We consider generalized Fibonacci sequences with recurrence
relation xn+p+1 = xn+p + xn, which have growth rates of the form
limn→∞ |xn|1/n that behave similarly to the golden ratio, (1 + √5)/2.
Following Makover and McGowan’s analysis of the random Fibonacci se-
quence, we find bounds for the value of E(|xn|)1/n for random sequences
given by xn+p+1 = ±xn+p + xn. Finally, we further generalize these ran-
dom sequences using two parameters, p and q, and we experimentally
observe how limn→∞ |xn|1/n contains surprising information about the
divisors of q + 1