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%I #12 Jan 30 2020 21:29:18
%S 1,1,0,1,2,1,1,4,5,3,7,16,16,16,40,66,65,99,211,288,329,603,1079,1372,
%T 1897,3529,5538,7219,11431,20076,29305,41141,68970,113103,162229,
%U 245454,411984,642006,939016,1491348,2444027,3715023,5619485,9095842,14510185,22008169,34300205,55456432,86830187,133182523,211375518,338423557,525898418,818766393,1308164859,2073414046,3226270813,5084761609,8117959191,12786484606
%N Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UU, 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 at (n,0) and never crossing the x-axis, i.e., staying at nonnegative altitude.
%F G.f.: (1+t)*(1-t^3-sqrt(1-2t^3-4t^4+t^6))/(2t^4).
%F D-finite with recurrence: +(n+4)*a(n) -a(n-1) +a(n-2) -2*n*a(n-3) +(-4*n+11)*a(n-4) +a(n-5) +(n-6)*a(n-6)=0. - _R. J. Mathar_, Jan 09 2020
%e a(7)=4 since we have the following 4 excursions of length 7: UHUHDDH, UHUHDHD, UHDHUHD and HUHUHDD.
%Y Cf. 329690.
%K nonn,walk
%O 0,5
%A _Valerie Roitner_, Dec 06 2019