On the average growth rate of random compositions of fibonacci and padovan recurrences

Abstract

An integer sequence {t n } defined by the random recurrence t 0=0 t 1=1, t 2=1, t n+1=t n-ξ +t n-1-ξ , n ≥ 2, where the random variable ξ is equal to 0 or 1 with the probabilities p and q respectively, is called a random composition of Fibonacci and Padovan recurrences. We show that lim n→∞ n√E(tn)is equal to the greatest absolute value of the roots of the algebraic equation λ 3=p λ 2+λ+q. © 2009 Springer Berlin Heidelberg.