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A345733
Numbers k such that there are two distinct abelian squares of order k in the tribonacci word A080843.
0
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
1,1
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
Cf. A080843.
Sequence in context: A085803 A168573 A139578 * A297729 A156662 A027443
KEYWORD
nonn
AUTHOR
Jeffrey Shallit, Jun 25 2021
STATUS
approved