%N Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UH, HU, HD and DH.
%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+2t^3-(1-t)*sqrt(1-4*t^2))/(2t^2(1-t)).
%F a(n)=1 for n odd, a(n)=C(n)+1 for n>0 even, where C(n) is the n-th Catalan number A000108, and a(0)=1.
%F D-finite with recurrence: +(n+2)*a(n) +2*(-n-1)*a(n-1) +(-3*n+4)*a(n-2) +8*(n-2)*a(n-3) +4*(-n+3)*a(n-4)=0. - _R. J. Mathar_, Jan 09 2020
%e a(4)=3 since we have 3 excursions of length 4, namely UUDD, UDUD and HHHH. More generally, for n=2k > 0 even we have all Dyck paths of semilength k and a path consisting only of horizontal steps H^n. For n odd, we only have the path H^n.
%Y Cf. A000108.
%A _Valerie Roitner_, Dec 12 2019