On The Circulant matrices with Ducci Sequences and Fibonacci Numbers

Abstract

A Ducci sequence generated by A=(a₁,a₂,...,a_{n})∈ℤⁿ is the sequence {A,DA,D²A,...} where the Ducci map D:ℤⁿ→ℤⁿ is defined by

D(A) = D(a₁,a₂,...,a_{n})
= (|a₂-a₁|,|a₃-a₂|,...,|a_{n}-a_{n-1}|,|a_{n}-a₁|).

In this study, we examine some properties of the matrices C_{n}, DC_{n}, D²C_{n}, where C_{n}=Circ(c₀,c₁,...,c_{n-1}) is a circulant matrix whose entries consist of Fibonacci numbers.