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Properties of Multidimensional Vector Zeckendorf Representations

Ivan Bortnovskyi, June Duvivier, Pedro Espinosa, Michael Lucas, Steven J. Miller, Tiancheng Pan, Arman Rysmakhanov, Iana Vranesko, Ren Watson, Steven Zanetti

Abstract

Zeckendorf's Theorem says that for all $k \geq 3$, 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 extend this result to the multidimensional context: letting the $k$-bonacci vectors $\vec{\mathbf{X}}_i \in \mathbb{Z}^{k-1}$ be given by $\vec{\mathbf{X}}_0=\vec{\mathbf{0}}$, $\vec{\mathbf{X}}_{-i}=\vec{\mathbf{e}}_i$ for $1 \leq i \leq k-1$, and $\vec{\mathbf{X}}_n=\sum_{i=1}^k \vec{\mathbf{X}}_{n-i}$ for all $n \in \mathbb{Z}$, they show that for all $k \geq 3$, every $\vec{\mathbf{v}} \in \mathbb{Z}^{k-1}$ has a unique $k$-bonacci vector Zeckendorf representation, a sum of distinct $k$-bonacci vectors where no $k$ consecutive $k$-bonacci vectors are present in the representation. Their proof provides an inductive algorithm for finding such representations. We present two improved algorithms for finding the $k$-bonacci vector Zeckendorf representation of $\vec{\mathbf{v}}$ and analyze their relative efficiency. We utilize a projection map $S_n:\mathbb Z^{k-1} \to \mathbb Z_{\geq 0}$, introduced in Anderson and Bicknell-Johnson work, that reduces the study of $k$-bonacci vector representations to the setting of $k$-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 $\vec{\mathbf{v}}$. Using this map and a bijection between $\mathbb Z^{k-1}$ and $\mathbb Z_{\geq 0}$, 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.

Properties of Multidimensional Vector Zeckendorf Representations

Abstract

Zeckendorf's Theorem says that for all , every nonnegative integer has a unique -Zeckendorf representation as a sum of distinct -bonacci numbers, where no consecutive -bonacci numbers are present in the representation. Anderson and Bicknell-Johnson extend this result to the multidimensional context: letting the -bonacci vectors be given by , for , and for all , they show that for all , every has a unique -bonacci vector Zeckendorf representation, a sum of distinct -bonacci vectors where no consecutive -bonacci vectors are present in the representation. Their proof provides an inductive algorithm for finding such representations. We present two improved algorithms for finding the -bonacci vector Zeckendorf representation of and analyze their relative efficiency. We utilize a projection map , introduced in Anderson and Bicknell-Johnson work, that reduces the study of -bonacci vector representations to the setting of -bonacci number representations, provided a lower bound is established for the most negatively indexed -bonacci vector present in the -bonacci vector Zeckendorf representation of . Using this map and a bijection between and , we further show that the number of and gaps between summands in -bonacci vector Zeckendorf representations exhibit the same properties as those in -Zeckendorf representations and that -bonacci vector Zeckendorf representations exhibit summand minimality.
Paper Structure (16 sections, 23 theorems, 43 equations, 1 figure)

This paper contains 16 sections, 23 theorems, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. History and Motivation
  3. Main Results
  4. Improved Algorithms for Finding Vector Representations
  5. Improved Algorithms for Finding Vector Representations
  6. Small Steps Algorithm
  7. Large Steps Algorithm
  8. Analysis of Large Steps Algorithm
  9. Relative Efficiency of Algorithms for Finding SRs
  10. Properties of Vector Representations
  11. Overview
  12. Distribution of the Number of Summands
  13. Distribution of Gaps Between Summands
  14. Summand Minimality
  15. Future Work
  16. ...and 1 more sections

Key Result

Theorem 1.2

Every nonnegative integer $n$ can be written uniquely as a sum of distinct $k$-bonacci numbers $n=\sum_{i \geq 2}c_i x_i$ such that $c_i\in \{0,1\}$ for all $i$ and no $k$ consecutive $c_i$'s are equal to $1$.

Figures (1)

  • Figure 2.4: Scatter plot of $j_{lsb}$ and the proposed upper bound with 1000 randomly generated vectors

Theorems & Definitions (49)

  • Definition 1.1: $k$-bonacci Sequence
  • Theorem 1.2: Zeckendorf
  • Definition 1.3: $k$-bonacci Number Greedy Algorithm
  • Definition 1.4: $k$-bonacci Vectors
  • Theorem 1.5
  • Definition 1.6: Satisfying Representation
  • Definition 1.7: $k$-bonacci Projection Map
  • Lemma 1.8
  • Lemma 1.9
  • Definition 1.10: Big-$O$ Notation
  • ...and 39 more