Identities for Tribonacci-related sequences

TL;DR

This paper explores identities connecting Tribonacci-related sequences, introduces a Tribomatrix extending Fibonacci matrices, and relates one sequence to determinants of matrix minors, revealing structural parallels to Fibonacci and Lucas sequences.

Contribution

It introduces the Tribomatrix and establishes new identities linking Tribonacci-related sequences and matrix determinants, expanding understanding of these sequences' algebraic properties.

Findings

Identified identities relating two Tribonacci-related sequences.

Defined the Tribomatrix extending Fibonacci matrices.

Connected one sequence to determinants of matrix minors.

Abstract

We establish some identities relating two sequences that are, as explained, related to the Tribonacci sequence. One of these sequences bears the same resemblance to the Tribonacci sequence as the Lucas sequence does to the Fibonacci sequence. Defining a matrix that we call Tribomatrix, which extends the Fibonacci matrix, we see that the other sequence is related to the sum of the determinants of the 2nd order principal minors of this matrix.