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Zeckendorf proved that every positive integer has a unique representation asa sum of non-consecutive Fibonacci numbers. A natural generalization of thistheorem 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 uniquerepresentation as a sum of $F_{n,m}$'s where the indexes of summands are atleast $n$ apart. We call this the $n$-decomposition. Griffiths showed aninteresting relationship between the Zeckendorf decomposition and the goldenstring. In this paper, we continue the work to show a relationship between the$n$-decomposition and the generalized golden string.

On Generalized Zeckendorf Decompositions and Generalized Golden Strings