A329687 - OEIS

%I #14 Feb 21 2020 07:58:58

%S 1,1,0,1,1,0,2,2,0,5,5,0,14,14,0,42,42,0,132,132,0,429,429,0,1430,

%T 1430,0,4862,4862,0,16796,16796,0,58786,58786,0,208012,208012,0,

%U 742900,742900,0,2674440,2674440,0,9694845,9694845,0,35357670,35357670,0,129644790,129644790,0,477638700

%N Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UU, UD, HH 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 on the x-axis and never crossing the x-axis, i.e., staying at nonnegative altitude.

%F G.f.: (1+t)*(1-sqrt(1-4t^3))/(2t^3).

%F D-finite with recurrence: +(n+3)*a(n) +(n+1)*a(n-1) +2*(-2*n+3)*a(n-3) +2*(-2*n+7)*a(n-4)=0. - _R. J. Mathar_, Feb 21 2020

%e a(6)=2 since we have the following two excursions of length 6: UHDUHD and UHUHDD.

%Y Cf. A000108.

%K nonn,walk

%O 0,7

%A _Valerie Roitner_, Nov 29 2019