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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.
1
0, 1, 2, 4, 7, 13, 23, 41, 72, 127, 223, 392, 688, 1208, 2120, 3721, 6530, 11460, 20111, 35293, 61935, 108689, 190736, 334719, 587391, 1030800, 1808928, 3174448, 5570768, 9776017, 17155714, 30106180, 52832663, 92714861, 162703239, 285524281, 501060184, 879299327
OFFSET
0,3
COMMENTS
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.
LINKS
Kassie Archer and Aaron Geary, Powers of permutations that avoid chains of patterns, arXiv:2312.14351 [math.CO], 2023.
FORMULA
G.f.: x/((1-x)*(1-x-x^2-x^4)).
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).
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.
a(n) = A274110(n+1) - 1.
MAPLE
a:= proc(n) option remember;
`if`(n<1, 0, 1+add(a(n-j), j=[1, 2, 4]))
end:
seq(a(n), n=0..37); # Alois P. Heinz, Dec 20 2023
MATHEMATICA
LinearRecurrence[{2, 0, -1, 1, -1}, {0, 1, 2, 4, 7}, 38] (* Stefano Spezia, Dec 21 2023 *)
CROSSREFS
Cf. A000071 (d_i in {1,2}), A077868 (d_i in {1,3}), A274110, A303666.
Partial sums of A181532.
Sequence in context: A136299 A208354 A003116 * A303666 A260917 A165648
KEYWORD
nonn,easy
AUTHOR
Kassie Archer, Dec 20 2023
STATUS
approved