Mechanical Proofs of Properties of the Tribonacci Word

Abstract

We implement a decision procedure for answering questions about a class of infinite words that might be called (for lack of a better name) "Tribonacci-automatic". This class includes, for example, the famous Tribonacci word T = 0102010010201 · · · , the fixed point of the morphism 0 → 01, 1 → 02, 2 → 0. We use it to reprove some old results about the Tribonacci word from the literature, such as assertions about the occurrences in T of squares, cubes, palindromes, and so forth. We also obtain some new results.

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

“…By convention, denote τ Let T [1, n] be the prefix of T of length n. In this paper, we consider the four functions below: In 2006, A.Glen [2] gave expressions of A(t m ). In 2014, H.Mousavi and J.Shallit [6] gave expressions of B(t m ) and D(t m ), which they proved by mechanical way. All of these results above only consider the squares or cubes in the prefixes of some special lengths: the Tribonacci numbers.…”

“…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].…”

Queens in Exile: Non-attacking Queens on Infinite Chess Boards

“…An alternative way to specify the positions of the exiled queens is by the sequence {S c : c 0} (A275895), which indicates which row contains the queen in column c (this the electronic journal of combinatorics 22 (2015), #P00 is well-defined, thanks to Theorem 23 below). The initial values of S c are 0, 2, 4, 1, 3,8,10,12,14,5,7,18,6,21,9,24,26,28,30,11,13,34, . .…”

Deciding game invariance

“…This result has been recently used to get positive results in combinatorics on words. Implementations to deal with the Fibonacci and Tribonacci numerations systems have been developped [8,25]. With these implementations (mostly relying on automata recognizing addition in these systems) many properties of the Fibonacci, Tribonacci and related infinite words are proved automatically on a laptop with computing time ranging from a few seconds to two hours.…”

“…With these implementations (mostly relying on automata recognizing addition in these systems) many properties of the Fibonacci, Tribonacci and related infinite words are proved automatically on a laptop with computing time ranging from a few seconds to two hours. The source code developped by the authors of [8,25] has not yet been publicly released. Also see [22] for an example about integer base systems.…”