Kernel words and gap sequence of the tribonacci sequence

The numbers of distinct and repeated squares and cubes in the Tribonacci sequence

“…Let Ker(ω) be the maximal kernel word occurring in factor ω, then by Theorem 4.3 in [3], Ker(ω) occurs in ω only once. Moreover, Property 2.1 (Theorem 4.11 in [3]). Ker(ω p ) = Ker(ω) p for all ω ∈ T and p ≥ 1.…”

“…The Tribonacci sequence T is the fixed point of the substitution σ(a, b, c) = (ab, ac, a). As a natural generalization of the Fibonacci sequence, the Tribonacci sequence has been studied extensively by many authors, see [3,6,7].…”

“…Taking "square" for example, the main difficulty is twofold: (1) The positions of all squares are not easy to be determined. We overcome this difficulty by using the "gap sequence" property of T, which we introduced and studied in [3]. (2) By the gap sequence property of T, we can find out all distinct squares in T [1, n].…”

Envelope Words and Return Words Sequences in the Period-doubling Sequence

“…Huang-Wen [6,7] determine the structure of the return word sequences of the Fibonacci sequence F and the Tribonacci sequence T, respectively. More precisely, for any factor ω of F (resp.…”

“…However, in the studies of the period-doubling sequence D, the techniques of kernel words fail. In fact, there is no kernel set in D, which satisfies the "uniqueness of kernel decomposition", see [6,7]. To overcome this difficulty, we introduce a new notion called "envelope word".…”