A Note on the Abelian Complexity of the Rudin-Shapiro Sequence

Let {r(n)}n & GE;0 be the Rudin-Shapiro sequence, and let rho(n):=max{ n-ary sumation j=ii+n-1r(j) divide i & GE;0}+1 be the abelian complexity function of the Rudin-Shapiro sequence. In this note, we show that the function rho(n) has many similarities with the classical summatory function Sr(n):= n-ary sumation i=0nr(i). In particular, we prove that for every positive integer n, 3 & LE;rho(n)n & LE;3. Moreover, the point set {rho(n)n:n & GE;1} is dense in [3,3].