Embedding Fibonacci Words into Fibonacci Word Patterns

Abstract

Given two initial words ω₁ ω₂, two sequences of Fibonacci words {ω_n} are defined recursively for n ≥ 3 by

$$ w_n^1 = w_{{n - 2}}^1w_{{n - 2}}^1 $$

(1)

(producing sequences labeled F ¹), and

$$ w_n^0 = w_{{n - 1}}^0w_{{n - 2}}^0 $$

(2)

(producing sequences labeled F ₀). Such sequences of words have been considered by many mathematicians (see [1]–[12]) and are related to Fibonacci trees [10], Fibonacci word patterns ([11] and [12]), golden sequences [10], the sequence [nθ] ([1]), symmetric words [4] and the well-known rabbit problem [8].