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Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UH, HD and DU.
1

%I #6 Dec 18 2019 09:03:43

%S 1,1,2,3,5,8,13,22,37,64,111,195,346,618,1114,2019,3683,6753,12440,

%T 23018,42750,79683,148993,279407,525382,990325,1870993,3542241,

%U 6719450,12769653,24308665,46348192,88501413,169228912,324018531,621159937,1192189832,2290699981,4406021067

%N Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UH, HD and DU.

%C The Motzkin step set is U=(1,1), H=(1,0) and D=(1,-1). An excursion is a path starting at (0,0), ending at (n,0) and never crossing the x-axis, i.e., staying at nonnegative altitude.

%H Andrei Asinowski, Cyril Banderier, and Valerie Roitner, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/several_patterns.pdf">Generating functions for lattice paths with several forbidden patterns</a>, preprint, 2019.

%F G.f.: (1-t-t^2+2t^3-2t^4-sqrt(1-2t-t^2+2t^3+t^4-4t^5+4t^6))/(2t^3(1-t)^2).

%e a(3)=3, since we have the following 3 excursions of length 3: UDH, HUD and HHH.

%Y Cf. A329695.

%K nonn,walk

%O 0,3

%A _Valerie Roitner_, Dec 16 2019