Integers Having F_2k in Both Zeckendorf And Chung-Graham Decompositions

2025/04/28 by Lucas Bustos, Hùng Việt Chu, Bustos, Lucas +9 · 1 citation
Physics and Astronomy · Computer Science · Mathematics · #Advanced Mathematical Theories and Applications #semigroups and automata theory #Advanced Combinatorial Mathematics

paper · pdf · doi:10.48550/arxiv.2504.20286

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 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.

Cited by

• Shifting Zeckendorf and Chung-Graham representations

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• Fixed-Term Decompositions Using Even-Indexed Fibonacci Numbers
• Zeckendorf's Theorem Using Indices in an Arithmetic Progression
• The Chung-Graham Expansion
• Zeckendorf Representations Using Negative Fibonacci Numbers
• On the number of summands in Zeckendorf decompositions