On (a,b) Pairs in Random Fibonacci Sequences

TL;DR

This paper investigates the properties of random walks in the infinite binary tree generated by a recursive rule involving Fibonacci-like operations, focusing on the occurrence probabilities of coprime pairs and their distribution.

Contribution

It extends previous work by quantifying the probability of a single (1,1) pair in a random walk and providing bounds on the frequency of any coprime pair at fixed depths.

Findings

Probability of exactly one (1,1) pair in the walk is determined.

Tight bounds are established for the occurrence of any coprime pair at fixed depths.

The study generalizes the understanding of pair distributions in the random Fibonacci tree.

Abstract

We study the random Fibonacci tree, which is an infinite binary tree with non-negative 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.