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%I #16 Oct 24 2024 15:09:16
%S 4,6,7,11,13,17,18,20,24,26,27,30,31,33,37,38,40,41,42,43,44,48,50,51,
%T 55,57,61,62,63,64,68,70,74,75,77,79,81,85,86,87,88,92,94,95,98,99,
%U 101,105,107,108,111,112,114,116,118,119,122,123,125,129,131,132
%N Orders of abelian cubes in the tribonacci word A080843.
%C 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.
%H Pierre Popoli, Jeffrey Shallit, and Manon Stipulanti, <a href="https://arxiv.org/abs/2410.02409">Additive word complexity and Walnut</a>, arXiv:2410.02409 [math.CO], 2024. See p. 17.
%F 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.
%e Here are the earliest-appearing abelian cubes of the first few orders:
%e n = 4: 2010.0102.0102
%e n = 6: 102010.010201.010201
%e n = 7: 0102010.0102010.1020100
%e n = 11: 02010010201.01020100102.01020100102
%Y Cf. A080843.
%K nonn
%O 1,1
%A _Jeffrey Shallit_, Jun 24 2021