A153070 Denominators 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.

1, 0, 1, 1, 11, 12, 107, 119, 10579, 42435, 53014, 95449, 721157, 15960903, 16682060, 49325023, 164657129, 4330410377, 4495067506, 53776152943, 58271220449, 636488357433, 694759577882, 6889324558371, 21362733252995

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, 1/12, 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[ {1, 0}, Denominator[ Table[ FromContinuedFraction[ Take[cfrac, j] ], {j, 1, nmax + 1} ] ] ]

Crossrefs

Cf. A006752, A014538, A054543, A104338, A118323, A153069 (numerators).

Sequence in context: A041258 A082262 A239463 * A193023 A278985 A071159

Adjacent sequences: A153067 A153068 A153069 * A153071 A153072 A153073

Keyword

nonn,frac,easy

Author

Stuart Clary, Dec 17 2008

Status

approved