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Numbers m such that the Fibonacci word (A003849) has an abelian cube of order m.
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%I #13 Oct 24 2024 12:34:08

%S 0,2,3,5,6,7,8,10,11,13,15,16,18,19,21,23,24,26,27,28,29,31,32,34,36,

%T 37,39,40,42,44,45,47,49,50,52,53,55,57,58,60,61,62,63,65,66,68,70,71,

%U 73,74,76,78,79,81,82,83,84,86,87,89,91,92,94,95,96,97,99

%N Numbers m such that the Fibonacci word (A003849) has an abelian cube of order m.

%C An abelian cube is three consecutive blocks x x' x'' of the same length and having the same number of occurrences of each letter. For example, "deeded" is an abelian cube. The order of an abelian cube x x' x'' is the length of x.

%H C. F. Du, H. Mousavi, L. Schaeffer, and Jeffrey Shallit, <a href="https://doi.org/10.1142/S0129054116500386">Decision Algorithms for Fibonacci-Automatic Words, III: Enumeration and Abelian Properties</a>, Intl. J. Found. Comput. Sci. 27 (2016), 943-963.

%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. 16.

%e For m = 1 neither 000 nor 111 appears in the Fibonacci word, so 1 is not a term.

%e But for m = 2 the word 101001 appears, so 2 is a term.

%K nonn

%O 1,2

%A _Jeffrey Shallit_, Jul 23 2020