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

Author

Jeffrey Shallit, Jun 24 2021

Keywords

• nonn

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.

Examples

			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
		

Links

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

Crossrefs

Cf. A080843.

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.