Some properties of circulant matrices with Ducci sequences

A Ducci sequence is the sequence {X, DX, (DX)-X-2,...} generated by n-tuples X = (x(1), x(2), ..., x(n)) is an element of Z(n), where DX = D (x(1), x(2), ..., x(n)) = (vertical bar x(2) - x(1)vertical bar, vertical bar x(3) - x(2)vertical bar, ..., vertical bar x(n) - x(1)vertical bar). Equivalently, the Ducci sequence of n-vector X may be defined as {(DX)-X-k}, where k = 0,1,2,... and (DX)-X-k denotes k times iterated vector X. In this study, we examine properties of the matrix D (C-n), which results from applying the Ducci map to the rows of the circulant matrix C-n, with first row (c(0), c(1), ... c(n-1)). (C) 2017 Elsevier Inc. All rights reserved.