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%I #12 Jan 30 2020 21:29:18
%S 1,1,1,3,4,6,12,20,33,61,109,191,349,639,1159,2133,3953,7311,13595,
%T 25417,47570,89272,168126,317226,599699,1136403,2157363,4102113,
%U 7813560,14906230,28476388,54475340,104347011,200113007,384207955,738468129,1420824404,2736345674,5274795212
%N Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UU, HH and DD.
%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^2-2*t^3-(1+t)*sqrt(1-2*t+t^2-4*t^3+4*t^4))/(2*t^4).
%F D-finite with recurrence: (n+4)*a(n) +(-n-4)*a(n-1) +(-n+2)*a(n-2) -3*n*a(n-3) +6*a(n-4) +4*(n-5)*a(n-5)=0. - _R. J. Mathar_, Jan 09 2020
%e a(4)=4 since we have 4 excursions of length 4, namely UHDH, UDUD, HUHD and HUDH.
%Y Cf. A329665, which counts meanders avoiding consecutive UU, HH and DD steps.
%K nonn,walk
%O 0,4
%A _Valerie Roitner_, Nov 26 2019
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