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 A292460 Expansion of (1 - x - x^2 - sqrt((1 - x - x^2)^2 - 4*x^3))/(2*x^3) in powers of x. 3
 1, 1, 2, 4, 8, 17, 37, 82, 185, 423, 978, 2283, 5373, 12735, 30372, 72832, 175502, 424748, 1032004, 2516347, 6155441, 15101701, 37150472, 91618049, 226460893, 560954047, 1392251012, 3461824644, 8622571758, 21511212261, 53745962199, 134474581374 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Number of U_{k}D-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are P-equivalent iff the positions of pattern P are identical in these paths. - Sergey Kirgizov, Apr 08 2018 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Jean-Luc Baril, Sergey Kirgizov and Armen Petrossian, Enumeration of Łukasiewicz paths modulo some patterns, arXiv:1804.01293 [math.CO], 2018. FORMULA G.f.: 1/(1-x-x^2-x^3/(1-x-x^2-x^3/(1-x-x^2-x^3/(1-x-x^2-x^3/(1-x-x^2-x^3/(... (continued fraction). a(n) = A004148(n+1). a(n) ~ 5^(1/4) * phi^(2*n + 4) / (2*sqrt(Pi)*n^(3/2)), where phi is the golden ratio (1+sqrt(5))/2. - Vaclav Kotesovec, Sep 17 2017 MATHEMATICA CoefficientList[Series[(1-x-x^2 -Sqrt[(1-x-x^2)^2 -4*x^3])/(2*x^3), {x, 0, 50}], x] (* G. C. Greubel, Aug 13 2018 *) PROG (PARI) x='x+O('x^50); Vec((1-x-x^2 -sqrt((1-x-x^2)^2 -4*x^3))/(2*x^3)) \\ G. C. Greubel, Aug 13 2018 (MAGMA) m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x-x^2 -Sqrt((1-x-x^2)^2 -4*x^3))/(2*x^3))); // G. C. Greubel, Aug 13 2018 CROSSREFS Cf. A001006, A004148, A292461. Sequence in context: A292461 A203019 A004148 * A085022 A003426 A179476 Adjacent sequences:  A292457 A292458 A292459 * A292461 A292462 A292463 KEYWORD nonn,changed AUTHOR Seiichi Manyama, Sep 16 2017 STATUS approved

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Last modified August 19 01:39 EDT 2018. Contains 313840 sequences. (Running on oeis4.)