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

Abstract

In this paper, we define a novel family of arithmetic sequences associated with the Fibonacci numbers. Consider the ordinary Fibonacci sequence {fn}n∈N0 having initial terms f0=0, and f1=1 and recurrence relation fn=fn−1+fn−2(n≥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 tfn∗, with initial terms tf0∗=(f0∗,f1∗,f2∗) and tf1∗=(f1∗,f2∗,f3∗), which is generated by the recurrence relation tfn∗=tfn−1∗+tfn−2∗(n≥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.

Keywords

• Binet’s formula
• Fibonacci-type sequences
• Tricomplex Fibonacci sequence
• generating function