Abstract
In this paper, we introduce a word-combinatorial interpretation of the biperiodic Fibonacci sequence. We study some properties of this new family of infinite words. Moreover, we associate to this family of words a family of curves with interesting patterns.
Highlights
The Fibonacci numbers and their generalizations have many interesting properties and combinatorial interpretations, see, e.g., [13]
The Fibonacci numbers Fn are defined by the recurrence relation Fn = Fn−1 + Fn−2, for all integer n ≥ 2, and with initial values F0 = 1 = F1
We study a word-combinatorial interpretation of the biperiodic Fibonacci sequence [12]
Summary
The Fibonacci numbers and their generalizations have many interesting properties and combinatorial interpretations, see, e.g., [13]. The word f can be associated with a curve from a drawing rule, which has geometric properties obtained from the combinatorial properties of f [4, 16]. The nth-curve of Fibonacci, denoted by Fn, is obtained by applying the odd-even drawing rule to the word fn. We study a word-combinatorial interpretation of the biperiodic Fibonacci sequence [12]. This problem was recently proposed by Ramírez et al [22]. In addition to this definition, we investigate some new combinatorial properties and we associate a family of curves with interesting geometric properties. These properties are obtained from the combinatorial properties of the word f(a,b)
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