A153068 Denominators 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.

1, 0, 1, 1, 4, 5, 9, 23, 32, 567, 599, 6557, 7156, 13713, 75721, 89434, 165155, 419744, 584899, 1004643, 1589542, 2594185, 6777912, 16150009, 22927921, 39077930, 62005851, 101083781, 264173413, 1157777433, 1421950846, 2579728279

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

Crossrefs

Cf. A086724, A153066, A153067.

Sequence in context: A240926 A041032 A041993 * A075116 A041079 A042715

Adjacent sequences: A153065 A153066 A153067 * A153069 A153070 A153071

Keyword

nonn,frac,easy

Author

Stuart Clary, Dec 17 2008

Status

approved