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%I #9 Dec 02 2021 17:55:37
%S 1,1,1,1,1,2,2,4,5,8,13,18,32,46,77,123,192,325,506,849,1375,2245,
%T 3750,6085,10206,16798,27936,46689,77389,130048,216717,363701,610657,
%U 1023965,1726537,2902221,4898323,8265964,13957522,23622321,39949012,67710936,114768860,194709672,330693182
%N Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UU, HH, 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.
%F G.f.: (1+t)*(1-t^2+t^3-sqrt(1-2*t^2-2*t^3+t^4-2*t^5+t^6))/(2*t^3).
%e a(7)=4 since we have the following 4 excursions of length 7: UHUDDUD, UHUDUDD, UDUHUDD and HUDUDUD.
%t CoefficientList[Series[(1 + x)*(1 - x^2 + x^3 - Sqrt[1 - 2 x^2 - 2 x^3 + x^4 - 2 x^5 + x^6])/(2 x^3), {x, 0, 50}], x] (* _Wesley Ivan Hurt_, Dec 02 2021 *)
%Y Cf. A329691, A329693.
%K nonn,walk
%O 0,6
%A _Valerie Roitner_, Dec 06 2019