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%I #9 Jan 25 2023 12:42:29
%S 1,2,4,9,20,46,107,252,599,1435,3460,8389,20437,49996,122758,302401,
%T 747114,1850696,4595370,11435380,28513149,71225270,178219696,
%U 446637759,1120946389,2817089354,7088656546,17858286741,45039810918,113711798916,287369435649,726905294670,1840328917065
%N Number of meanders of length n with Motzkin-steps avoiding the consecutive steps UU.
%C The Motzkin step set is U=(1,1), H=(1,0) and D=(1,-1). A meander is a path starting at (0,0) and never crossing the x-axis, i.e., staying at nonnegative altitude.
%F G.f.: -(1+t)*(1-t-3*t^2-sqrt(1-2*t-t^2-2*t^3+t^4))/(2*t^2*(1-2*t-2*t^2)).
%F D-finite with recurrence (n+2)*a(n) +(-3*n-5)*a(n-1) +(-3*n+2)*a(n-2) +(5*n+2)*a(n-3) +(11*n-19)*a(n-4) +(9*n-32)*a(n-5) +2*a(n-6) +2*(-n+6)*a(n-7)=0. - _R. J. Mathar_, Jan 25 2023
%e a(2)=4 since we have 4 meanders of length 2 avoiding UU, namely UH, UD, HU and HH.
%Y Cf. A004148 (shifted by 1) which counts excursions avoiding consecutive UU steps. See also A329673 and A329674 which count meanders avoiding consecutive HH and DD respectively.
%K nonn,walk
%O 0,2
%A _Valerie Roitner_, Nov 26 2019