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%I #20 Mar 06 2022 09:40:03
%S 1,1,1,2,1,2,4,2,5,10,5,14,28,14,42,84,42,132,264,132,429,858,429,
%T 1430,2860,1430,4862,9724,4862,16796,33592,16796,58786,117572,58786,
%U 208012,416024,208012,742900,1485800,742900,2674440,5348880,2674440,9694845,19389690,9694845,35357670
%N Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UU, 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 on the x-axis and never crossing the x-axis, i.e., staying at nonnegative altitude.
%F G.f.: (1+t)*(1+t-2t^3-(1+t)*sqrt(1-4t^3))/(2t^4).
%F D-finite with recurrence: -(n+4)*(5*n^2-11*n-6)*a(n) +36*(-n+1)*a(n-1) +36*(n-2)*a(n-2) +2*(2*n-5)*(5*n^2-n-12)*a(n-3)=0. - _R. J. Mathar_, Jan 09 2020
%F a(n) ~ (sqrt(3)*(4 + 2^(1/3) + 2^(5/3)) - sqrt(3)*(-8 + 2^(1/3) + 2^(5/3)) * cos(2*Pi*n/3) + 3*(2^(1/3) - 2^(5/3)) * sin(2*Pi*n/3)) * 2^(2*n/3 - 1) / (sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Nov 19 2021
%e a(5)=2 since we have 2 such excursions of length 5, namely UHUDD and UDHUD.
%Y Cf. A000108.
%K nonn,walk
%O 0,4
%A _Valerie Roitner_, Nov 29 2019
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