

A153073


Numerators of the convergents of the continued fraction for L(3, chi4), where L(s, chi4) is the Dirichlet Lfunction for the nonprincipal character modulo 4.


3



0, 1, 0, 1, 31, 125, 156, 2933, 61749, 64682, 126431, 317544, 443975, 1205494, 1649469, 6153901, 38572875, 121872526, 3451003603, 3572876129, 14169631990, 31912140109, 46081772099, 124075684307, 2651671142546
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OFFSET

2,5


REFERENCES

Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 175, 284 and 287.
Bruce C. Berndt, "Ramanujan's Notebooks, Part II", SpringerVerlag, 1989. See page 293, Entry 25 (iii).


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(3, chi4) = Sum_{k>=1} chi4(k) k^{3} = 1  1/3^3 + 1/5^3  1/7^3 + 1/9^3  1/11^3 + 1/13^3  1/15^3 + ...
Series: L(3, chi4) = Sum_{k>=0} tanh((2k+1) pi/2)/(2k+1)^3. [Ramanujan; see Berndt, page 293]
Closed form: L(3, chi4) = Pi^3/32.


EXAMPLE

L(3, chi4) = 0.9689461462593693804836348458469186... = [0; 1, 31, 4, 1, 18, 21, 1, 1, 2, 1, 2, 1, 3, 6, 3, 28, ...], the convergents of which are 0/1, 1/0, [0/1], 1/1, 31/32, 125/129, 156/161, 2933/3027, 61749/63728, 64682/66755, 126431/130483, 317544/327721, 443975/458204, ..., with brackets marking index 0. Those prior to index 0 are for initializing the recurrence.


MATHEMATICA

nmax = 100; cfrac = ContinuedFraction[Pi^3/32, nmax + 1]; Join[ {0, 1}, Numerator[ Table[ FromContinuedFraction[ Take[cfrac, j] ], {j, 1, nmax + 1} ] ] ]


CROSSREFS

Cf. A153071, A153072, A153074.
Sequence in context: A068021 A131992 A042884 * A042886 A042888 A183836
Adjacent sequences: A153070 A153071 A153072 * A153074 A153075 A153076


KEYWORD

nonn,frac,easy,changed


AUTHOR

Stuart Clary, Dec 17 2008


STATUS

approved



