On the Periodicity of Certain Recursive Sequences

In 2000, Viswanath showed that random Fibonacci sequences grow exponentially and calculated the rate at which they grow assuming the coin flipped was fair. In this paper, we explore the Fibonacci sequences generated by finite, repeating sequences of pluses and minuses. The main results of this paper will be to show the necessary conditions for a sequence to be periodic, as well as to show all the possible periods of the sequences. It will be clear that the set of periodic random Fibonacci sequences is a subset of measure 0 of random Fibonacci sequences. In his 2000 paper, Viswanath [4] examined the growth properties of random Fibonacci sequences generated with a fair coin flip; his paper resulted in a new mathematical constant, 1.13198824.... His result was a specific, computed case of an earlier result due to Furstenberg and Kesten [2]. Here we will examine the properties of the subset of random sequences that are periodic. We will show that these sequences can only occur when the plus minus sequence generated by the coin flips, which corresponds to a Bernoulli sequence, is itself periodic. This, of course, happens with probability 0, but we will study the properties of these finite sequences of pluses and minuses acting on the integers. Suppose we have a recursion, R1 defined as R1 : Xn+1 = Xn−1 + Xn. We can think of Xn+1 as a vector, −→ X n+1 = ( Xn+1 Xn ) = ( Xn−1 + Xn Xn ) = ( 1 1 1 0 )( Xn Xn−1 ) . Let M1 = ( 1 1 1 0 ) , then − → X n+1 = M1 ( Xn Xn−1 ) . Similarly, if we define a second recursion, R2 as R2 : Xn+1 = Xn−1 −Xn, we have −→ X n+1 = ( Xn+1 Xn ) = ( Xn−1 −Xn Xn ) = ( −1 1 1 0 )( Xn Xn−1 ) .