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%I #28 Nov 25 2019 08:04:39
%S 1,1,0,1,2,2,4,8,12,21,40,69,122,227,412,747,1386,2567,4744,8851,
%T 16566,31004,58268,109858,207368,392331,744072,1413291,2688822,
%U 5124738,9781492,18694896,35780444,68566567,131546440,252661515,485806614,935017790,1801327884,3473467328,6703610548
%N Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UD, HH 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.
%F G.f.: (t+1)*(1 - t - sqrt(4*t^4 - 4*t^3 + t^2 - 2*t + 1))/(2*t^3).
%e a(4)=2 as one has 2 excursions of length 4, namely: HUHD and UHDH.
%Y Cf. A004149 (avoiding UD and DU).
%Y Cf. A329666, A329665.
%K nonn,walk
%O 0,5
%A _Valerie Roitner_, Nov 19 2019
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