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 A329666 #28 Oct 24 2023 12:51:14
%S A329666 1,1,1,3,4,7,15,26,50,102,196,392,800,1609,3290,6786,13973,28998,
%T A329666 60469,126295,264945,557594,1176004,2487485,5274110,11204631,23854581,
%U A329666 50881939,108715072,232671125,498724913,1070525053,2301048452,4952319218,10671175097,23020363339
%N A329666 Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UU and HH.
%C A329666 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.
%C A329666 a(n) also counts excursions avoiding the consecutive steps HH and DD. This can easily be seen by time reversal.
%C A329666 a(n) also counts excursions avoiding the consecutive steps HH and DU.
%H A329666 Michael De Vlieger, <a href="/A329666/b329666.txt">Table of n, a(n) for n = 0..2857</a>
%H A329666 Helmut Prodinger, <a href="https://arxiv.org/abs/2310.12497">Motzkin paths of bounded height with two forbidden contiguous subwords of length two</a>, arXiv:2310.12497 [math.CO], 2023.
%F A329666 G.f.: (1/2)*(1 - t^3 - t^2 - sqrt(t^6 + 2*t^5 - 3*t^4 - 6*t^3 - 2*t^2 + 1))/t^3.
%F A329666 a(0) = a(1) = a(2) = 1; a(n) = a(n-2) + a(n-3) + Sum_{k=0..n-3} a(k) * a(n-k-3). - _Ilya Gutkovskiy_, Nov 09 2021
%F A329666 D-finite with recurrence (n+3)*a(n) +2*-n*a(n-2) +3*(-2*n+3)*a(n-3) +3*(-n+3)*a(n-4) +(2*n-9)*a(n-5) +(n-6)*a(n-6)=0. - _R. J. Mathar_, Jan 25 2023
%e A329666 a(3)=3 as there are 3 excursions of length 3, namely: UDH, UHD and HUD.
%t A329666 CoefficientList[Series[(1/2)*(1 - x^3 - x^2 - Sqrt[x^6 + 2*x^5 - 3*x^4 - 6*x^3 - 2*x^2 + 1])/x^3, {x, 0, 40}], x] (* _Michael De Vlieger_, Oct 24 2023 *)
%Y A329666 See also A329667, A329668, A329669, which count meanders with the same step set and forbidden consecutive steps "UU and HH", "HH and DU" as well as "HH and DD" respectively.
%K A329666 nonn,walk
%O A329666 0,4
%A A329666 _Valerie Roitner_, Nov 19 2019