Ambiguity in the m-bonacci numeration system

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