%I #14 Nov 23 2023 10:39:03

%S 1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,

%T 0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,

%U 0,1,0,0,1,0,0,1

%N Number of excursions of length n with Motzkin-steps consisting only of consecutive steps UH, HD and DU.

%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 2 (namely 1, 1) and a period of length 3 (namely 0, 1, 0).

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1).

%F G.f.: (1+t-t^4)/(1-t^3).

%F a(n) = A173857(n). - _R. J. Mathar_, Dec 06 2019

%e For n=3k we have one excursion of this type, namely (UHD)^k. Furthermore, we have a(1)=1, since the excursion H is allowed.

%Y Cf. A329682, A329683, A329684.

%Y Essentially the same as A079978 and A022003.

%K nonn,walk,easy

%O 0

%A _Valerie Roitner_, Nov 29 2019