Fixed-Term Decompositions Using Even-Indexed Fibonacci Numbers

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 ≥ 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.