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A153074
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Denominators 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.
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3
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1, 0, 1, 1, 32, 129, 161, 3027, 63728, 66755, 130483, 327721, 458204, 1244129, 1702333, 6351128, 39809101, 125778431, 3561605169, 3687383600, 14623755969, 32934895538, 47558651507, 128052198552, 2736654821099
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OFFSET
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-2,5
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REFERENCES
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Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 175, 284 and 287.
Bruce C. Berndt, "Ramanujan's Notebooks, Part II", Springer-Verlag, 1989. See page 293, Entry 25 (iii).
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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MATHEMATICA
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nmax = 100; cfrac = ContinuedFraction[Pi^3/32, nmax + 1]; Join[ {1, 0}, Denominator[ Table[ FromContinuedFraction[ Take[cfrac, j] ], {j, 1, nmax + 1} ] ] ]
Join[{1, 0}, Convergents[Pi^3/32, 30]//Denominator] (* Harvey P. Dale, Mar 05 2019 *)
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CROSSREFS
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KEYWORD
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nonn,frac,easy
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AUTHOR
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STATUS
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approved
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