On Generalized Zeckendorf Decompositions and Generalized Golden Strings

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.