Some matrices with Padovan Q-matrix property Supunnee Sompong, Natthaphat Wora-Ngon, Areeya Piranan, and Natnapa Wongkaentow
Citation: AIP Conference Proceedings 1905, 030035 (2017); View online: https://doi.org/10.1063/1.5012181 View Table of Contents: http://aip.scitation.org/toc/apc/1905/1 Published by the American Institute of Physics
Some Matrices with Padovan Q-Matrix Property Supunnee Sompong1,a), Natthaphat Wora-Ngon2,b), Areeya Piranan3,c) and Natnapa Wongkaentow4,d) 1,2,3,4
Department of Mathematics and Statistics, Faculty of Science and Technology, Sakon Nakhon Rajabhat University 47000, Thailand a)
Corresponding author:
[email protected] b)
[email protected] c)
[email protected] d)
[email protected]
Abstract. Recently, there are so many studies in the matrix sequences which is considered in the different types of numbers such as Fibonacci, Lucas, Pell, Padovan, and Perrin. In this research, we are interested to discover the new matrices which have similar properties to Padovan Q matrix. Moreover, some important relation of Padovan sequences is presented.
INTRODUCTION One famous integer sequences in mathematics is the Padovan sequence [1-2]. The Padovan sequence is the sequence of integers Pn defined by the initial values P0 0, P1 0, P2 1 and the recurrence relation [4-9]
Pn-2 Pn-3
Pn
for all n t 3.
ሺͳሻ
Here are the first few Padovan sequence.
n Pn
0
1
TABLE 1. The Padovan sequence 2 3 4 5 6 7 8
9
10
0
0
1
3
4
0
1
1
1
2
2
…
In 2013, K. Sokhuma, [6] studies and investigated the Padovan sequence, Padovan Q matrix, n power th
of Q matrix and some of their simple properties. In 2015, P. Seenukul, et al. [3] found the new matrices of
3 u 3 , which have similar properties to Padovan Q Matrix and its generalized relation. In this research, we interested to discover the new matrices which have similar properties to Padovan Q matrix. Moreover, some important relation of Padovan sequences is presented.
Preliminaries In this section we will list some concepts and some previous results which are needed for this paper.
Proceedings of the 13th IMT-GT International Conference on Mathematics, Statistics and their Applications (ICMSA2017) AIP Conf. Proc. 1905, 030035-1–030035-6; https://doi.org/10.1063/1.5012181 Published by AIP Publishing. 978-0-7354-1595-9/$30.00
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Theorem 1.1 [6] Let Pn be the n Padovan sequences, then for any n N , we have; th
If
ª0 1 0 º «0 0 1 » , then Q n « » «¬1 1 0 »¼
Q
ª Pn1 «P « n «¬ Pn1
Pn1 Pn 2 Pn3
Pn º Pn 1 »» Pn 2 »¼
(2)
Theorem 1.2 [3] Let Pn be the n Padovan sequences, then for any n N , we have; th
1. If
2. If
3. If
4. If
5. If
* 1
ª0 0 1 º «1 0 0» , then Q* n 1 « » «¬1 1 0»¼
ª Pn 2 «P « n 1 «¬ Pn 3
Pn Pn 1 Pn 1
Pn 1 º Pn »» Pn 2 »¼
* 2
ª0 0 1 º «1 0 1 » , then Q* n 2 « » «¬0 1 0»¼
ª Pn 1 «P « n 1 «¬ Pn
Pn Pn 2 Pn 1
Pn 1 º Pn 3 »» Pn 2 »¼
(4)
* 3
ª0 1 1 º «1 0 0» , then Q* n 3 « » «¬0 1 0»¼
ª Pn 2 «P « n 1 «¬ Pn
Pn 3 Pn 2 Pn 1
Pn 1 º Pn »» Pn 1 »¼
(5)
* 4
ª0 1 0º «1 0 1 » , then « » «¬1 0 0»¼
Q
ª Pn 2 «P « n 3 «¬ Pn 1
Pn 1 Pn 2 Pn
Pn º Pn 1 »» Pn 1 »¼
(6)
* 5
ª0 1 1 º «0 0 1 » , then « » «¬1 0 0 »¼
Q
ª Pn 2 «P « n «¬ Pn 1
Pn1 Pn1 Pn
Pn3 º Pn 1 »» Pn 2 »¼
(7)
Q
Q
Q
Q
Q
* n 4
* n 5
(3)
Main Results In this section we found the new 3 u 3 matrices which have similar properties to Padovan Q Matrix. Moreover, some relation of Padovan numbers are obtained. In this work, we construct and rearrange the element 0 and 1 in matrices. Then we have the following results. th
Theorem 2.1 Let Pn be the n Padovan sequences, we have;
If
Q1
ª0 1 0 º «0 1 1 » , then Q n 1 « » «¬1 0 1 »¼
ª P2 n1 «P « 2 n1 «¬ P2 n
P2 n P2 n 2 P2 n 1
Proof. Our proof starts with the principle of mathematical induction on
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P2 n 1 º P2 n3 »» , n t 1 . P2 n 2 »¼ n.
(8)
For
n 1 it is easy to see that Q11
So, it is true for
ª0 1 0 º «0 1 1 » « » ¬«1 0 1 »¼
ª P1 «P « 3 ¬« P2
P2 P4 P3
ª P2 1 1 « « P2 1 1 « «¬ P2 1
P3 º P5 »» P4 ¼»
P2 1 P2 1 2 P2 1 1
P2 1 1 º » P2 1 3 » » P2 1 2 »¼
n 1
Suppose that it is holds for
n k. That is k 1
Q
ª P2 k 1 «P « 2 k 1 «¬ P2 k
P2 k 1 º P2k 3 »» P2 k 2 »¼
P2 k P2 k 2 P2 k 1
n k 1.
Next, we will prove it for
By the laws of exponents hold for the
k 1 1
Q
k 1
Q Q1
Q1k 1
Q1k Q1. We have
P2 k 1 º ª0 1 0 º ª P2 k 1 P2 k «P »« » « 2 k 1 P2 k 2 P2 k 3 » «0 1 1 » «¬ P2 k P2 k 1 P2 k 2 »¼ «¬1 0 1 »¼ P2 k P2 k 1 º ª P2 k 1 P2 k 1 P2 k «P » « 2 k 3 P2 k 1 P2 k 2 P2 k 2 P2 k 3 » «¬ P2 k 2 P2 k P2 k 1 P2 k 1 P2 k 2 »¼ ª P2 k 1 «P « 2 k 3 ¬« P2 k 2
P2 k 2 P2 k 4 P2 k 3
ª P2 k 1 1 « « P2 k 1 1 « «¬ P2 k 1 Therefore, it is true for every integer The Theorem is completed.
P2 k 3 º P2 k 5 »» P2 k 4 ¼»
P2 k 1 P2 k 1 2 P2 k 1 1
P2 k 1 1 º » P2 k 1 3 » » P2 k 1 2 »¼
n t 1.
Next, we have the following Theorems. For proving of these Theorems are omitted, since it similar to the proving of Theorem 2.1. Then we have; th
Theorem 2.2 Let Pn be the n Padovan sequences, we have;
If
Q2
ª1 0 1 º «1 0 0» , then Q n 2 « » «¬0 1 1 »¼
ª P2 n 2 «P « 2n «¬ P2 n1
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P2 n1 P2 n 1 P2 n
P2 n3 º P2 n 1 »» , n t 1 . P2 n 2 »¼
(9)
th
Theorem 2.3 Let Pn be the n Padovan sequences, we have;
If
Q3
ª1 0 1 º «1 1 0» , then Q n 3 « » «¬0 1 0»¼
ª P2 n 2 «P « 2 n 3 «¬ P2 n1
P2 n 1 P2 n 2 P2 n
P2 n º P2 n1 »» , n t 1 . P2 n 1 »¼
(10)
P2 n 1 P2 n 2 P2 n3
P2 n º P2 n 1 »» , n t 1 . P2 n2 »¼
(11)
P2 n P2 n 1 P2 n1
P2 n 1 º P2 n »» , n t 1 . P2 n 2 »¼
(12)
P2 n3 P2 n 2 P2 n1
P2 n1 º P2 n »» , n t 1 . P2 n 1 »¼
(13)
th
Theorem 2.4 Let Pn be the n Padovan sequences, we have;
If
Q4
ª0 0 1 º «1 1 0» , then Q n 4 « » «¬0 1 1 »¼
ª P2 n1 «P « 2n «¬ P2 n1
th
Theorem 2.5 Let Pn be the n Padovan sequences, we have;
If
Q5
ª1 1 0 º «0 0 1 » , then Q n 5 « » «¬1 0 1 »¼
ª P2 n 2 «P « 2 n1 «¬ P2 n3
th
Theorem 2.6 Let Pn be the n Padovan sequences, we have;
If
Q6
ª1 1 0 º «0 1 1 » , then Q n 6 « » «¬1 0 0»¼
Proposition 2.7 For all integer m, n such that
ª P2 n 2 «P « 2 n1 «¬ P2 n
0 m d n. We have the following relations;
1.
P2 n
P2 m1P2 nm P2 m P2 nm 2 P2 m1P2 n m 1
(14)
2.
P2 n
P2 m P2 nm 1 P2 m1P2 nm 1 P2 m2 P2 nm
(15)
3.
P2 n1
P2 m1P2 nm 1 P2 m P2 nm 3 P2 m1P2 nm 2
(16)
4.
P2 n1
P2 m P2 nm P2 m1P2 nm 2 P2 m 2 P2 nm 1
(17)
Proof. From Theorem 2.1 and the laws of exponent for the square matrix. We have,
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Q1n ª P2 n 1 «P « 2 n 1 «¬ P2 n
P2 n P2 n 2 P2 n 1
P2 n 1 º P2 n 3 »» P2 n 2 »¼
Q1mQ1n m ª P2 m 1 «P « 2 m1 «¬ P2 m
P2 m P2 m 2 P2 m 1
ª a11 a12 «a « 21 a22 «¬ a31 a32
P2 m 1 º ª P2 n m 1 « P2 m3 »» « P2 n m 1 « P2 m 2 »¼ « P ¬ 2 n m
P2 n m P2 n m 2 P2 n m 1
P2 n m 1 º » P2 n m 3 » » P2 n m 2 »¼
a13 º a23 »» a33 »¼
where,
a11
P2 m 1 P2 n m 1 P2 m P2 n m 1 P2 m 1P2 n m
a12
P2 m 1 P2 n m P2 m P2 n m 2 P2 m 1 P2 n m 1
a13
P2 m 1 P2 n m 1 P2 m P2 n m 3 P2 m 1 P2 n m 2
a21
P2 m 1 P2 n m 1 P2 m 2 P2 n m 1 P2 m 3 P2 n m
a22
P2 m 1 P2 n m P2 m 2 P2 n m 2 P2 m 3 P2 n m 1
a23
P2 m 1 P2 n m 1 P2 m 2 P2 n m 3 P2 m 3 P2 n m 2
a31
P2 m P2 n m 1 P2 m 1 P2 n m 1 P2 m 2 P2 n m
a32
P2 m P2 n m P2 m 1 P2 n m 2 P2 m 2 P2 n m 1
a33
P2 m P2 n m 1 P2 m 1 P2 n m 3 P2 m 2 P2 n m 2 .
P2 n
P2 m 1 P2 n m P2 m P2 n m 2 P2 m 1 P2 n m 1 ,
P2 n
P2 m P2 n m 1 P2 m 1 P2 n m 1 P2 m 2 P2 n m ,
P2 n 1
P2 m 1 P2 n m 1 P2 m P2 n m 3 P2 m 1 P2 n m 2
P2 n1
P2 m P2 nm P2 m1P2 nm 2 P2 m2 P2 nm 1.
Thus,
and
This completes the proof.
Remark 2.8 In Proposition 2.7, if 1.
P2 n
2.
P2 n1
3.
P2 n
4.
P2 n1
m 2 in (14), (16) and m 1 in (15) and (17), then
P3 P2 n2 P4 P2 n2 P5 P2 n3
P3 P2 n3 P4 P2 n1 P5 P2 n2 P2 P2 n3 P3 P2 n1 P4 P2 n2
P2 P2 n2 P3 P2 n P4 P2 n1
0 P2n2 1 P2n2 1 P2n3 P2n2 P2n3 , 0 P2 n3 1 P2n1 1 P2n2 P2 n1 P2 n2 , 1 P2 n3 0 P2 n1 1 P2 n2 P2 n2 P2 n3 , 1 P2n2 0 P2n 1 P2n1 P2n1 P2n2 .
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Conclusions
3 u 3 matrices which have similar properties to Padovan Q matrix but also some relation for Padovan number P2n and P2 n 1. Moreover, we conjecture that this concept could be extended to n u n matrices, where n t 4 . In this work, we found not only the new six matrices of
ACKNOWLEDGMENTS The authors would like to thank the referees for useful comments and suggestions on the manuscript.
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