Matrix Sequences in terms of Padovan and Perrin Numbers

Abstract

The first main idea of this paper is to develop the matrix sequences that 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.

1. Introduction and Preliminaries

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., [1–4] and the references cited therein). On the other hand, the matrix sequences have taken so much interest in different types of numbers (cf. [5–7]). Therefore, a new matrix sequence related to less known numbers it is worth studying. 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. Actually the most important difference with some other similar studies is, herein, that the study contains three-dimensional matrices instead of two as given in Fibonacci, Lucas, and Pell.

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. It is also clear that the ratio has so many applications in, especially, physics, engineering, architecture, and so forth [8, 9]. In a similar manner, the ratio of two consecutive Padovan and Perrin numbers converges to 𝛼_𝑃=³12+16233+³12−16233 (1) that is named as Plastic constant and was firstly defined in 1924 by Gérard Cordonnier. He described applications to architecture; in 1958, he gave a lecture tour that illustrated the use of the Plastic constant in many buildings and monuments. The smallest Pisot number is the positive root of the characteristic equation 𝑋³−𝑋−1=0 known as the Plastic constant. This is also the characteristic equation of the recurrence equations (2) and (3), and the Plastic constant is one of its roots which is the unique real root.

Although 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 {𝑅_𝑛}_𝑛∈ℕ and Padovan {𝑃_𝑛}_𝑛∈ℕ sequences as in the forms 𝑅_𝑛+3=𝑅_𝑛+1+𝑅_𝑛,where𝑅₀=3,𝑅₁=0,𝑅₂𝑃=2,_𝑛+3=𝑃_𝑛+1+𝑃_𝑛,where𝑃₀=𝑃₁=𝑃₂=1, (2)(3) respectively. It is well known that the relationship between {𝑅_𝑛}_𝑛∈ℕ and {𝑃_𝑛}_𝑛∈ℕ is presented by 𝑅_𝑛=3𝑃_𝑛−5+2𝑃_𝑛−4. (4)

This paper is divided into two sections except the first one. In Section 2, the matrix sequences of Padovan and Perrin numbers will be defined for the first time in the literature. Then, 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. In Section 3, we will present the relationship between these matrix sequences. Since we are studying three-dimensional matrices and so sequences for Padovan and Perrin numbers, there exist some difficulties in the meaning of the investigation of properties of Padovan and Perrin numbers. However, by the results in Sections 2 and 3 of this paper, we have a great opportunity to compare and obtain some new properties over these numbers. This is the main aim of this paper.

2. The Matrix Sequences of Padovan and Perrin Numbers

In this section, we will mainly focus on the matrix sequences of Padovan and Perrin numbers to get some important results. In fact, as a middle step, we will also present the related Binet formulas, summations, and generating functions. Besides, the new Binet formulas will be used in Section 3.

Hence, in the following, we will firstly define the Padovan and Perrin matrix sequences.

Definition 1. For 𝑛∈ℕ, the Padovan (𝒫_𝑛) and Perrin matrix sequences (ℛ_𝑛) are defined by 𝒫_𝑛+3=𝒫_𝑛+1+𝒫_𝑛,ℛ_𝑛+3=ℛ_𝑛+1+ℛ_𝑛, (5)(6) respectively, with initial conditions 𝒫₀=100010001,𝒫₁=,𝒫010001110₂=,ℛ001110011₀=42−3−3122−11,ℛ₁=,ℛ−3122−1113−1₂=.2−1113−1−103 (7)
In Definition 1, the matrix 𝒫₁ is a matrix analogue of the Fibonacci Q-matrix which exists for Padovan numbers.
The first main result gives the 𝑛th general terms of the sequences in (5) and (6) via Padovan and Perrin numbers as in the following.

Theorem 2. For any integer 𝑛≥0, one has the matrix sequences 𝒫_𝑛=𝑃_𝑛−5𝑃_𝑛−3𝑃_𝑛−4𝑃_𝑛−4𝑃_𝑛−2𝑃_𝑛−3𝑃_𝑛−3𝑃_𝑛−1𝑃_𝑛−2ℛ_𝑛,=𝑅_𝑛−5𝑅_𝑛−3𝑅_𝑛−4𝑅_𝑛−4𝑅_𝑛−2𝑅_𝑛−3𝑅_𝑛−3𝑅_𝑛−1𝑅_𝑛−2, (8)(9) respectively.

Proof. The proof will be done by induction steps.
First of all, let us consider (3) and then fix 𝑛=2,𝑛=1,𝑛=0,𝑛=−1,𝑛=−2 (10) in it. Thus we obtain the equalities 𝑃₋₁=𝑃₋₃=𝑃₋₄=0 and 𝑃₋₂=𝑃₋₅=1 which gives the following first step of the induction: 𝒫₀=𝑃₋₅𝑃₋₃𝑃₋₄𝑃₋₄𝑃₋₂𝑃₋₃𝑃₋₃𝑃₋₁𝑃₋₂=100010001. (11) Secondly, again considering (10) and initial condition for (3), we also get 𝒫₁=010001110. (12)
Actually, by iterating this procedure and assuming the equation in (8) holds for all 𝑛=𝑘∈ℤ⁺, we can end up the proof if we manage to show that the case also holds for 𝑛=𝑘+1: 𝒫_𝑘+1=𝒫_𝑘−1+𝒫_𝑘−2=𝑃_𝑘−6𝑃_𝑘−4𝑃_𝑘−5𝑃_𝑘−5𝑃_𝑘−3𝑃_𝑘−4𝑃_𝑘−4𝑃_𝑘−2𝑃_𝑘−3+𝑃_𝑘−7𝑃_𝑘−5𝑃_𝑘−6𝑃_𝑘−6𝑃_𝑘−4𝑃_𝑘−5𝑃_𝑘−5𝑃_𝑘−3𝑃_𝑘−4=𝑃_𝑘−4𝑃_𝑘−2𝑃_𝑘−3𝑃_𝑘−3𝑃_𝑘−1𝑃_𝑘−2𝑃_𝑘−2𝑃_𝑘𝑃_𝑘−1. (13)
Hence that is the result.
For the truthness of the Perrin matrix sequence, we need to follow almost the same approximation by considering (2). Similarly as in the above case, the final step of the induction can be obtained by ℛ_𝑘+1=ℛ_𝑘−1+ℛ_𝑘−2 as follows:ℛ_𝑘+1=𝑅_𝑘−4𝑅_𝑘−2𝑅_𝑘−3𝑅_𝑘−3𝑅_𝑘−1𝑅_𝑘−2𝑅_𝑘−2𝑅_𝑘𝑅_𝑘−1. (14)
This completes the proof.

Theorem 3. For every 𝑛∈ℕ, one can write the Binet formulas for the Padovan and Perrin matrix sequences as the form 𝒫_𝑛=𝐴₁𝑥^𝑛+𝐵₁𝑦^𝑛+𝐶₁𝑧^𝑛,ℛ_𝑛=𝐴₂𝑥^𝑛+𝐵₂𝑦^𝑛+𝐶₂𝑧^𝑛, (15) where 𝐴₁=𝑥𝒫₂+𝑥²𝒫₁+𝒫₀𝑥𝑥−𝑦(𝑥−𝑧),𝐵₁=𝑦𝒫₂+𝑦²𝒫₁+𝒫₀𝑦,𝐶𝑦−𝑥𝑦−𝑧₁=𝑧𝒫₂+𝑧²𝒫₁+𝒫₀𝑧(,𝐴𝑧−𝑥)𝑧−𝑦₂=𝑥ℛ₂+𝑥²ℛ₁+ℛ₀𝑥𝑥−𝑦(𝑥−𝑧),𝐵₂=𝑦ℛ₂+𝑦²ℛ₁+ℛ₀𝑦,𝐶𝑦−𝑥𝑦−𝑧₂=𝑧ℛ₂+𝑧²ℛ₁+ℛ₀𝑧(),𝑧−𝑥𝑧−𝑦 (16) such that 𝑥, 𝑦, and 𝑧 are roots of characteristic equations of (5) and (6).

Proof. We note that the proof will be based on the recurrence relations (5) and (6) in Definition 1. As in the previous result, we will only show the truthness of the Binet formula for Padovan matrix sequence and will omit the proof of the same formula for Perrin matrix sequence since they have the same characteristic equations.
So let us consider (5). By the assumption, the roots of the characteristic equation of (5) are 𝑥, 𝑦, and 𝑧. Hence its general solution of it is given by 𝒫_𝑛=𝐴₁𝑥^𝑛+𝐵₁𝑦^𝑛+𝐶₁𝑧^𝑛. (17) Using initial conditions in Definition 1 and also applying fundamental linear algebra operations, we clearly get the matrices 𝐴₁, 𝐵₁, and 𝐶₁, as desired. This implies the formula for 𝒫_𝑛.

In [3], the authors obtained the Binet formulas for Padovan and Perrin numbers. Now as a different approximation and so as a consequence of Theorems 2 and 3, in the following corollary, we will present the formulas for these numbers via related matrix sequences. In fact, in the proof of this corollary, we will just compare the linear combination of the 3rd row and 2nd column entries of the matrices:(i)𝐴₁, 𝐵₁, and 𝐶₁ with the matrix 𝒫_𝑛 in (8) and, similarly,(ii) 𝐴₂, 𝐵₂, and 𝐶₂ with the matrix ℛ_𝑛 in (9).

Corollary 4. The Binet formulas for Padovan and Perrin numbers in terms of their matrix sequences are given by 𝑃_𝑛−1=𝑥^𝑛+3()+𝑦𝑥−𝑦𝑥−𝑧^𝑛+3+𝑧𝑦−𝑥𝑦−𝑧_𝑛+3(),𝑅𝑧−𝑥𝑧−𝑦_𝑛−1=𝑥^𝑛−1+𝑦^𝑛−1+𝑧^𝑛−1, (18) where 𝑛>0.

Proof. For the first part of the proof, by taking into account Definition 1 and Theorem 3, we can write𝒫_𝑛=𝐴₁𝑥^𝑛+𝐵₁𝑦^𝑛+𝐶₁𝑧^𝑛=𝑥𝒫₂+𝑥²𝒫₁+𝒫₀𝑥()𝑥𝑥−𝑦𝑥−𝑧^𝑛+𝑦𝒫₂+𝑦²𝒫₁+𝒫₀𝑦𝑦𝑦−𝑥𝑦−𝑧^𝑛+𝑧𝒫₂+𝑧²𝒫₁+𝒫₀𝑧()𝑧𝑧−𝑥𝑧−𝑦^𝑛=𝑥^𝑛−1𝑥−𝑦(𝑥−𝑧)1𝑥²𝑥𝑥𝑥+1𝑥²𝑥²𝑥+𝑥²+𝑦𝑥+1_𝑛−1𝑦−𝑥𝑦−𝑧1𝑦²𝑦𝑦𝑦+1𝑦²𝑦²𝑦+𝑦²+𝑧𝑦+1_𝑛−1(𝑧−𝑥)𝑧−𝑦1𝑧²𝑧𝑧𝑧+1𝑧²𝑧²𝑧+𝑧².𝑧+1 (19) Herein, since 𝑥, 𝑦, and 𝑧 are roots of the characteristic equation 𝑋³−𝑋−1=0, we clearly have 𝒫_𝑛=𝑥^𝑛−1𝑥−𝑦(𝑥−𝑧)1𝑥²𝑥𝑥𝑥³𝑥²𝑥²𝑥⁴𝑥³+𝑦^𝑛−1𝑦−𝑥𝑦−𝑧1𝑦²𝑦𝑦𝑦³𝑦²𝑦²𝑦⁴𝑦³+𝑧^𝑛−1(𝑧−𝑥)𝑧−𝑦1𝑧²𝑧𝑧𝑧³𝑧²𝑧²𝑧⁴𝑧³. (20) Also, by Theorem 2, we obtain 𝑃_𝑛−5𝑃_𝑛−3𝑃_𝑛−4𝑃_𝑛−4𝑃_𝑛−2𝑃_𝑛−3𝑃_𝑛−3𝑃_𝑛−1𝑃_𝑛−2=𝑥^𝑛−1𝑥−𝑦(𝑥−𝑧)1𝑥²𝑥𝑥𝑥³𝑥²𝑥²𝑥⁴𝑥³+𝑦^𝑛−1𝑦−𝑥𝑦−𝑧1𝑦²𝑦𝑦𝑦³𝑦²𝑦²𝑦⁴𝑦³+𝑧^𝑛−1(𝑧−𝑥)𝑧−𝑦1𝑧²𝑧𝑧𝑧³𝑧²𝑧²𝑧⁴𝑧³. (21) Now, if we compare the 3rd row and 2nd column entries with the matrices in the above equation, then we get 𝑃_𝑛−1=𝑥^𝑛+3()+𝑦𝑥−𝑦𝑥−𝑧^𝑛+3+𝑧𝑦−𝑥𝑦−𝑧()^𝑛+3.𝑧−𝑥𝑧−𝑦 (22) For the second part of the proof, in a similar manner, by again taking into account Theorem 3 and Definition 1, we can write ℛ_𝑛=𝐴₂𝑥^𝑛+𝐵₂𝑦^𝑛+𝐶₂𝑧^𝑛=𝑥ℛ₂+𝑥²ℛ₁+ℛ₀𝑥()𝑥𝑥−𝑦𝑥−𝑧^𝑛+𝑦ℛ₂+𝑦²ℛ₁+ℛ₀𝑦𝑦𝑦−𝑥𝑦−𝑧^𝑛+𝑧ℛ₂+𝑧²ℛ₁+ℛ₀𝑧()𝑧𝑧−𝑥𝑧−𝑦^𝑛=𝑥^𝑛−1()×𝑥−𝑦𝑥−𝑧2𝑥−3𝑥²+4−𝑥+𝑥²+2𝑥+2𝑥²−3𝑥+2𝑥²−33𝑥−𝑥²+1−𝑥+𝑥²+2−𝑥+𝑥²+23𝑥²−13𝑥−𝑥²+𝑦+1_𝑛−1×𝑦−𝑥𝑦−𝑧2𝑦−3𝑦²+4−𝑦+𝑦²+2𝑦+2𝑦²−3𝑦+2𝑦²−33𝑦−𝑦²+1−𝑦+𝑦²+2−𝑦+𝑦²+23𝑦²−13𝑦−𝑦²+𝑧+1_𝑛−1()×𝑧−𝑥𝑧−𝑦2𝑧−3𝑧²+4−𝑧+𝑧²+2𝑧+2𝑧²−3𝑧+2𝑧²−33𝑧−𝑧²+1−𝑧+𝑧²+2−𝑧+𝑧²+23𝑧²−13𝑧−𝑧².+1 (23) Herein, since 𝑥+𝑦+𝑧=0 and 𝑥𝑦𝑧=1, we also clearly get ℛ_𝑛=𝑥^𝑛−12𝑥²+()×1/𝑥2𝑥−3𝑥²+4−𝑥+𝑥²+2𝑥+2𝑥²−3𝑥+2𝑥²−33𝑥−𝑥²+1−𝑥+𝑥²+2−𝑥+𝑥²+23𝑥²−13𝑥−𝑥²+𝑦+1_𝑛−12𝑦²+×1/𝑦2𝑦−3𝑦²+4−𝑦+𝑦²+2𝑦+2𝑦²−3𝑦+2𝑦²−33𝑦−𝑦²+1−𝑦+𝑦²+2−𝑦+𝑦²+23𝑦²−13𝑦−𝑦²+𝑧+1_𝑛−12𝑧²+()×1/𝑧2𝑧−3𝑧²+4−𝑧+𝑧²+2𝑧+2𝑧²−3𝑧+2𝑧²−33𝑧−𝑧²+1−𝑧+𝑧²+2−𝑧+𝑧²+23𝑧²−13𝑧−𝑧².+1 (24) Moreover, by Theorem 2, we obtain 𝑅_𝑛−5𝑅_𝑛−3𝑅_𝑛−4𝑅_𝑛−4𝑅_𝑛−2𝑅_𝑛−3𝑅_𝑛−3𝑅_𝑛−1𝑅_𝑛−2=𝑥^𝑛−12𝑥²+()×1/𝑥2𝑥−3𝑥²+4−𝑥+𝑥²+2𝑥+2𝑥²−3𝑥+2𝑥²−33𝑥−𝑥²+1−𝑥+𝑥²+2−𝑥+𝑥²+23𝑥²−13𝑥−𝑥²+𝑦+1_𝑛−12𝑦²+×1/𝑦2𝑦−3𝑦²+4−𝑦+𝑦²+2𝑦+2𝑦²−3𝑦+2𝑦²−33𝑦−𝑦²+1−𝑦+𝑦²+2−𝑦+𝑦²+23𝑦²−13𝑦−𝑦²+𝑧+1_𝑛−12𝑧²+()×1/𝑧2𝑧−3𝑧²+4−𝑧+𝑧²+2𝑧+2𝑧²−3𝑧+2𝑧²−33𝑧−𝑧²+1−𝑧+𝑧²+2−𝑧+𝑧²+23𝑧²−13𝑧−𝑧².+1 (25) Now, if we compare the 3rd row and 2nd column entries with the matrices in the above equation, then we get 𝑅_𝑛−1=3𝑥³−𝑥2𝑥³𝑥+1^𝑛−1+3𝑦³−𝑦2𝑦³𝑦+1^𝑛−1+3𝑧³−𝑧2𝑧³𝑧+1^𝑛−1. (26) Finally, since 𝑥, 𝑦, and 𝑧 are roots of the characteristic equation 𝑋³−𝑋−1=0, we can replace 𝑥+1, 𝑦+1, and 𝑧+1 by 𝑥³,𝑦³, and 𝑧³. Then we conclude that 𝑅_𝑛−1=𝑥^𝑛−1+𝑦^𝑛−1+𝑧^𝑛−1, (27) as required.

Now, for Padovan and Perrin matrix sequences, we give the summations according to specified rules as we depicted at the beginning of this section.

Theorem 5. For 𝑚>𝑗≥0, there exist ^𝑛−1_𝑖=0𝒫_{𝑚𝑖+𝑗}=𝒫_{𝑚𝑛+𝑚+𝑗}+𝒫_{𝑚𝑛−𝑚+𝑗}+1−𝑅_𝑚𝒫_{𝑚𝑛+𝑗}−𝒫_𝑚+𝑗𝑅_𝑚−𝑅_−𝑚−𝒫_𝑚−𝑗−𝑅_𝑚−1𝒫_𝑗𝑅_𝑚−𝑅_{−𝑚𝑛−1𝑖=0},ℛ_{𝑚𝑖+𝑗}=ℛ_{𝑚𝑛+𝑚+𝑗}+ℛ_{𝑚𝑛−𝑚+𝑗}+1−𝑅_𝑚ℛ_{𝑚𝑛+𝑗}−ℛ_𝑚+𝑗𝑅_𝑚−𝑅_−𝑚−ℛ_𝑚−𝑗−𝑅_𝑚−1ℛ_𝑗𝑅_𝑚−𝑅_−𝑚. (28)(29)

Proof. The main point of the proof will be touched just the result Theorem 3, in other words the Binet formulas of related matrix sequences. Differently from previous results, we will consider the proof over Perrin matrix sequence and will omit the case of Padovan. Thus, ^𝑛−1_𝑖=0ℛ_{𝑚𝑖+𝑗}=^𝑛−1_𝑖=0𝐴₂𝑥^{𝑚𝑖+𝑗}+𝐵₂𝑦^{𝑚𝑖+𝑗}+𝐶₂𝑧^{𝑚𝑖+𝑗}=𝐴₂𝑥^𝑗𝑥^𝑚𝑛−1𝑥^𝑚−1+𝐵₂𝑦^𝑗𝑦^𝑚𝑛−1𝑦^𝑚−1+𝐶₂𝑧^𝑗𝑧^𝑚𝑛−1𝑧^𝑚.−1 (30) Herein, simplifying the last equality will be implied in (29) as required.

If we state almost the same explanation as in Corollary 4, then the following result will be clear for the summations of Padovan and Perrin numbers as a consequence of Theorem 5.

Corollary 6. For 𝑚>𝑗>0, one has ^𝑛−1_𝑖=0𝑃_{𝑚𝑖+𝑗−1}=𝑃_{𝑚𝑛+𝑚+𝑗−1}+𝑃_{𝑚𝑛−𝑚+𝑗−1}+1−𝑅_𝑚𝑃_{𝑚𝑛+𝑗−1}𝑅_𝑚−𝑅_−𝑚−𝑃_{𝑚+𝑗−1}+𝑃_{𝑚−𝑗−1}−𝑅_𝑚−1𝑃_𝑗−1𝑅_𝑚−𝑅_{−𝑚𝑛−1𝑖=0},𝑅_{𝑚𝑖+𝑗−1}=𝑅_{𝑚𝑛+𝑚+𝑗−1}+𝑅_{𝑚𝑛−𝑚+𝑗−1}+1−𝑅_𝑚𝑅_{𝑚𝑛+𝑗−1}𝑅_𝑚−𝑅_−𝑚−𝑅_{𝑚+𝑗−1}+𝑅_{𝑚−𝑗−1}−𝑅_𝑚−1𝑅_𝑗−1𝑅_𝑚−𝑅_−𝑚. (31)

As we noted at the beginning of this section, the other aim of this paper is to present generating functions of our new matrix sequences. The next result deals with it.

Theorem 7. For Padovan and Perrin matrix sequences, one has the generating functions ^∞_𝑖=0𝒫_𝑖𝑥^𝑖=11−𝑥²−𝑥³1−𝑥²𝑥𝑥²𝑥²1𝑥𝑥𝑥+𝑥²1_∞𝑖=0,ℛ_𝑖𝑥^𝑖=11−𝑥²−𝑥³×4−3𝑥−2𝑥²2+𝑥−3𝑥²−3+2𝑥+4𝑥²−3+2𝑥+4𝑥²1−𝑥+2𝑥²2+𝑥−3𝑥²2+𝑥−3𝑥²−1+3𝑥+𝑥²1−𝑥+2𝑥², (32)(33) respectively.

Proof. We will again omit Padovan case since the proof will be quite similar.
Assume that 𝐺(𝑥) is the generating function for the sequence {ℛ_𝑛}_𝑛∈ℕ. Then we have 𝐺(𝑥)=^∞_𝑖=0ℛ_𝑖𝑥^𝑖=ℛ₀+ℛ₁𝑥+ℛ₂𝑥²+^∞_𝑖=3ℛ_𝑖−2+ℛ_𝑖−3𝑥^𝑖. (34) From Definition 1, we obtain 𝐺(𝑥)=ℛ₀+ℛ₁𝑥+ℛ₂𝑥²−ℛ₀𝑥²+𝑥^2∞_𝑖=0ℛ_𝑖𝑥^𝑖+𝑥^3∞_𝑖=0ℛ_𝑖𝑥^𝑖=ℛ₀+ℛ₁𝑥+ℛ₂−ℛ₀𝑥²+𝑥²𝐺(𝑥)+𝑥³𝐺(𝑥). (35) Now, rearrangement of the above equation will imply that 𝐺(𝑥)=ℛ₀+ℛ₁𝑥+ℛ₂−ℛ₀𝑥²1−𝑥²−𝑥³ (36) which equals the ∑^∞_𝑖=0ℛ_𝑖𝑥^𝑖 in the theorem.
Hence that is the result.

In [10], the authors obtained the generating functions for Padovan and Perrin numbers. However, herein, we will obtain these functions in terms of Padovan and Perrin matrix sequences as a consequence of Theorem 7. To do that we will again compare the 3rd row and 2nd column entries with the matrices in Theorem 7. Hence we have the following corollary.

Corollary 8. There always exist ^∞_𝑖=0𝑃_𝑖−1𝑥^𝑖=𝑥+𝑥²1−𝑥²−𝑥³,^∞_𝑖=0𝑅_𝑖−1𝑥^𝑖=−1+3𝑥+𝑥²1−𝑥²−𝑥³. (37)

3. Relationships between New Matrix Sequences

The following proposition (which will be needed for some of our results in this section) expresses that there always exist some interpasses between the Padovan and Perrin matrix sequences. In fact its proof can be seen directly by considering Theorem 2 and the equality in (4).

Proposition 9. For the matrix sequences (𝒫_𝑛)_𝑛∈ℕ and (ℛ_𝑛)_𝑛∈ℕ, one has the following equalities:(i)ℛ_𝑛=3𝒫_𝑛−5+2𝒫_𝑛−4,𝑓𝑜𝑟𝑛>4, (ii)ℛ₀𝒫_𝑛=2𝒫_𝑛−2+𝒫_𝑛−5=ℛ_𝑛,𝑓𝑜𝑟𝑛>4, (iii)ℛ₀𝒫_𝑛=𝒫_𝑛ℛ₀=ℛ_𝑛.

Remark 10. We remark that the interpass between Padovan and Perrin numbers was stated in (4) as the expression of a Perrin number in terms of Padovan numbers. In addition to this, by taking into account Proposition 9, one can also obtain𝑃_𝑛−1=1𝑅23^𝑛−3+8𝑅_𝑛−2+10𝑅_𝑛−1 (38) as a new interpass for the same numbers. Notice the relation in (38) based on the expression of a Padovan number in terms of Perrin numbers.

Theorem 11. For 𝑚,𝑛∈ℕ, the following equalities hold:(i)𝒫_𝑚𝒫_𝑛=𝒫_𝑚+𝑛, (ii)𝒫_𝑚ℛ_𝑛=ℛ_𝑛𝒫_𝑚=ℛ_𝑚+𝑛, (iii)ℛ_𝑚ℛ_𝑛=ℛ_𝑛ℛ_𝑚=4𝒫_{𝑚+𝑛−4}+4𝒫_{𝑚+𝑛−7}+𝒫_{𝑚+𝑛−10}, for 𝑚,𝑛>4,(iv)ℛ_𝑚ℛ_𝑛=ℛ_𝑛ℛ_𝑚=2ℛ_{𝑚+𝑛−2}+ℛ_{𝑚+𝑛−5}, where 𝑚>4 or 𝑛>4.

Proof. (i) From Theorem 3 with its assumptions, we can have 𝒫_𝑚𝒫_𝑛=𝐴₁𝑥^𝑚+𝐵₁𝑦^𝑚+𝐶₁𝑧^𝑚×𝐴₁𝑥^𝑛+𝐵₁𝑦^𝑛+𝐶₁𝑧^𝑛=𝐴₁𝐴₁𝑥^𝑚+𝑛+𝐴₁𝐵₁𝑥^𝑚𝑦^𝑛+𝐴₁𝐶₁𝑥^𝑚𝑧^𝑛+𝐵₁𝐴₁𝑦^𝑚𝑥^𝑛+𝐵₁𝐵₁𝑦^𝑚+𝑛+𝐵₁𝐶₁𝑦^𝑚𝑧^𝑛+𝐶₁𝐴₁𝑧^𝑚𝑥^𝑛+𝐶₁𝐵₁𝑧^𝑚𝑦^𝑛+𝐶₁𝐶₁𝑧^𝑚+𝑛. (39) Herein, since 𝑥+𝑦+𝑧=0 and 𝑥𝑦𝑧=1, simple matrix calculations imply that 𝐴²₁=𝐴₁, 𝐵²₁=𝐵₁, 𝐶²₁=𝐶₁, and 𝐴₁𝐵₁=𝐴₁𝐶₁=𝐵₁𝐴₁=𝐵₁𝐶₁=𝐶₁𝐴₁=𝐶₁𝐵₁=[0]. (40) Then we obtain 𝒫_𝑚𝒫_𝑛=𝐴₁𝑥^𝑚+𝑛+𝐵₁𝑦^𝑚+𝑛+𝐶₁𝑧^𝑚+𝑛=𝒫_𝑚+𝑛. (41) (ii) Here, we will just show the truthness of the equality 𝒫_𝑚ℛ_𝑛=ℛ_𝑚+𝑛 since the other can be done similarly. Now, by Proposition 9(iii), we write 𝒫_𝑚ℛ_𝑛=𝒫_𝑚𝒫_𝑛ℛ₀. (42) On the other hand, by (i) and again Proposition 9(iii), we finally have 𝒫_𝑚ℛ_𝑛=𝒫_𝑚+𝑛ℛ₀=ℛ_𝑚+𝑛.
(iii) As in (ii), we will just show the first equality of this condition. So, by Proposition 9(ii), we have ℛ_𝑚ℛ_𝑛=2𝒫_𝑚−2+𝒫_𝑚−52𝒫_𝑛−2+𝒫_𝑛−5. (43) It is easy to see that one can use (i) in this latest equality. Thus, applying sufficient operations, we then obtain ℛ_𝑚ℛ_𝑛=4𝒫_{𝑛+𝑚−4}+4𝒫_{𝑚+𝑛−7}+𝒫_{𝑚+𝑛−10}, (44) as desired.
The final part of the proof can be seen similarly as in the proof of (iii).

Comparing matrix entries and then using Theorems 2 and 11, we obtain the following result.

Corollary 12. One has the following identities for Padovan and Perrin numbers:(i)𝑃_𝑚−3𝑃_𝑛−3+𝑃_𝑚−1𝑃_𝑛−2+𝑃_𝑚−2𝑃_𝑛−1=𝑃_{𝑚+𝑛−1}, (ii)𝑃_𝑚−3𝑅_𝑛−3+𝑃_𝑚−1𝑅_𝑛−2+𝑃_𝑚−2𝑅_𝑛−1=𝑅_{𝑚+𝑛−1}, (iii)𝑅_𝑚−3𝑅_𝑛−3+𝑅_𝑚−1𝑅_𝑛−2+𝑅_𝑚−2𝑅_𝑛−1=4𝑃_{𝑚+𝑛−5}+4𝑃_{𝑚+𝑛−8}+𝑃_{𝑚+𝑛−11},(iv)𝑅_𝑚−3𝑅_𝑛−3+𝑅_𝑚−1𝑅_𝑛−2+𝑅_𝑚−2𝑅_𝑛−1=2𝑅_{𝑚+𝑛−3}+𝑅_{𝑚+𝑛−6}.

Proof. (i) By using Theorems 2 and 11, we have 𝒫_𝑚𝒫_𝑛=𝒫_𝑚+𝑛 which can be written mathematically as 𝑃_𝑚−5𝑃_𝑚−3𝑃_𝑚−4𝑃_𝑚−4𝑃_𝑚−2𝑃_𝑚−3𝑃_𝑚−3𝑃_𝑚−1𝑃_𝑚−2𝑃_𝑛−5𝑃_𝑛−3𝑃_𝑛−4𝑃_𝑛−4𝑃_𝑛−2𝑃_𝑛−3𝑃_𝑛−3𝑃_𝑛−1𝑃_𝑛−2=𝑃_{𝑚+𝑛−5}𝑃_{𝑚+𝑛−3}𝑃_{𝑚+𝑛−4}𝑃_{𝑚+𝑛−4}𝑃_{𝑚+𝑛−2}𝑃_{𝑚+𝑛−3}𝑃_{𝑚+𝑛−3}𝑃_m+𝑛−1𝑃_{𝑚+𝑛−2}. (45)
Now, by firstly multiplying the left-hand side matrices and then by comparing the 3rd rows and 2nd columns entries, we finally obtain the required equation in (i).
The proofs of the remaining conditions can be done similarly by considering again Theorems 11 and 2.
Hence that is the result.

In the light of the above results, the following theorems provide us the convenience to obtain the powers of Padovan and Perrin matrix sequences.

Theorem 13. For 𝑚,𝑛,r∈ℕ and 𝑛≥𝑟, the following equalities hold:(i)𝒫^𝑚_𝑛=𝒫_𝑚𝑛,(ii)𝒫^𝑚_𝑛+1=𝒫^𝑚₁𝒫_𝑚𝑛,(iii)𝒫_𝑛−𝑟𝒫_𝑛+𝑟=𝒫²_𝑛=𝒫^𝑛₂.

Proof. (i) We actually can write 𝒫^𝑚_𝑛=𝒫_𝑛𝒫_𝑛⋯𝒫_𝑛 (𝑚-times). Now, by Theorem 11(i), we clearly obtain 𝒫_𝑚𝑛 as the next step of this equality.
(ii) Let us consider the left-hand side of the equality. As a similar approximation in (i), we write 𝒫^𝑚_𝑛+1=𝒫_𝑛+1𝒫_𝑛+1⋯𝒫_𝑛+1=𝒫_𝑚(𝑛+1)=𝒫_𝑚𝒫_𝑚𝑛=𝒫₁𝒫_𝑚−1𝒫_𝑚𝑛. (46) Similarly, we can write 𝒫_𝑚−1=𝒫₁𝒫_𝑚−2. By iterative processes, we hence obtain 𝒫^𝑚_𝑛+1=𝒫₁𝒫₁⋯𝒫_1𝑚times𝒫_𝑚𝑛=𝒫^𝑚₁𝒫_𝑚𝑛. (47)
The proof of (iii) can be seen quite similarly as the proof of (ii).

Theorem 14. For 𝑚,𝑛,𝑟∈ℕ and 𝑛≥𝑟, the equalities ℛ_𝑛−𝑟ℛ_𝑛+𝑟=ℛ²_𝑛,ℛ^𝑚_𝑛=ℛ^𝑚₀𝒫_𝑚𝑛 (48) always hold.

Proof. In the first part of the proof, we mainly consider Theorem 3, in other words, the Binet formula of Perrin matrix sequence. Hence we can write ℛ_𝑛−𝑟ℛ_𝑛+𝑟−ℛ²_𝑛=𝐴₂𝑥^𝑛−𝑟+𝐵₂𝑦^𝑛−𝑟+𝐶₂𝑧^𝑛−𝑟×𝐴₂𝑥^𝑛+𝑟+𝐵₂𝑦^𝑛+𝑟+𝐶₂𝑧^𝑛+𝑟−𝐴₂𝑥^𝑛+𝐵₂𝑦^𝑛+𝐶₂𝑧^𝑛2, (49) where 𝐴₂,  𝐵₂, and 𝐶₂  and 𝑥, 𝑦, and 𝑧 are as given in Theorem 3. By applying some elementary operations, we obtain ℛ_𝑛−𝑟ℛ_𝑛+𝑟−ℛ²_𝑛=𝐷, (50) where 𝐷=𝐴₂𝐵₂𝑥^𝑛−𝑟𝑦^𝑛−𝑟𝑥^𝑟−𝑦^𝑟2+𝐴₂𝐶₂𝑥^𝑛−𝑟𝑧^𝑛−𝑟𝑥^𝑟−𝑧^𝑟2+𝐵₂𝐶₂𝑦^𝑛−𝑟𝑧^𝑛−𝑟𝑦^𝑟−𝑧^𝑟2. (51) Eventually, by 𝐴₂𝐵₂=𝐴₂𝐶₂=𝐵₂𝐶₂=[0]_3×3, we get ℛ_𝑛−𝑟ℛ_𝑛+𝑟=ℛ²_𝑛, as required.
Secondly, let us consider the right-hand side of the equality ℛ^𝑚_𝑛=ℛ^𝑚₀𝒫_𝑚𝑛. By Theorem 13(i), we have ℛ^𝑚₀𝒫_𝑚𝑛=ℛ₀ℛ₀⋯ℛ_0𝑚times𝒫_𝑛𝒫_𝑛⋯𝒫_{𝑛𝑚times}. (52)
By iterating usage of Proposition 9(iii), we finally obtain ℛ^𝑚₀𝒫_𝑚𝑛=ℛ₀𝒫_𝑛ℛ₀𝒫_𝑛⋯ℛ₀𝒫_𝑛=ℛ_𝑛ℛ_𝑛⋯ℛ_𝑛=ℛ^𝑚_𝑛. (53)
Hence that is the result.

Conflict of Interests

We declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This study is a part of Nazmiye Yilmaz’s Ph.D. Thesis. Thanks are due to the editor and reviewers for their interests and valuable comments.