A129659 Denominators 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.

1, 0, 1, 1, 8, 9, 17, 26, 43, 69, 388, 457, 845, 8062, 33093, 438271, 1786177, 2224448, 6235073, 170571419, 176806492, 5121153195, 5297959687, 15717072569, 36732104825, 125913387044, 288558878913, 2145825539435, 2434384418348

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..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)).

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[ {1, 0}, Denominator[ Table[ FromContinuedFraction[ Take[cfrac, j] ], {j, 1, nmax + 1} ] ] ]

Crossrefs

Cf. A129404, A129405, A129406, A129407, A129408, A129409, A129410, A129411.

Cf. A129658, A129660, A129661, A129662, A129663, A129664, A129665.

Sequence in context: A274406 A261454 A022098 * A041130 A041307 A042289

Adjacent sequences: A129656 A129657 A129658 * A129660 A129661 A129662

Keyword

nonn,frac,easy

Author

Stuart Clary, Apr 30 2007

Status

approved