A269121 Number of sequences with n copies each of 1,2,3 avoiding the pattern 123.

1, 5, 43, 374, 3199, 26945, 224296, 1850672, 15168751, 123700631, 1004849563, 8137906034, 65749071892, 530209766156, 4269268590280, 34334967161024, 275866609658095, 2214734413345667, 17769222356447761, 142492124882039978, 1142168892887289319

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

a(n) = A006480(n) - A047911(n).

From Seiichi Manyama, Apr 30 2026: (Start)

a(n) = [x^(3*n)] (1-x) * (1+x)^(3*n) * (Sum_{k=0..n} x^k)^3.

a(n) = 2^(3*n) - 3 * Sum_{k=0..n-1} binomial(3*n,k). (End)

Example

a(0) = 1: the empty sequence.
a(1) = 5: 132, 213, 231, 312, 321.
a(2) = 43: 113322, 131322, 133122, 133212, 133221, 211332, 213132, 213312, 213321, 221133, 221313, 221331, 223113, 223131, 223311, 231132, 231312, 231321, 232113, 232131, 232311, 233112, 233121, 233211, 311322, 313122, 313212, 313221, 321132, 321312, 321321, 322113, 322131, 322311, 323112, 323121, 323211, 331122, 331212, 331221, 332112, 332121, 332211.

Prog

(PARI) a(n) = 2^(3*n)-3*sum(k=0, n-1, binomial(3*n, k)); \\ Seiichi Manyama, Apr 30 2026

Crossrefs

Row n=3 of A269129.

Cf. A006480, A047911, A087809.

Sequence in context: A038140 A178826 A082311 * A326884 A241707 A322246

Adjacent sequences: A269118 A269119 A269120 * A269122 A269123 A269124

Keyword

nonn

Author

Alois P. Heinz, Feb 19 2016

Status

approved