OFFSET
1,1
COMMENTS
An abelian square is a word of the form x x' where x' is a permutation of x, like the English word "reappear". The order of an abelian square x x' is the length of x.
The tribonacci word has abelian squares of all orders. If we consider two abelian squares x x' and y y' to be the same if y is a permutation of x, then some orders have only 1 abelian square (up to this equivalence), while others have 2, and these are the only possibilities. There is a 463-state automaton that recognizes the tribonacci representation of those terms k in this sequence. All this can be proved with the Walnut theorem prover.
EXAMPLE
For k = 15, the two distinct abelian squares are 100102010102010.010201001020101 and 020102010010201.010201001020102.
CROSSREFS
KEYWORD
nonn
AUTHOR
Jeffrey Shallit, Jun 25 2021
STATUS
approved