Properties of Multidimensional Vector Zeckendorf Representations

Abstract

A generalization of Zeckendorf’s theorem states for integer k≥3, every nonnegative integer has a unique k-Zeckendorf representation as a sum of distinct k-bonacci numbers, where no k consecutive k-bonacci numbers are present in the representation. Anderson and Bicknell Johnson extended this to the multidimensional context: letting the k-bonacci vectorsX→i∈Zk−1 be given by X→0=0→, X→−i=e→i for 1≤i≤k−1, and X→n=∑i=1kX→n−i for all n∈Z, they proved for k≥3, every v→∈Zk−1 has a unique k-bonacci vector Zeckendorf representation, a sum of distinct k-bonacci vectors where no k consecutive k-bonacci vectors are in the representation. We present two improved algorithms for finding the k-bonacci vector Zeckendorf representation of v→ and analyze their relative efficiency. We reduce the study of k-bonacci vector representations to k-bonacci number representations, provided a lower bound is established for the most negatively indexed k-bonacci vector present in the k-bonacci vector Zeckendorf representation of v→. We further show that the number of and gaps between summands in k-bonacci vector Zeckendorf representations exhibit the same properties as those in k-Zeckendorf representations and that k-bonacci vector Zeckendorf representations exhibit summand minimality.