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%I #7 Dec 06 2019 11:54:35
%S 1,1,1,2,2,3,4,6,9,13,21,31,50,78,123,200,315,517,831,1355,2224,3620,
%T 5995,9835,16291,27004,44734,74625,124078,207437,346765,580418,974358,
%U 1634622,2750502,4628758,7800544,13164287,22223486,37579843,63571333,107659948,182479796,309478532
%N Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UH, HH, 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 at (n,0) and never crossing the x-axis, i.e., staying at nonnegative altitude.
%F G.f.: (1+t)*(1+t-t^2-t^4-(1+t)*sqrt(1-2t^2-2t^3+t^4-2t^5+t^6))/(2t^3).
%e a(5)=3 since we have 3 excursions of length 5, namely UUDDH, HUUDD and UDHUD.
%Y Cf. A329691, A329692.
%K nonn,walk
%O 0,4
%A _Valerie Roitner_, Dec 06 2019