Fixed-Term Decompositions Using Even-Indexed Fibonacci Numbers

TL;DR

This paper explores the unique decomposition of positive integers into even-indexed Fibonacci numbers, characterizing those integers that lack certain Fibonacci terms, extending prior Zeckendorf-related research.

Contribution

It provides a complete characterization of integers without even-indexed Fibonacci numbers in their decompositions, building on previous Zeckendorf decomposition studies.

Findings

Characterization of integers without even-indexed Fibonacci numbers

Extension of Zeckendorf decomposition research

Connections to prior Fibonacci decomposition work

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.