Properties of Fibonacci languages

Abstract

The Fibonacci language F-u,F-nu is the set of all Fibonacci words, where the first word and the second word in the Fibonacci sequence are u and nu, respectively. We show that the language F-u,F-nu is context-free free. We also show that F-u,F-nu is not dense if the word u nu contains at least two distinct letters. Let w(i) denote the ith Fibonacci word. When considering the Fibonacci language F-a,F-b for two distinct letters a and b, we show that for k greater than or equal to 2 and 1 less than or equal to i < k, the word w(i)w(i) is not a prefix of the Fibonacci word w(k). We also show that for k greater than or equal to 2, the language F-k = {w(nk) / n greater than or equal to 1} is a code. (C) 2000 Elsevier Science B.V. All rights reserved

The solutions of a linear homogeneous recurrence relation with unit coefficients

“…However, in this paper, we proposed a new approach to solve the recurrence relation (6), which can be generalized to some general cases.…”

“…i=0 a(n, k, i) satisfies the recurrence relation (6) and 7. The values of x n (4) are shown in Table 2 in Appendix.…”

“…After this counting problem about the growth of rabbit population has been posed, the properties of the Fibonacci sequence, Fibonacci matrices and related sequences have been studied broadly and applied in many fields. For example, Turner [5] discussed the expression of the Fibonacci word patterns; Yu and Zhao [6] analyzed and proved the properties of Fibonacci languages; Filipponi [2] showed how the m-by-m symmetric tridiagonal Toeplitz matrices are Fibonacci matrices; Lin [4] discussed the properties of the Fibonacci matrices and the relations between Fibonacci matrices and the units of the field 2(θ); Zhang [7] established some properties of the generalized Fibonacci sequences. In this paper, we focus our attention on the solutions of the recursive functions from three terms to k-terms which generalize the well-known formula of the solution of (1).…”

Two-dimensional Fibonacci Words: Tandem Repeats and Factor Complexity

“…Like Fibonacci numbers, Fibonacci words/arrays are very exciting. For a detailed study of Fibonacci words [14,15] can be referred. Continuing the work done in [1] on Fibonacci arrays, we count the exact number of tandems in a given Fibonacci array f m,n .…”

Representing Nonterminating Rewriting with $\mathbf{F}_2^μ$

“…In the second reduction sequence, each string is called a Fibonacci word, and the set of all such words is known to be context-free free, i.e. any infinite subset can not be described by a context-free language [25]. We will show how to represent the second reduction sequence in Section 6.…”