ {0, 1}, Numerator[ Table[ FromContinuedFraction[ Take[cfrac, j] ], {j, 1, nmax + 1} ] ] ]
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CROSSREFS
| Cf. A129404, A129405, A129406, A129407, A129408, A129409, A129410, A129411.
Cf. A129659, A129660, A129661, A129662, A129663, A129664, A129665.
Sequence in context: A070424 A022097 A041100 * A041693 A042001 A020690
Adjacent sequences: A129655 A129656 A129657 * A129659 A129660 A129661
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KEYWORD
| nonn,frac,easy
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AUTHOR
| Stuart Clary (clary(AT)uakron.edu), Apr 30, 2007
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