A345733 Numbers k such that there are two distinct abelian squares of order k in the tribonacci word A080843.

15, 34, 59, 90, 96, 97, 102, 134, 137, 170, 171, 172, 178, 183, 215, 240, 252, 259, 262, 289, 321, 333, 364, 370, 371, 387, 389, 391, 402, 408, 411, 445, 457, 470, 482, 489, 516, 519, 538, 556, 557, 563, 594, 600, 601, 606, 638, 665, 674, 675, 676, 682, 687

Offset: 1

Author

Jeffrey Shallit, Jun 25 2021

Keywords

• nonn

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.

Examples

			For k = 15, the two distinct abelian squares are 100102010102010.010201001020101 and 020102010010201.010201001020102.
		

Crossrefs

Cf. A080843.