A345733 - OEIS

%I #22 Jul 21 2021 09:34:04

%S 15,34,59,90,96,97,102,134,137,170,171,172,178,183,215,240,252,259,

%T 262,289,321,333,364,370,371,387,389,391,402,408,411,445,457,470,482,

%U 489,516,519,538,556,557,563,594,600,601,606,638,665,674,675,676,682,687

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

%C 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.

%C 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.

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

%Y Cf. A080843.

%K nonn

%O 1,1

%A _Jeffrey Shallit_, Jun 25 2021