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%I #18 Jul 24 2023 11:33:07
%S 1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,
%T 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,
%U 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2
%N Number of excursions of length n with Motzkin-steps forbidding all consecutive steps of length 2 except 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 on the x-axis and never crossing the x-axis, i.e., staying at nonnegative altitude.
%C This sequence is periodic with a pre-period of length 3 (namely 1, 1, 1) and a period of length 1 (namely 2).
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).
%F G.f.: (1 + t^3)/(1 - t).
%e For n >= 3 we always have two allowed excursions, namely UH^(n-2)D and H^n.
%e For n = 0, 1, 2 we have one meander each, namely the empty walk, H and HH.
%Y Cf. A329680, A329682, A329684.
%K nonn,walk,easy
%O 0,4
%A _Valerie Roitner_, Nov 29 2019
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