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%I #38 Mar 07 2023 11:20:14
%S 1,1,2,4,8,17,37,82,185,423,978,2283,5373,12735,30372,72832,175502,
%T 424748,1032004,2516347,6155441,15101701,37150472,91618049,226460893,
%U 560954047,1392251012,3461824644,8622571758,21511212261,53745962199,134474581374
%N Expansion of (1 - x - x^2 - sqrt((1 - x - x^2)^2 - 4*x^3))/(2*x^3) in powers of x.
%C 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
%H G. C. Greubel, <a href="/A292460/b292460.txt">Table of n, a(n) for n = 0..1000</a>
%H Jean-Luc Baril, Sergey Kirgizov and Armen Petrossian, <a href="https://arxiv.org/abs/1804.01293">Enumeration of Łukasiewicz paths modulo some patterns</a>, arXiv:1804.01293 [math.CO], 2018.
%H Jean-Luc Baril and José Luis Ramírez, <a href="https://arxiv.org/abs/2302.12741">Descent distribution on Catalan words avoiding ordered pairs of Relations</a>, arXiv:2302.12741 [math.CO], 2023.
%F 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).
%F a(n) = A004148(n+1).
%F 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
%F D-finite with recurrence: (n+3)*a(n) +(-2*n-3)*a(n-1) -n*a(n-2) +(-2*n+3)*a(n-3) +(n-3)*a(n-4)=0. - _R. J. Mathar_, Jan 23 2020
%F a(0) = a(1) = 1; a(n) = a(n-1) + a(n-2) + Sum_{k=0..n-3} a(k) * a(n-k-3). - _Ilya Gutkovskiy_, Nov 09 2021
%t 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 *)
%o (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
%o (Magma) m:=25; R<x>:=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
%Y Cf. A001006, A004148, A292461.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Sep 16 2017
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