Abstract

We exhibit and study various regularity properties of the sequence (R(n))_n1 which counts the number of different representations of the positive integer n in the Fibonacci numeration system. The regularity properties in question are observed by representing the sequence as a two-dimensional array consisting of an infinite number of rows L₁,L₂,L₃,… where each L_k contains f_k-1 (the k-1st Fibonacci number) entries of the sequence (R(n)). We give a purely combinatorial recursive algorithm for generating each row L_k from previous rows L_j with j<k. We then show that for each positive integer m, and for all k2m, the number of occurrences of m in L_k is a constant rk(m) depending only on m. The function rk(m) has many interesting number theoretic properties and is intimately connected to the Euler φ-function.

Keywords: Numeration systems; Fibonacci numbers; Generalized Euclidean algorithm