Abstract
The first main idea of this paper is to develop thematrix sequencesthat represent Padovan and Perrin numbers. Then, by taking into account matrix properties of these new matrix sequences, some behaviours of Padovan and Perrin numbers will be investigated. Moreover, some important relationships between Padovan and Perrin matrix sequences will be presented.
Highlights
Introduction and PreliminariesThere are so many studies in the literature that concern the special number sequences such as Fibonacci, Lucas, Pell, Jacobsthal, Padovan, and Perrin
In Fibonacci numbers, there clearly exists the term Golden ratio which is defined as the ratio of two consecutive Fibonacci numbers that converges to α = (1 + √5)/2
The ratio of two consecutive Padovan and Perrin numbers converges to αP
Summary
There are so many studies in the literature that concern the special number sequences such as Fibonacci, Lucas, Pell, Jacobsthal, Padovan, and Perrin (see, e.g., [1,2,3,4] and the references cited therein). In the light of this thought, the goal of this paper is to define the related matrix sequences for Padovan and Perrin numbers for the first time in the literature. The study of Perrin numbers started in the beginning of the 19 century under different names, the master study was published in 2006 by Shannon et al [3] In this reference, the authors defined the Perrin {Rn}n∈N and Padovan {Pn}n∈N sequences as in the forms. By giving the generating functions, the Binet formulas, and summation formulas over these new matrix sequences, we will obtain some fundamental properties on Padovan and Perrin numbers.
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