Fixed-Term Decompositions Using Even-Indexed Fibonacci Numbers

Hung Viet Chu; Aney Manish Kanji; Zachary Louis Vasseur

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arXiv · 2501.03231

Fixed-Term Decompositions Using Even-Indexed Fibonacci Numbers

Hung Viet Chu·Aney Manish Kanji·Zachary Louis Vasseur

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Abstract

As a variant of Zeckendorf's theorem, Chung and Graham proved that every positive integer can be uniquely decomposed into a sum of even-indexed Fibonacci numbers, whose coefficients are either $0, 1$, or $2$ so that between two coefficients $2$, there must be a coefficient $0$. This paper characterizes all positive integers that do not have $F_{2k}$ ($k\ge 1$) in their decompositions. This continues the work of Kimberling, Carlitz et al., Dekking, and Griffiths, to name a few, who studied such a characterization for Zeckendorf decomposition.

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BibTeX RIS

Hung Viet Chu, Aney Manish Kanji, Zachary Louis Vasseur. 2024-12-03. Fixed-Term Decompositions Using Even-Indexed Fibonacci Numbers. https://arxiv.org/abs/2501.03231

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