This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A329672 #9 Jan 25 2023 12:42:29 %S A329672 1,2,4,9,20,46,107,252,599,1435,3460,8389,20437,49996,122758,302401, %T A329672 747114,1850696,4595370,11435380,28513149,71225270,178219696, %U A329672 446637759,1120946389,2817089354,7088656546,17858286741,45039810918,113711798916,287369435649,726905294670,1840328917065 %N A329672 Number of meanders of length n with Motzkin-steps avoiding the consecutive steps UU. %C A329672 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 A329672 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 A329672 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 A329672 a(2)=4 since we have 4 meanders of length 2 avoiding UU, namely UH, UD, HU and HH. %Y A329672 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 A329672 nonn,walk %O A329672 0,2 %A A329672 _Valerie Roitner_, Nov 26 2019