A129658 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.

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

Offset

-2,5

References

Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 176 and 292

Links

Table of n, a(n) for n=-2..26.

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} 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)).

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.

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} ] ] ]

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

Keyword

nonn,frac,easy

Author

Stuart Clary, Apr 30 2007

Status

approved