%I
%S 1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U 1,1,1
%N Number of excursions of length n with Motzkinsteps forbidding all consecutive steps of length 2 except UD 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 on the xaxis and never crossing the xaxis, i.e., staying at nonnegative altitude.
%C This sequence is periodic with a preperiod of length 3 (namely 1, 1, 2) and a period of length 1 (namely 1).
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).
%F G.f.: (1+t^2t^3)/(1t).
%e a(2)=2 since UD and HH are allowed. For n different from 2, only the excursion H^n is allowed.
%Y Cf. A329680, A329682, A329683.
%Y Essentially the same as A294619, A261143 and A141044.
%K nonn,walk,easy
%O 0,3
%A _Valerie Roitner_, Nov 29 2019
