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 A356696 #25 Aug 29 2024 15:03:13
%S A356696 2,1,1,2,5,14,42,128,384,1123,3204,8955,24629,66913,180127,481568,
%T A356696 1280855,3393644,8965476,23633702,62197602,163483201,429300366,
%U A356696 1126514817,2954438135,7745187919,20297902537,53182073798,139315427369,364898425658,955648284654
%N A356696 a(n) = Fibonacci(2n-1) - 2^n + binomial(n,2) + 2.
%C A356696 If n>0, number of Catalan words of length n avoiding the pattern 1234. The unrestricted Catalan words of length n are enumerated by A000108(n-1).
%H A356696 Paolo Xausa, <a href="/A356696/b356696.txt">Table of n, a(n) for n = 0..1000</a>
%H A356696 Toufik Mansour and Mark Shattuck, <a href="https://doi.org/10.2298/FIL1703543M">Avoidance of classical patterns by Catalan sequences</a>, Filomat 31, No. 3, 543-558 (2017). Corollary 2.5.
%H A356696 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (8,-25,39,-32,13,-2).
%F A356696 G.f.: ( 2-15*x+43*x^2-59*x^3+39*x^4-9*x^5 ) / ( (2*x-1)*(x^2-3*x+1)*(x-1)^3 ).
%t A356696 LinearRecurrence[{8, -25, 39, -32, 13, -2}, {2, 1, 1, 2, 5, 14}, 50] (* or *)
%t A356696 A356696[n_] := Fibonacci[2*n - 1] - 2^n + Binomial[n, 2] + 2;
%t A356696 Array[A356696, 50, 0] (* _Paolo Xausa_, Aug 29 2024 *)
%Y A356696 Cf. A000045, A000108, A000325 (avoiding 123), A307464, A307465, A307466.
%K A356696 nonn,easy
%O A356696 0,1
%A A356696 _R. J. Mathar_, Aug 23 2022