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A368299 a(n) is the number of permutations pi of [n] that avoid {231,321} so that pi^4 avoids 132.

%I A368299 #42 Jan 16 2024 16:20:10
%S A368299 0,1,2,4,7,13,23,41,72,127,223,392,688,1208,2120,3721,6530,11460,
%T A368299 20111,35293,61935,108689,190736,334719,587391,1030800,1808928,
%U A368299 3174448,5570768,9776017,17155714,30106180,52832663,92714861,162703239,285524281,501060184,879299327
%N A368299 a(n) is the number of permutations pi of [n] that avoid {231,321} so that pi^4 avoids 132.
%C A368299 Number of compositions of n of the form d_1+d_2+...+d_k=n where d_i is in {1,2,4} if i>1 and d_1 is any positive integer.
%H A368299 Alois P. Heinz, <a href="/A368299/b368299.txt">Table of n, a(n) for n = 0..2000</a>
%H A368299 Kassie Archer and Aaron Geary, <a href="https://arxiv.org/abs/2312.14351">Powers of permutations that avoid chains of patterns</a>, arXiv:2312.14351 [math.CO], 2023.
%H A368299 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,1,-1).
%F A368299 G.f.: x/((1-x)*(1-x-x^2-x^4)).
%F A368299 a(n) = Sum_{m=0..n-1} Sum_{r=0..floor(m/4)} Sum_{j=0..floor((m-4*r)/2)} binomial(m-3*r-j,r)*binomial(m-4*r-j,j).
%F A368299 a(n) = 1+a(n-1)+a(n-2)+a(n-4) where a(0)=0, a(1)=1, a(2)=2, a(3)=4.
%F A368299 a(n) = A274110(n+1) - 1.
%p A368299 a:= proc(n) option remember;
%p A368299      `if`(n<1, 0, 1+add(a(n-j), j=[1, 2, 4]))
%p A368299     end:
%p A368299 seq(a(n), n=0..37);  # _Alois P. Heinz_, Dec 20 2023
%t A368299 LinearRecurrence[{2,0,-1,1,-1},{0,1,2,4,7},38] (* _Stefano Spezia_, Dec 21 2023 *)
%Y A368299 Cf. A000071 (d_i in {1,2}), A077868 (d_i in {1,3}), A274110, A303666.
%Y A368299 Partial sums of A181532.
%K A368299 nonn,easy
%O A368299 0,3
%A A368299 _Kassie Archer_, Dec 20 2023