A153067 Numerators of the convergents of the continued fraction for L(2, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3.

0, 1, 0, 1, 3, 4, 7, 18, 25, 443, 468, 5123, 5591, 10714, 59161, 69875, 129036, 327947, 456983, 784930, 1241913, 2026843, 5295599, 12618041, 17913640, 30531681, 48445321, 78977002, 206399325, 904574302, 1110973627, 2015547929

Offset

-2,5

Links

Table of n, a(n) for n=-2..29.

Formula

chi3(k) = Kronecker(-3, k); chi3(k) is 0, 1, -1 when k reduced modulo 3 is 0, 1, 2, respectively; chi3 is A102283.

Series: L(2, chi3) = Sum_{k=1..infinity} chi3(k) k^{-2} = 1 - 1/2^2 + 1/4^2 - 1/5^2 + 1/7^2 - 1/8^2 + 1/10^2 - 1/11^2 + ...

Example

L(2, chi3) = 0.781302412896486296867187429624092... = [0; 1, 3, 1, 1, 2, 1, 17, 1, 10, 1, 1, 5, 1, 1, 2, 1, ...], the convergents of which are 0/1, 1/0, [0/1], 1/1, 3/4, 4/5, 7/9, 18/23, 25/32, 443/567, 468/599, 5123/6557, 5591/7156, ..., with brackets marking index 0. Those prior to index 0 are for initializing the recurrence.

Mathematica

nmax = 100; cfrac = ContinuedFraction[(Zeta[2, 1/3] - Zeta[2, 2/3])/9, nmax + 1]; Join[ {0, 1}, Numerator[ Table[ FromContinuedFraction[ Take[cfrac, j] ], {j, 1, nmax + 1} ] ] ]

Crossrefs

Cf. A086724, A153066, A153068.

Sequence in context: A344783 A093611 A042375 * A041593 A258740 A109749

Adjacent sequences: A153064 A153065 A153066 * A153068 A153069 A153070

Keyword

nonn,frac,easy

Author

Stuart Clary, Dec 17 2008

Status

approved