Abelian complexity function of the Tribonacci word

Turek, Ondřej

Journal of integer sequences, Tome 18 (2015) no. 3.

Résumé

Summary: According to a result of Richomme, Saari and Zamboni, the abelian complexity of the Tribonacci word satisfies $\rho ^{ab}(n) \in {3, 4, 5, 6, 7}$ for each $n \in $ N. In this paper we derive an automaton that evaluates the function $\rho ^{ab}(n)$ explicitly. The automaton takes the Tribonacci representation of $n$ as its input; therefore, $(\rho ^{ab}(n))_{n\in N}$ is an automatic sequence in a generalized sense. Since our evaluation of $\rho ^{ab}(n)$ uses $O(\log n)$ operations, it is fast even for large values of $n$. Our result also leads to a solution of an open problem proposed by Richomme et al. concerning the characterization of those $n$ for which $\rho ^{ab}(n) = c$ with $c$ belonging to ${4, 5, 6, 7}$. In addition, we apply the same approach on the 4-bonacci word. In this way we find a description of the abelian complexity of the 4-bonacci word, too.

Détail

Classification : 11B85, 68R15
Keywords: abelian complexity, tribonacci word, finite automaton, 4-bonacci word