The Hurt-Sada Array and Zeckendorf Representations

TL;DR

This paper explores the Hurt-Sada array and Sada's sequence, revealing their deep connections with the golden ratio and Zeckendorf representation, and introduces related sequences.

Contribution

It establishes the relationship between the Hurt-Sada array, Sada's sequence, the golden ratio, and Zeckendorf representation, providing new insights into their interconnections.

Findings

The Hurt-Sada array relates to the golden ratio.

Sada's sequence connects with Zeckendorf representation.

Related sequences exhibit similar properties.

Abstract

Wesley Ivan Hurt and Ali Sada both independently proposed studying an infinite array where the $0$'th row consists of the non-negative integers $0,1,2,\ldots$ in increasing order. Thereafter the $n$'th row is formed from the $(n-1)$'th row by "jumping" the single entry $n$ by $n$ places to the right. Sada also defined a sequence $s(n)$ defined to be the first number that $n$ jumps over. In this note I show how the Hurt-Sada array and Sada's sequence are intimately connected with the golden ratio $\varphi$ and Zeckendorf representation. I also consider a number of related sequences.