On circulant and $r$-circulant matrices with Ducci sequences and Lucas numbers

Abstract

A Ducci sequence is a sequence $\{S, DS, D^{2}S, ...\}$, where the map $D: {\mathbb{Z}}^{n} \rightarrow {{\mathbb{Z}}^{n}}$ takes each $S=(s_1, s_2, s_3, ..., s_{n-1}, s_n)$ to $(\vert s_1-s_2 \vert, \vert s_2-s_3\vert, ..., \vert s_{n-1}-s_n \vert, \vert s_n-s_1 \vert)$.  In this paper, we study norms of $r$-circulant matrices $Circ_{r}{(D{{L}})}$ and $Circ_{r}{(D^{2}{{L}})}$, where ${L}$ is an $n$-tuple of Lucas numbers. Then we examine some properties of circulant matrices $Circ{(D{{L}})}$ and $Circ{(D^{2}{{L}})}$.