A153069 Numerators of the convergents of the continued fraction for Catalan's constant L(2, chi4), where L(s, chi4) is the Dirichlet L-function for the non-principal character modulo 4.

0, 1, 0, 1, 10, 11, 98, 109, 9690, 38869, 48559, 87428, 660555, 14619638, 15280193, 45180024, 150820265, 3966506914, 4117327179, 49257105883, 53374433062, 583001436503, 636375869565, 6310384262588, 19567528657329

Offset

-2,5

Links

Table of n, a(n) for n=-2..22.

Formula

chi4(k) = Kronecker(-4, k); chi4(k) is 0, 1, 0, -1 when k reduced modulo 4 is 0, 1, 2, 3, respectively; chi4 is A101455.

Series: L(2, chi4) = Sum_{k>=1} chi4(k) k^{-2} = 1 - 1/3^2 + 1/5^2 - 1/7^2 + 1/9^2 - 1/11^2 + 1/13^2 - 1/15^2 + ...

Example

L(2, chi4) = 0.91596559417721901505460351493238411... = [0; 1, 10, 1, 8, 1, 88, 4, 1, 1, 7, 22, 1, 2, 3, 26, ...], the convergents of which are 0/1, 1/0, [0/1], 1/1, 10/11, 11/12, 98/107, 109/119, 9690/10579, 38869/42435, 48559/53014, 87428/95449, 660555/721157, ..., with brackets marking index 0. Those prior to index 0 are for initializing the recurrence.

Mathematica

nmax = 100; cfrac = ContinuedFraction[Catalan, nmax + 1]; Join[ {0, 1}, Numerator[ Table[ FromContinuedFraction[ Take[cfrac, j] ], {j, 1, nmax + 1} ] ] ]

Crossrefs

Cf. A006752, A014538, A054543, A104338, A118323, A153070 (denominators).

Sequence in context: A046851 A045953 A136830 * A332703 A178569 A332795

Adjacent sequences: A153066 A153067 A153068 * A153070 A153071 A153072

Keyword

nonn,frac,easy

Author

Stuart Clary, Dec 17 2008

Status

approved