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 A383433 #6 May 03 2025 21:42:52
%S A383433 1,1,0,0,2,12,76,556,4592,42328,431184,4812936,58440200,767098296,
%T A383433 10826066776,163496360680,2631146363384,44953977477160,
%U A383433 812713132751832,15501004918724712,311078390317974872,6552553451281418472,144550752700158416920,3332886257051337065128,80168754370190239256408
%N A383433 Number of king permutations on n elements avoiding the mesh pattern (12, {(0,0),(0,2),(1,0),(1,2),(2,1)}).
%C A383433 A permutation p(1)p(2)...p(n) is a king permutation if |p(i+1)-p(i)|>1 for each 0<i<n. a(n) is the number of king permutations of length n that avoid the mesh pattern 12 with squares (0,0), (0,2), (1,0), (1,2), and (2,1) shaded.
%H A383433 Dan Li and Philip B. Zhang, <a href="https://arxiv.org/abs/2411.18131">Distributions of mesh patterns of short lengths on king permutations</a>, arXiv:2411.18131 [math.CO], 2024. See Theorem 4.7 at page 21.
%F A383433 G.f.: (2*A(t) - t - 1)/(A(t) - t) where A(t)=Sum_{n >= 0} n!*t^n*(1-t)^n/(1+t)^n is the g.f. for king permutations given by A002464.
%e A383433 For n = 4 the a(4) = 2 solutions are the two permutations 2413 and 3142.
%e A383433 For n = 5 the a(5) = 12 solutions are these 12 permutations: 13524, 14253, 24153, 25314, 31524, 35142, 35241, 41352, 42513, 42531, 52413, 53142.
%Y A383433 Cf. A002464, A382644, A382645, A382651, A383040, A383107, A383312, A383406, A383407, A383408.
%K A383433 nonn,easy
%O A383433 0,5
%A A383433 _Dan Li_, Apr 27 2025