Abelian Complexity Function of the Tribonacci Word

According to a result of Richomme, Saari and Zamboni, the abelian complexity of the Tribonacci word satisfies rho(ab)(n) is an element of {3, 4, 5, 6, 7} for each n is an element of 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))The N is an automatic sequence in a generalized sense. Since our evaluation of pab (n) uses 0(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) 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.