A345717 Orders of abelian cubes in the tribonacci word A080843.

4, 6, 7, 11, 13, 17, 18, 20, 24, 26, 27, 30, 31, 33, 37, 38, 40, 41, 42, 43, 44, 48, 50, 51, 55, 57, 61, 62, 63, 64, 68, 70, 74, 75, 77, 79, 81, 85, 86, 87, 88, 92, 94, 95, 98, 99, 101, 105, 107, 108, 111, 112, 114, 116, 118, 119, 122, 123, 125, 129, 131, 132

Offset

1,1

Comments

An abelian cube is a word of the form x x' x'', where x' and x'' are permutations of x, like the English word "deeded". The order of an abelian cube is the length of x.

Links

Table of n, a(n) for n=1..62.

Pierre Popoli, Jeffrey Shallit, and Manon Stipulanti, Additive word complexity and Walnut, arXiv:2410.02409 [math.CO], 2024. See p. 17.

Formula

There is a deterministic finite automaton of 1169 states that takes n in its tribonacci representation as input and accepts if and only if there is an abelian cube of order n. It can be obtained with the Walnut theorem-prover.

Example

Here are the earliest-appearing abelian cubes of the first few orders:
n = 4:  2010.0102.0102
n = 6: 102010.010201.010201
n = 7: 0102010.0102010.1020100
n = 11: 02010010201.01020100102.01020100102

Crossrefs

Cf. A080843.

Sequence in context: A287157 A102140 A047289 * A022436 A102139 A167228

Adjacent sequences: A345714 A345715 A345716 * A345718 A345719 A345720

Keyword

nonn

Author

Jeffrey Shallit, Jun 24 2021

Status

approved