Integers Having $F_{2k}$ in Both Zeckendorf And Chung-Graham Decompositions

Abstract

Zeckendorf's theorem states that every positive integer can be uniquely decomposed into nonadjacent Fibonacci numbers. On the other hand, Chung and Graham proved that every positive integer can be uniquely written as a sum of even-indexed Fibonacci numbers with coefficients $0,1$, or $2$ such that between two coefficients $2$, there is a coefficient $0$. We discover a correspondence between a lexicographically ordered sublist of Zeckendorf decompositions and letters in the golden string $\mathcal{S}$. Likewise, we identify a dual correspondence for Chung-Graham decompositions. We then use these correspondences to give the set of all positive integers having $F_{2k}$ in both of their Zeckendorf and Chung-Graham decompositions.