Growth rate of binary words avoiding xxxR

Consider the set of those binary words with no non-empty factors of the form xxx R. Du, Mousavi, Schaeffer, and Shallit asked whether this set of words grows polynomially or exponentially with length. In this paper, we demonstrate the existence of upper and lower bounds on the number of such words of length n, where each of these bounds is asymptotically equivalent to a (different) function of the form Cn lg n+c , where C, c are constants and lg n denotes the base-2 logarithm of n.