Zeckendorf Representations of Positive and Negative Integers by Pell Numbers

Abstract

Zeckendorf’s Theorem [14] for Fibonacci numbers, which is also given in Lekkerkerker [12], is stated by Brown [1], [2] for an integer N > 0 and a sequence {un} (where u n = F n + 1, the (n + l)th Fibonacci number, n ≥ 1) as follows: Zeckendorf’s Theorem: Every positive integer N has one and only one representation in the form $$ N = \sum\limits_{{i = 1}}^{\infty } {{\alpha_i}{u_i}} $$ (1.1) where each α i is a binary digit (i.e., with value 0 or 1) and $$ {\alpha_i}\;{\alpha_{{i + 1}}} = 0\;for\;i \geqslant 1. $$ (1.2)