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%I #43 Nov 11 2022 04:36:31
%S 1,1,1,2,3,4,7,11,16,27,43,65,108,173,267,440,707,1105,1812,2917,4597,
%T 7514,12111,19196,31307,50503,80380,130883,211263,337284,548547,
%U 885831,1417582,2303413,3720995,5965622,9686617,15652239,25130844,40783083,65913927
%N Expansion of 1/(1-x-x^3*c(x^3)), c(x) the g.f. of A000108.
%C Hankel transform is A010892(n+1).
%C Row sums of A165408.
%C Number of UF-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are UF-equivalent iff the positions of pattern UF are identical in these paths. This also works for the pattern FU. - _Sergey Kirgizov_, Apr 08 2018
%C a(n) is the total number of lattice paths from (0,0) to (n-2*i,i) that do not go above the diagonal x=y using steps in {(1,0), (0,1)}. - _Alois P. Heinz_, Sep 20 2022
%H Alois P. Heinz, <a href="/A165407/b165407.txt">Table of n, a(n) for n = 0..2000</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 J.-L. Baril and A. Petrossian, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Baril/baril3.html">Equivalence Classes of Motzkin Paths Modulo a Pattern of Length at Most Two</a>, J. Int. Seq. 18 (2015) 15.7.1
%F G.f.: 2/(1 - 2*x + sqrt(1-4*x^3)) = 1/(1-x-x^3/(1-x^3/(1-x^3/(1-x^3/(1-.... (continued fraction).
%F a(n) = Sum_{k=0..n} if(n<=3k, C((n+k)/2,k)*((3*k-n)/2 + 1)*(1+(-1)^(n-k) )/(2*(k+1)).
%F a(n) = Sum_{k=0..n+1} Fibonacci(n-k+1)*(0^k - A000108((k-2)/3)*(1+2*cos(2*Pi*(k-2)/3))/3).
%F (n+1)*a(n) = (n+1)*a(n-1) + (n+1)*a(n-2) +2*(2*n-7)*a(n-3) -2*(2*n-7)*a(n-4) -2*(2*n-7)*a(n-5). - _R. J. Mathar_, Nov 15 2011
%F a(3*n) = A026726(n); a(3*n+1) = A026671(n); a(3*n+2) = A026674(n+1). - _Philippe Deléham_, Feb 01 2014
%F Limit_{n->oo} a(n+1)/a(n) = A001622. - _Alois P. Heinz_, Sep 15 2022
%p b:= proc(x, y) option remember; `if`(y<=x, `if`(x=0, 1,
%p b(x-1, y)+`if`(y>0, b(x, y-1), 0)), 0)
%p end:
%p a:= n-> add(b(n-2*i, i), i=0..n/3):
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Sep 20 2022
%t b[x_, y_]:= b[x, y]= If[y<=x, If[x==0, 1, b[x-1, y] +If[y>0, b[x, y-1], 0]], 0];
%t a[n_] := Sum[b[n-2*i, i], {i, 0, n/3}];
%t Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Oct 08 2022, after _Alois P. Heinz_ *)
%t CoefficientList[Series[(Sqrt[1-4*x^3] -1+2*x)/(2*x*(1-x-x^2)), {x,0,50}], x] (* _G. C. Greubel_, Nov 09 2022 *)
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 50); Coefficients(R!( (Sqrt(1-4*x^3) -1+2*x)/(2*x*(1-x-x^2)) )); // _G. C. Greubel_, Nov 09 2022
%o (SageMath)
%o def A165407_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( 2/(1-2*x+sqrt(1-4*x^3)) ).list()
%o A165407_list(50) # _G. C. Greubel_, Nov 09 2022
%Y Cf. A000045, A000108, A001622, A010892, A165408.
%Y Trisections give: A026726, A026671, A026674.
%K nonn,easy
%O 0,4
%A _Paul Barry_, Sep 17 2009
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