The Properties of the Lucas Triangle and its Relationship to other Integer Sequences

Abstract

This study focuses on the construction and analysis of the Lucas Triangle as a recursive triangular array of integers related to Pascal’s Triangle and Lucas numbers. The study establishes the formation of the Lucas Triangle and investigates its mathematical properties through the development and proof of various theorems, propositions, and lemmas. Several combinatorial and divisibility properties of the triangle are derived, including closed-form representations and row sum identities. Furthermore, the study explores how the entries of the Lucas Triangle generate significant integer sequences such as natural numbers, odd numbers, square numbers, Lucas numbers, Fibonacci numbers, central binomial coefficients, triangular numbers, Catalan numbers, and square pyramidal numbers. To support computation and visualization, Python programming in Google Colab was utilized to generate and analyze the entries of the Lucas Triangle. The findings reveal that the Lucas Triangle possesses rich recursive and combinatorial structures that establish meaningful relationships among various integer sequences. The study contributes to the broader investigation of triangular arrays and recursive number patterns in discrete mathematics and number theory.