%I #9 Jan 25 2023 12:44:06
%S 1,2,4,10,24,60,152,388,1000,2592,6752,17664,46368,122080,322240,
%T 852464,2259552,5999552,15954560,42486592,113282048,302386304,
%U 807999744,2161077120,5785032448,15498450944,41551965184,111478804480,299274439680,803905119232,2160632498176,5810087371520
%N Number of meanders of length n with Motzkin-steps avoiding the consecutive steps HH.
%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.: -(1-2*t-2*t^2-sqrt(1-4*t^2-8*t^3-4*t^4))/(2*t*(1-2*t-2*t^2)).
%F D-finite with recurrence (n+1)*a(n) -2*a(n-1) -4*n*a(n-2) +8*(-n+2)*a(n-3) +4*(-n+3)*a(n-4)=0. - _R. J. Mathar_, Jan 25 2023
%e a(2)=4 since we have 4 meanders of length 2 avoiding HH, namely UU, UH, UD and HU.
%Y Cf. A104545 which counts excursions avoiding consecutive HH steps. Cf. A329672 and A329674 which count meanders avoiding consecutive UU and DD respectively.
%K nonn,walk
%O 0,2
%A _Valerie Roitner_, Nov 26 2019