%I
%S 1,1,1,1,2,3,6,10,20,36,72,136,273,532,1074,2137,4342,8766,17925,
%T 36574,75234,154749,320038,662490,1376653,2864534,5977603,12492157,
%U 26165052,54882573,115329739,242683876,511456452,1079252975,2280413318,4823955728,10216401353,21659426346
%N Number of excursions of length n with Motzkinsteps avoiding the consecutive steps UH, HU and HH.
%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 xaxis, i.e., staying at nonnegative altitude.
%H Andrei Asinowski, Cyril Banderier, and Valerie Roitner, <a href="https://lipn.univparis13.fr/~banderier/Papers/several_patterns.pdf">Generating functions for lattice paths with several forbidden patterns</a>, preprint, 2019.
%F G.f.: (1+t^3sqrt(14t^22t^3+t^6))/(2t^2).
%F G.f. A(x) satisfies: A(x) = x + 1 / (1  x^2 * A(x)).  _Ilya Gutkovskiy_, Nov 03 2021
%e a(5)=3 since we have the following 3 excursions of length 5: UUDDH, UUHDH and UDUDH.
%Y Cf. A329700.
%K nonn,walk
%O 0,5
%A _Valerie Roitner_, Dec 16 2019
