Arithmetic Triangles and Pascal-Type Recurrence Relations

Abstract

The Arithmetic Triangle, commonly known as Pascal's Triangle, has been an object of interest for mathematicians since antiquity. The entries in the Arithmetic Triangle display interesting patterns while also having much combinatorial significance. We recount some of these patterns and exhibit a new construct on the Arithmetic Triangle: alternating products. Then, we generalize the construction of the Arithmetic Triangle by applying the same Pascal-type recurrence relation to different sets of seed values. We show that these Generalized Arithmetic Triangles still display many of the same interesting patterns as The Arithmetic Triangle, but with slight modifications determined by the seed values. By creating an order on the elements of Pascal's Triangle which captures the Pascal-type recurrence relation, we consider Pascal's Triangle through the lens of Order Theory. We conclude by considering the algebraic super-structure of the collection of all Generalized Arithmetic Triangles and see that there is a natural way to form the Arithmetic Triangle Group.