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%I
%S 1,1,1,2,3,5,9,16,30,56,108,208,409,805,1606,3211,6479,13108,26691,
%T 54499,111808,229983,474787,982528,2039143,4241187,8842137,18469760,
%U 38657209,81047625,170212312,358013615,754140328,1590709427,3359666293,7104369046,15040357081,31875827699
%N Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UH, HH and HD.
%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)*(1-t^3-sqrt(1-4t^2-2t^3+t^6))/(2t^2).
%e a(4)=3 since we have 3 excursions of length 4, namely UUDD, UDUD and HUDH.
%Y Cf. A329699.
%K nonn,walk
%O 0,4
%A _Valerie Roitner_, Dec 16 2019
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