# Article ID: MTJPAM-D-21-00025

## Title: Lucas Cube vs Zeckendorf’s Lucas Code

Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-21-00025; Volume 3 / Issue 2 / Year 2021, Pages 47-50

Document Type: Research Paper

Author(s): Sadjia Abbad a , Hacène Belbachir b , Ryma Ould-Mohamed c

aUSTHB, Faculty of Mathematics, RECITS Laboratory, BP 32 El-Alia, Bab-Ezzouar 16111, Algiers, Algeria. Saad Dahlab University, BP 270, Route de Soumaa, Blida, Algeria

bUSTHB, Faculty of Mathematics, RECITS Laboratory, BP 32 El-Alia, Bab-Ezzouar 16111, Algiers, Algeria

cUSTHB, Faculty of Mathematics, RECITS Laboratory, BP 32 El-Alia, Bab-Ezzouar 16111, Algiers, Algeria. University of Algiers 1, 2 Rue Didouche Mourad, Alger Ctre 16000, Algiers, Algeria

Received: 29 March 2021, Accepted: 12 April 2021, Published: 15 May 2021.

Corresponding Author: Ryma Ould-Mohamed (Email address: ryma.ouldmohamed@gmail.com)

Full Text: PDF

Abstract

The theorem of Zeckendorf states that every positive integer n can be uniquely decomposed as a sum of non consecutive Lucas numbers in the form n = ∑i biLi, where bi ∈ {0,1} and satisfy b0b2 = 0. The Lucas string is a binary string that do not contain two consecutive 1’s in a circular way. In this note, we derive a bijection between the set of Zeckendorf’s Lucas codes and the set of vertices of the Lucas’ strings.

Keywords: Fibonacci numbers, Lucas numbers, Fibonacci cube, Lucas cube

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