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%I #8 Dec 10 2016 03:04:32
%S 0,1,0,1,31,125,156,2933,61749,64682,126431,317544,443975,1205494,
%T 1649469,6153901,38572875,121872526,3451003603,3572876129,14169631990,
%U 31912140109,46081772099,124075684307,2651671142546
%N Numerators of the convergents of the continued fraction for L(3, chi4), where L(s, chi4) is the Dirichlet L-function for the non-principal character modulo 4.
%D Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 175, 284 and 287.
%D Bruce C. Berndt, "Ramanujan's Notebooks, Part II", Springer-Verlag, 1989. See page 293, Entry 25 (iii).
%F 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.
%F 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 + ...
%F Series: L(3, chi4) = Sum_{k>=0} tanh((2k+1) pi/2)/(2k+1)^3. [Ramanujan; see Berndt, page 293]
%F Closed form: L(3, chi4) = Pi^3/32.
%e 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.
%t nmax = 100; cfrac = ContinuedFraction[Pi^3/32, nmax + 1]; Join[ {0, 1}, Numerator[ Table[ FromContinuedFraction[ Take[cfrac, j] ], {j, 1, nmax + 1} ] ] ]
%Y Cf. A153071, A153072, A153074.
%K nonn,frac,easy
%O -2,5
%A _Stuart Clary_, Dec 17 2008
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