The fractal nature of the Fibonomial triangle

TL;DR

This paper explores the fractal structure of Fibonomial coefficients modulo two and three, revealing self-similar patterns and providing three different proofs for these properties.

Contribution

It demonstrates that the Fibonomial triangle exhibits fractal behavior modulo two and three, with three distinct proofs for these phenomena.

Findings

Fibonomial triangle shows self-similarity modulo two.

Subtriangle duplication pattern identified and proven.

Similar structure observed modulo three.

Abstract

It is well known that Pascal's triangle exhibits fractal behavior when reduced modulo a prime. We show that the triangle of Fibonomial coefficients has a similar nature modulo two. Specifically, for any $m \ge 0$, the subtriangle consisting of the first $3 \cdot 2^m$ rows is duplicated on the left and right sides of the next $3 \cdot 2^m$ rows, with an inverted triangle of zeros in between. We give three proofs of this fact. The first uses a combinatorial interpretation of the Fibonomials due to Sagan and Savage. The second employs an analogue of Lucas' congruence for the parity of binomial coefficients. The final one is inductive. We also use induction to show that the Fibonomial triangle has a similar structure modulo three. We end with some open questions.