Matrix Sequences in terms of Padovan and Perrin Numbers

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 newmatrix 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 somany 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


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][2][3][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][6][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  = (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 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  3 −  − 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 {  } ∈N and Padovan {  } ∈N sequences as in the forms respectively.It is well known that the relationship between {  } ∈N and {  } ∈N is presented by 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 threedimensional 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.

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, the Padovan (P  ) and Perrin matrix sequences (R  ) are defined by respectively, with initial conditions ) , In Definition 1, the matrix P 1 is a matrix analogue of the Fibonacci Q-matrix which exists for Padovan numbers.
The first main result gives the th general terms of the sequences in ( 5) and ( 6) via Padovan and Perrin numbers as in the following.Theorem 2. For any integer  ≥ 0, one has the matrix sequences respectively.
Proof.The proof will be done by induction steps.First of all, let us consider (3) and then fix in it.Thus we obtain the equalities  −1 =  −3 =  −4 = 0 and  −2 =  −5 = 1 which gives the following first step of the induction: Secondly, again considering (10) and initial condition for (3), we also get Actually, by iterating this procedure and assuming the equation in ( 8) holds for all  =  ∈ Z + , we can end up the proof if we manage to show that the case also holds for  =  + 1: 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 +1 = R −1 + R −2 as follows: ) . ( This completes the proof.
Theorem 3.For every  ∈ N, one can write the Binet formulas for the Padovan and Perrin matrix sequences as the form where such that , , and  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 , , and .Hence its general solution of it is given by Using initial conditions in Definition 1 and also applying fundamental linear algebra operations, we clearly get the matrices  1 ,  1 , and  1 , as desired.This implies the formula for P  .
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)  1 ,  1 , and  1 with the matrix P  in (8) and, similarly, (ii)  2 ,  2 , and  2 with the matrix R  in (9).

Corollary 4. The Binet formulas for Padovan and Perrin numbers in terms of their matrix sequences are given by
where  > 0.
Proof.For the first part of the proof, by taking into account Definition 1 and Theorem 3, we can write ) .
(20) Also, by Theorem 2, we obtain ) . (21) Now, if we compare the 3rd row and 2nd column entries with the matrices in the above equation, then we get For the second part of the proof, in a similar manner, by again taking into account Theorem 3 and Definition 1, we can write ) .
Herein, since  +  +  = 0 and  = 1, we also clearly get Moreover, by Theorem 2, we obtain Now, if we compare the 3rd row and 2nd column entries with the matrices in the above equation, then we get Finally, since , , and  are roots of the characteristic equation  3 −  − 1 = 0, we can replace  + 1,  + 1, and  + 1 by  3 ,  3 , and  3 .Then we conclude that 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. −1 (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, 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  >  > 0, one has 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
respectively.
Proof.We will again omit Padovan case since the proof will be quite similar.Assume that () is the generating function for the sequence {R  } ∈N .Then we have From Definition 1, we obtain Now, rearrangement of the above equation will imply that which equals the ∑ ∞ =0 R    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.