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Tricomplex Fibonacci Numbers: A New Family of Fibonacci-Type Sequences

by Eudes A. Costa, Paula M. M. C. Catarino, Francival S. Monteiro, Vitor M. A. Souza and Douglas C. Santos

Mathematics 2024, 12(23), 3723; https://doi.org/10.3390/math12233723 - 27 Nov 2024

Cited by 4 | Viewed by 1739

Abstract

In this paper, we define a novel family of arithmetic sequences associated with the Fibonacci numbers. Consider the ordinary Fibonacci sequence

{f_{n}}_{n \in N_{0}}

having initial terms

f_{0} = 0

, and

f_{1} = 1

and [...] Read more.

In this paper, we define a novel family of arithmetic sequences associated with the Fibonacci numbers. Consider the ordinary Fibonacci sequence

{f_{n}}_{n \in N_{0}}

having initial terms

f_{0} = 0

, and

f_{1} = 1

and recurrence relation

f_{n} = f_{n - 1} + f_{n - 2} (n \geq 2)

. In many studies, authors worked on the generalizations of integer sequences in different ways, some by preserving the initial terms, others by preserving the recurrence relation, and some for numeric sets other than positive integers. Here, we will follow the third path. So, in this article, we study a new extension

t f_{n}^{*}

, with initial terms

t f_{0}^{*} = (f_{0}^{*}, f_{1}^{*}, f_{2}^{*})

and

t f_{1}^{*} = (f_{1}^{*}, f_{2}^{*}, f_{3}^{*})

, which is generated by the recurrence relation

t f_{n}^{*} = t f_{n - 1}^{*} + t f_{n - 2}^{*} (n \geq 2)

, the Fibonacci-type sequence. The aim of this paper is to define Tricomplex Fibonacci numbers as an extension of the Fibonacci sequence and to examine some of their properties such as the recurrence relation, summation formula and generating function, and some classical identities. Full article

(This article belongs to the Section E1: Mathematics and Computer Science)

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