Ambiguity in the m-bonacci numeration system

Abstract

We study the properties of the function R (m) (n) defined as the number of representations of an integer n as a sum of distinct m-Bonacci numbers F (m) k, given by F i (m) =2 i-1, for i∈ { 1, 2, …, m}, F k+m (m) =F k+m-1 (m) +F k+m-2 (m) + ⋯ + F k (m), for k ≥ 1. We give a matrix formula for calculating R (m) (n) from the greedy expansion of n. We determine the maximum of R (m) (n) for n with greedy expansion of fixed length k, i.e. for F (m) k ≤ n<F (m) k+1. Unlike the Fibonacci case m=2, the values of the maxima are not related to the sequence (F (m) k) k ≥ 1. We describe the palindromic structure of the sequence (R (m) (n)) n ∈ ℕ, which is richer than in the case of Fibonacci numeration system.