On Generalized Zeckendorf Decompositions and Generalized Golden Strings

TL;DR

This paper explores the relationship between generalized Zeckendorf decompositions, which extend Fibonacci representations, and generalized golden strings, revealing new connections in number theory and combinatorics.

Contribution

It establishes a link between the $n$-decomposition of integers and generalized golden strings, extending previous work on Fibonacci-based representations.

Findings

Demonstrates a relationship between $n$-decompositions and generalized golden strings

Extends Griffiths' work on Zeckendorf decompositions and golden strings

Provides new insights into number representations and combinatorial structures

Abstract

Zeckendorf proved that every positive integer has a unique representation as a sum of non-consecutive Fibonacci numbers. A natural generalization of this theorem is to look at the sequence defined as follows: for $n\ge 2$, let $F_{n,1} = F_{n,2} = \cdots = F_{n,n} = 1$ and $F_{n, m+1} = F_{n, m} + F_{n, m+1-n}$ for all $m\ge n$. It is known that every positive integer has a unique representation as a sum of $F_{n,m}$'s where the indexes of summands are at least $n$ apart. We call this the $n$-decomposition. Griffiths showed an interesting relationship between the Zeckendorf decomposition and the golden string. In this paper, we continue the work to show a relationship between the $n$-decomposition and the generalized golden string.