The Structure of a Bernoulli Process Variation of the Fibonacci Sequence

TL;DR

This paper explores a Bernoulli process-based variation of the Fibonacci sequence, representing its structure with a binary tree, and establishes a key reflection property linking tree traversal to sequence terms, with applications to Fibonacci expansions and puzzles.

Contribution

It introduces a binary tree model for Bernoulli variations of the Fibonacci sequence and proves a reflection property connecting tree traversal to sequence values, advancing understanding of such stochastic recurrences.

Findings

Binary tree representation of Bernoulli Fibonacci variations

Reflection of binary codes yields sequence terms

Connections to Fibonacci expansions and the Three Hat Puzzle

Abstract

We consider the structure of a variation of the Fibonacci sequence which is determined by a Bernoulli process. The associated structure of all Bernoulli variations of the Fibonacci sequence can be represented by a directed binary tree, which we denote X, with vertex labels representing the specific state of the recurrence variation. Since X is a binary tree, we can consider the term of a sequence variation given by a finite traversal of X represented by a binary code t. We then prove that the traversal of X that is the reflection of the digits of t gives exactly the integer term corresponding to t. We consider how to further this result with the statement of an additional conjecture. Finally, we give connections to Fibonacci expansions, the Stern-Brocot tree, and we apply our methods to the Three Hat Problem as seen in ``Puzzle Corner'' of the ``Technology Review'' magazine.