Research Article | Open Access
Matrix Sequences in terms of Padovan and Perrin Numbers
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 . 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 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. . In this reference, the authors defined the Perrin and Padovan sequences as in the forms respectively. It is well known that the relationship between and 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 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 , the Padovan and Perrin matrix sequences are defined by
respectively, with initial conditions
In Definition 1, the matrix 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 , 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 and 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 , we can end up the proof if we manage to show that the case also holds for :
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 as follows:
This completes the proof.
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 , , and , as desired. This implies the formula for .
In , 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 rd row and nd column entries of the matrices:(i), , and with the matrix in (8) and, similarly,(ii) , , and with the matrix in (9).
Corollary 4. The Binet formulas for Padovan and Perrin numbers in terms of their matrix sequences are given by where .
Proof. For the first part of the proof, by taking into account Definition 1 and Theorem 3, we can write Herein, since , , and are roots of the characteristic equation , we clearly have Also, by Theorem 2, we obtain Now, if we compare the rd row and nd 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 and , we also clearly get Moreover, by Theorem 2, we obtain Now, if we compare the rd row and nd column entries with the matrices in the above equation, then we get Finally, since , , and are roots of the characteristic equation , we can replace , , and by ,, and . 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.
Theorem 5. For , there exist
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.
Corollary 6. For , 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 . Then we have From Definition 1, we obtain Now, rearrangement of the above equation will imply that which equals the in the theorem.
Hence that is the result.
In , 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 rd row and nd column entries with the matrices in Theorem 7. Hence we have the following corollary.
Corollary 8. There always exist
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 and , one has the following equalities:(i), (ii), (iii).
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 obtain 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 , the following equalities hold:(i), (ii), (iii), for ,(iv), where or .
Proof. (i) From Theorem 3 with its assumptions, we can have
Herein, since and , simple matrix calculations imply that , , , and
Then we obtain
(ii) Here, we will just show the truthness of the equality since the other can be done similarly. Now, by Proposition 9(iii), we write
On the other hand, by (i) and again Proposition 9(iii), we finally have .
(iii) As in (ii), we will just show the first equality of this condition. So, by Proposition 9(ii), we have It is easy to see that one can use in this latest equality. Thus, applying sufficient operations, we then obtain as desired.
The final part of the proof can be seen similarly as in the proof of (iii).
Corollary 12. One has the following identities for Padovan and Perrin numbers:(i), (ii), (iii),(iv).
Proof. (i) By using Theorems 2 and 11, we have which can be written mathematically as
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 and , the following equalities hold:(i),(ii),(iii).
Proof. We actually can write (-times). Now, by Theorem 11(i), we clearly obtain as the next step of this equality.
(ii) Let us consider the left-hand side of the equality. As a similar approximation in , we write Similarly, we can write . By iterative processes, we hence obtain
The proof of can be seen quite similarly as the proof of .
Theorem 14. For and , the equalities 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
where , , and and , , and are as given in Theorem 3. By applying some elementary operations, we obtain
Eventually, by , we get , as required.
Secondly, let us consider the right-hand side of the equality . By Theorem 13(i), we have
By iterating usage of Proposition 9(iii), we finally obtain
Hence that is the result.
Conflict of Interests
We declare that there is no conflict of interests regarding the publication of this paper.
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.
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Copyright © 2013 Nazmiye Yilmaz and Necati Taskara. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.