Some properties of circulant matrices with Ducci sequences

A Ducci sequence is the sequence {X,DX,D2X,...} generated by n-tuples X=(x1,x2,...,xn)∈Zn, whereDX=D(x1,x2,...,xn)=(|x2−x1|,|x3−x2|,...,|xn−x1|). Equivalently, the Ducci sequence of n-vector X may be defined as {DkX}, where k=0,1,2,... and DkX denotes k times iterated vector X. In this study, we examine properties of the matrix D(Cn), which results from applying the Ducci map to the rows of the circulant matrix Cn with first row (c0,c1,...cn−1).