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 A308435 #31 Jan 27 2025 10:18:56
%S A308435 1,2,4,9,20,45,102,233,535,1234,2857,6636,15456,36085,84424,197883,
%T A308435 464585,1092348,2571770,6062109,14305022,33789777,79887365,189031914,
%U A308435 447639473,1060798484,2515512091,5968826698,14171068794,33662866431,80005478832,190237068767,452548530595
%N A308435 Peak- and valleyless Motzkin meanders.
%C A308435 a(n) is the number of Motzkin meanders that avoid UD and DU. A Motzkin meander is a lattice paths that starts at (0,0), uses steps U=1, H=0, D=-1, and never goes below the x-axis.
%H A308435 Michael De Vlieger, <a href="/A308435/b308435.txt">Table of n, a(n) for n = 0..2616</a>
%H A308435 Andrei Asinowski, Axel Bacher, Cyril Banderier, and Bernhard Gittenberger, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/patterns2019.pdf">Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata</a>, Algorithmica (2019).
%H A308435 Andrei Asinowski, Axel Bacher, Cyril Banderier, and Bernhard Gittenberger, <a href="https://doi.org/10.4230/LIPIcs.AofA.2018.10">Analytic Combinatorics of Lattice Paths with Forbidden Patterns: Asymptotic Aspects and Borges's Theorem</a>, 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018).
%H A308435 Andrei Asinowski, Cyril Banderier, and Valerie Roitner, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/several_patterns.pdf">Generating functions for lattice paths with several forbidden patterns</a>, (2019).
%H A308435 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.
%H A308435 Helmut Prodinger, <a href="https://arxiv.org/abs/2501.13645">Cornerless, peakless, valleyless Motzkin paths (regular and skew) and applications to bargraphs</a>, arXiv:2501.13645 [math.CO], 2025. See p. 8.
%F A308435 G.f.: -(1+t-sqrt((1-t^4)/(1-2*t-t^2)))/(2*t^2).
%F A308435 D-finite with recurrence (n+2)*a(n) +(-2*n-3)*a(n-1) +(-n-1)*a(n-2) +(-n+4)*a(n-4) +(2*n-9)*a(n-5) +(n-5)*a(n-6)=0. - _R. J. Mathar_, Jan 25 2023
%e A308435 For n=3, the a(3)=9 such meanders are UUU, UUH, UHU, UHH, UHD, HUU, HUH, HHU, HHH.
%t A308435 CoefficientList[Series[-(1+x-Sqrt[(1-x^4)/(1-2*x-x^2)])/(2*x^2), {x, 0, 40}], x] (* _Vaclav Kotesovec_, Jun 05 2019 *)
%o A308435 (PARI) my(t='t + O('t^40)); Vec(-(1+t-sqrt((1-t^4)/(1-2*t-t^2)))/(2*t^2)) \\ _Michel Marcus_, May 27 2019
%Y A308435 Cf. A004149.
%K A308435 nonn
%O A308435 0,2
%A A308435 _Andrei Asinowski_, May 27 2019