Quadratic Approximation of Generalized Tribonacci Sequences

TL;DR

This paper develops a quadratic approximation for generalized Tribonacci sequences, enabling matrix representation and re-derivation of key identities and formulas, including Cassini-type and Binet's formula for quaternions.

Contribution

It introduces a quadratic approximation method for generalized Tribonacci sequences and applies it to derive matrix forms and classical identities.

Findings

Quadratic approximation of generalized Tribonacci sequences.

Matrix form of the sequence's companion matrix powers.

Re-derivation of Cassini-type and Binet's formulas.

Abstract

In this paper, we give quadratic approximation of generalized Tribonacci sequence $\{V_{n}\}_{n\geq0}$ defined by Eq. (\ref{eq:7}) and use this result to give the matrix form of the $n$-th power of a companion matrix of $\{V_{n}\}_{n\geq0}$. Then we re-prove the cubic identity or Cassini-type formula for $\{V_{n}\}_{n\geq0}$ and the Binet's formula of the generalized Tribonacci quaternions.