The Hurt-Sada Array and Zeckendorf Representations

TL;DR

The paper investigates the Hurt-Sada array, generated by iteratively shifting the unique entry $n$ in row $n-1$ by $n$ positions, and studies Sada's sequence $s(n)$ as the first entry jumped over. It unveils deep connections to the golden ratio $\varphi$ and Zeckendorf representations, employing finite automata and the Walnut theorem prover to derive exact formulas and structural theorems for key sequences, including $p(n)$, $s(n)$, $t(n)$, $d(n)$, and $d'(n)$. It further characterizes row structure via explicit $b(n)$ and $c(n)$, analyzes antidiagonals with Run parameters $h(n),h'(n),r(n)$ (identifying $r(n)$ with OEIS A026272 and giving formulas for $h$ and $h'$), and demonstrates how automata-based verification yields rigorous first-order results. The work showcases the power of combining Zeckendorf representations with finite automata and the Walnut prover to discover and prove new combinatorial-number-theoretic properties, highlighting rich connections between the Hurt-Sada construction, the golden ratio, and related integer sequences.

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.

Figures (4)

• Figure 1: Automaton accepting even numbers in Zeckendorf representation.
• Figure 2: The automaton that decides if $d(n)\geq n$.
• Figure 3: The automaton that decides if $d'(n)\geq n$.
• Figure 4: Automaton that computes $r(n)$.

Theorems & Definitions (22)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof