The trinomial transform triangle

TL;DR

This paper explores the properties of Pascal-like triangles generated by the trinomial transform, focusing on ternary linear recurrent sequences and their sums, providing new insights into their structure and behavior.

Contribution

It introduces the analysis of Pascal-like triangles under the trinomial transform, specifically for ternary linear recurrent sequences, including sum and alternating sum formulas.

Findings

Derived formulas for sums of columns in trinomial transform triangles

Established properties of alternating sums in these triangles

Provided examples illustrating the structure of the trinomial transform triangle

Abstract

The trinomial transform of a sequence is a generalization of the well-known binomial transform, replacing binomial coefficients with trinomial coefficients. We examine Pascal-like triangles under trinomial transform, focusing on the ternary linear recurrent sequences. We determine the sums and alternating sums of the elements in columns, and we give some examples of the trinomial transform triangle.