# On (a,b) Pairs in Random Fibonacci Sequences

### J.C. Saunders (Ben-Gurion University)

*Thursday, May 23, 2019,
11:10 – 12:00*,
**-101**

**Abstract:**

We deal with the random Fibonacci tree, which is an inﬁnite binary tree with nonnegative integers at each node. The root consists of the number 1 with a single child, also the number 1. We deﬁne the tree recursively in the following way: if x is the parent of y, then y has two children, namely | x−y | and x+y. This tree was studied by Benoit Rittaud who proved that any pair of integers a,b that are coprime occur as a parent-child pair inﬁnitely often. We extend his results by determining the probability that a random inﬁnite walk in this tree contains exactly one pair (1,1), that being at the root of the tree. Also, we give tight upper and lower bounds on the number of occurrences of any speciﬁc coprime pair (a,b) at any given ﬁxed depth in the tree. |