The Hurt-Sada Array and Zeckendorf Representations

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 $0,1,2,\dots$ 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 $\ddot{I}†$ and Zeckendorf representation. I also consider a number of related sequences.