[Paper Review] The unifying formula for all Tribonacci-type octonions sequences and their properties

Gamaliel Cerda-Morales|arXiv (Cornell University)|Jul 10, 2018

Advanced Mathematical Theories and Applications16 references4 citations

TL;DR

This paper introduces a unifying formula for generalized Tribonacci-type octonion sequences by extending the concept of generalized Tribonacci numbers to octonion algebra. It derives a Binet-style formula, generating function, and key identities for these octonions, unifying known sequences like Fibonacci, Pell, and Jacobsthal octonions under a single framework based on third-order linear recurrence relations in non-associative 8-dimensional algebras.

ABSTRACT

Various families of octonion number sequences (such as Fibonacci octonion, Pell octonion and Jacobsthal octonion) have been established by a number of authors in many different ways. In addition, formulas and identities involving these number sequences have been presented. In this paper, we aim at establishing new classes of octonion numbers associated with the generalized Tribonacci numbers. We introduce the Tribonacci and generalized Tribonacci octonions (such as Narayana octonion, Padovan octonion and third-order Jacobsthal octonion) and give some of their properties. We derive the relations between generalized Tribonacci numbers and Tribonacci octonions.

Motivation & Objective

• To generalize the concept of Tribonacci numbers to octonion algebras, extending known sequences like Fibonacci and Pell octonions.
• To establish a unified framework for all Tribonacci-type octonion sequences using generalized Tribonacci numbers.
• To derive closed-form expressions (Binet-style formula) and generating functions for generalized Tribonacci octonions.
• To explore algebraic and number-theoretic properties of these octonions, including recurrence identities and characteristic equations.
• To unify diverse octonion sequences (e.g., Narayana, Padovan, third-order Jacobsthal) under a single algebraic structure.

Proposed method

• Define generalized Tribonacci numbers via a third-order linear recurrence: $ V_n = rV_{n-1} + sV_{n-2} + tV_{n-3} $ with arbitrary initial values and real parameters.
• Construct generalized Tribonacci octonions as $ O_{v,n} = V_n e_0 + V_{n+1} e_1 + \cdots + V_{n+7} e_7 $, embedding the sequence into the octonion algebra.
• Use the roots $ \alpha, \omega_1, \omega_2 $ of the characteristic equation $ x^3 - rx^2 - sx - t = 0 $ to derive a Binet-style formula for $ O_{v,n} $.
• Apply techniques from Horadam and Iyer on Fibonacci quaternions, adapted to octonions, using symmetric functions of roots and algebraic identities.
• Derive the generating function for generalized Tribonacci octonions using the closed-form expression and properties of the roots.
• Establish recurrence identities by manipulating the characteristic equation and octonion multiplication rules, proving consistency with the linear recurrence.

Experimental results

Research questions

• RQ1 How can generalized Tribonacci numbers be extended to form a coherent family of octonion sequences?
• RQ2 What is the closed-form Binet-style formula for generalized Tribonacci octonions?
• RQ3 How do the algebraic properties of octonions influence the structure of these sequences?
• RQ4 What identities and generating functions characterize the generalized Tribonacci octonion sequence?
• RQ5 Can known sequences like Jacobsthal, Padovan, and Narayana octonions be unified under this generalized framework?

Key findings

• A unifying Binet-style formula is derived for generalized Tribonacci octonions: $ O_{v,n} = \frac{P\underline{\alpha}\alpha^n}{(\alpha-\omega_1)(\alpha-\omega_2)} - \frac{Q\underline{\omega_1}\omega_1^n}{(\alpha-\omega_1)(\omega_1-\omega_2)} + \frac{R\underline{\omega_2}\omega_2^n}{(\alpha-\omega_2)(\omega_1-\omega_2)} $, where $ P, Q, R $ are defined by initial values and roots of the characteristic equation.
• The generating function for generalized Tribonacci octonions is constructed using the closed-form expression and properties of the roots $ \alpha, \omega_1, \omega_2 $.
• The recurrence identity $ \alpha^2 O_{v,n+2} + \alpha(sO_{v,n+1} + tO_{v,n}) + tO_{v,n+1} = P\underline{\alpha}\alpha^{n+2} $ is proven, linking octonion sequences to the characteristic equation.
• When $ r=s=t=1 $, $ V_0=0, V_1=V_2=1 $, the formula reduces to the classic Tribonacci octonion, confirming consistency with known cases.
• The framework successfully unifies various octonion sequences, including Narayana, Padovan, and third-order Jacobsthal octonions, under a single algebraic structure.
• The method generalizes prior work on Fibonacci and Lucas quaternions to octonions, extending the Horadam-Iyer approach to higher-dimensional non-associative algebras.