TETRANACCI MATRIX VIA PASCAL’S MATRIX

  Bulletin of Mathematics

  ISSN Printed: 2087-5126; Online: 2355-8202

Vol. 09, No. 01 (2017), pp. 1–7. https://talenta.usu.ac.id

TETRANACCI MATRIX VIA PASCAL’S

MATRIX

_Abstract.

Mirfaturiqa, Sri Gemawati and M. D. H. Gamal

  The Pascal’s matrix expressed by P n is the n×n lower triangular matrix

whose entry is a number on the Pascal’s triangle. Then, the tetranacci matrix is

expressed by M n with each entry is a tetranacci number. In this article, we discuss

About the relationship of the Pascal’s matrix and the tetranacci matrix. Then from

the relationship of this two matrices we obtain a new matrix, i.e. the E n matrix so

_E that the Pascal’s matrix can be expressed as P n n n .

  = M

  1. INTRODUCTION Pascal’s triangle is a geometry rule that contains a binomial coefficient arrangement that looks like a triangle. Many articles discuss the properties of Pascal’s triangle, the relationship between the triangles Pascal and the Fibonacci number, the Pascal’s matrix, and the Pascal’s matrix relationship with the Fibonacci matrix. Varnadore [7] discusses the relationship of the Pascal’s triangle with the Fibonacci number, he obtains that the sum of every number of Pascal’s triangles diagonally produces Fibonacci numbers. Pascal’s triangle can be expressed as a square matrix with each entry being a number on Pascal’s triangle so that this matrix is called Pascal’s matrix. Brawer and Pirovino [1] provide a form of matrix representation of the Pascal’s triangle and discuss the algebraic form of the Pascal’s matrix. Then, Received 24-04-2017, Accepted 20-07-2017.

  2010 Mathematics Subject Classification

: 11B39, 11Hxx

Key words and Phrases : Tetranacci sequence, Tetranacci matrix, Pascal’s matrix.

  

Mirfaturiqa, Sri Gemawati & M. D. H.Gamal – Tetranacci matrix via pascal’s matrix

  Call and Veleman [2] define the n × n Pascal’s matrix as the lower triangular Pascal’s matrix.

  Similar to Pascal’s triangle, the Fibonacci number can also be expressed as a square matrix. Lee et al. [4] define the Fibonacci matrix then from the definition of the Fibonacci matrix obtained the factorization form of the Fibonacci matrix and the eigenvalues of the Fibonacci symmetric matrix. Lee et al. [5] and Zhang and Wang [8] discuss the relationship of Pascal matrices with the Fibonacci matrix and obtain the different results. The same idea on the Pascal’s triangle and the Fibonacci number, the tribonacci number can also be expressed as a matrix.

  Sabeth et al. [6] define a matrix of the tribonacci expressed by T n and discusses the relationship of Pascal’s matrix and the tribonacci matrix. They derive two new matrices R n and A n , using the two matrices so that

  R T the P n Pascal’s matrix can be expressed by P n = T n n and P n = A n n . In this article, discusses the relationship between the Pascal’s matrix and the tetranacci matrix. In addition, the matrix of tetranacci and a new matrix E n are defined to obtain the relationship of Pascal matrix and the tetranacci matrix.

  2. PASCAL’S MATRIX AND TETRANACCI MATRIX In this section, a definition of Pascal’s matrix and tetranacci matrix are given. Pascal’s triangle can be express into a square matrix of Pascal matrices.The Pascal’s matrix is defines as follows [1, 2]. Definition 1.1

  For every natural number n, P n is the n×n Pascal’s matrix with entries P = [p ] for every i, j = 1, 2, 3, · · · , n, where

  n i,j ₍ i−

  1

  , if i ≥ j,

  j−

  1

  p

  i,j =

  (1) 0, otherwise. In similar way as the Fibonacci numbers and tribonacci numbers, tetranacci numbers can also be represented in the form of a square matrix. Then, with a similar idea to definition of the Fibonacci matrix in Lee et al. [5], the n × n tetranacci matrix is defined as follows.

  Definition 1.2 For every natural number n, M n is the n × n matrix tetranacci with each entries the M n = [m i,j ], ∀i, j = 1, 2, 3, · · · , n, where ₍

  M ,

  i−j +3 if i ≥ j,

  m = (2)

  i,j 0, otherwise.

  

Mirfaturiqa, Sri Gemawati & M. D. H.Gamal – Tetranacci matrix via pascal’s matrix

  From the definition 1.2, the matrix M n matrix is the lower triangular matrix with the main diagonal is 1 and the determinant value of the M n tetranacci matrix is the result of the multiplication of the diagonal entries so as to obtain det(M ) = 1. Since det (M ) 6= 0 so the M matrix tetranacci

  

n n n

has an inverse.

  From the calculation results obtain invers matrix tertanacci M as

  6 following. _{   −}

  1 _{−  − −   }

  1 1 

  1

  1

  1

1 M   .

  = (3)

  6 _{   − − − −   } − 1 − 1 −

  1

  1

  1

  1

  1

  1

  1 − 1 − 1 − 1 −

  1

  1 Based on the observations from equation (3), it can be conclude that for every entry of the inverse matrix the M apply i.e. each column entry m i,j

  6

  pattern is 1, -1, -1, -1 and -1. This is true for the n × n tetranacci matrix, the column entry m i,j pattern will not change. Thus, the general form of the inverse tetranacci matrix is n × n, For every natural number n inverse _{− h i ′}

  1

  matrix of the tetranacci matrix defined with entries M = m for every

  n i,j

  i, j = 1, 2, 3, · · · , n, where _{ ′  } 1, if i = j, m

  = − 1, if i − 4 ≤ j ≤ i − 1, (4)

  i,j _{ } 0, otherwise.

  3. TETRANACCI MATRIX VIS PASCAL’S MATRIX In a similar idea as in [5, 6], we define the matrix E n and we have the relations between the Pascal’s matrix and the tetranacci matrix as follows.

  Definition 1.3 For every natural number n, E n is the n × n matrix with entries E n = [e i,j ] for every i, j = 1, 2, 3, · · · , n, where i − 1 i − 2 i − 3 i − 4 i − 5 e = − − − − . (5)

  i,j

  j − j − j − j − j −

  1

  1

  1

  1

  1 From the definition 1.3, we see that e = 1, e = 0, ∀j ≥ 2; e =

  1,1 1,2 2,1

  0, e 2,2 = 1, e 2,j = 0, ∀j ≥ 3; e 3,1 = −1, e 3,2 = e 3,3 = 1, e 3,j = 0, ∀j ≥ 4; e 4,1 = −

  2, e = 0, e = 2, e = 1, e = 0, ∀j ≥ 5; e i, = −3, ∀j ≥ 5 and

  4,2 4,3 4,4 4,j

  1

  ∀i, j ≥ 2, e i,j = e i− 1,j−1 + e i− 1,j .

  Using the definition of P n as in (1.1), M n as in (1.2) and E n as in (1.3), Mirfaturiqa, Sri Gemawati & M. D. H.Gamal – Tetranacci matrix via pascal’s matrix

  Theorem 1.1 Given an E n matrix, so for every natural number n with the

  P n Pascal’s matrix defined by equation (1) and the M n tetranacci matrix E defined in equation (2), it can be express is that P n = M n n .

  Proof. For every natural number n, M matrix tetranacci is an invertible

  n

  matrix. It will be proven that ₋

  1 M P = E . (6)

n n

n

  Then, note that the left-hand side in the equation (6), if ∀i = 1 and _{′ ′} ∀j ≥

  2, then m = m = 0. So, ∀i, j = 1, we have

  i,j 1,j _{X ′} n n _{X ′} m p m p . k,j = k, = 1 = e 1,1 i,k

  1

1,k

k =1 k =1 _′

  If ∀i = 1 and ∀j ≥ 2, then m = 0 dan p = 0. So, ∀i = 1 and ∀j ≥ 2, we

  1,j 1,j

  have _{X ′} n n _{X ′} m p m p .

  

k,j = k, = 0 = e

i,k 1 1,j

1,k

k =1 k =1

  So, from equation (4) and (1), ∀i ≥ 4 and ∀j ≥ 2, we have _{X ′ ′ ′ ′ ′ ′} n m p p p p p p ,

  k,j = m i,j + m i− + m i− + m i− + m i− i,k i,i i,i− 1,j i,i− 2,j i,i− 3,j i,i− 4,j

  1

  2

  3

  4 k =1

  i − 1 (i − 1) − 1 (i − 2) − 1 (i − 3) − 1 − − −

  = j − 1 j − 1 j − 1 j −

  1 (i − 4) − 1

  − , j −

  1 i − 1 i − 2 i − 3 i − 4 i − 5 = − − − − , j − j − j − j − j −

  1

  1

  1

  1

  1 = e i,j . ₋

1 P

  Therefore, we have M n = E n , the proof is completed. So, for every

  n

  natural number n and we defined the matrix E n and the matrix M n we E have P n = M n n .

  Mirfaturiqa, Sri Gemawati & M. D. H.Gamal – Tetranacci matrix via pascal’s matrix

  From the theorem 1.1, we have that combinatorial identity involving the tetranacci numbers as follows. For every 1 ≤ k ≤ n and n ≥ j, we have that _X n n − 1 k − 1 k − 2 k − 3 k − 4 i − 5 M − − − − .

  = n−k

• 3

  1

  j − 1 j − 1 j − 1 j − 1 j − 1 j −

  k =j

  (7) E

  Note that for the entry of p n,j in the matrix P n = M n n has the form _X n p = m e = M e . (8)

  n,j n,k k,j n−k +3 k,j k

=j

So that from the equation (8), then the equation (7) the following holds.

  In particular, if for j = 1, with attention to each entry of the first column of the matrix E n and for p n,j = p n, , we have

  1 _X n

  p = M e ,

  1 k =1 _X n

  n, 1 n−k +3 k,

  e e e e e , = M n + M n + M n + M n− + (M n− + · · · + M ) k,

• 2 1,1 +1 2,1 3,1 1 4,1

  2

  3

  1 k =5

  − M − − = M n n 2M n− 3(M n− + · · · + M ).

• 2

  1

  

2

  3 If for j = 2, with attention to each entry of the second column of the matrix

  E

  n and for p n,j = p n, , we have

  2 _X n

  M p n, = n−k e k, ,

  2 +3

  2 k =2 _X n

  e e e e , = M n +1 2,2 + M n 3,2 + M n−

  1 4,2 + (M n− 2 + · · · + M 3 ) k,

  2 k =5

  = M + M + (M + · · · + M )(e + e ), ∀k ≥ 5.

  n +1 n n−

  2 3 k− 1,j−1 k− 1,j So, the proof follows.

  From the definition of E n as in (5), we have that matrix E n is an invertible matrix. The general form for each entry of the inverse matrix E

  n

  

Mirfaturiqa, Sri Gemawati & M. D. H.Gamal – Tetranacci matrix via pascal’s matrix

₋

  1

  as follows. For every natural number n, E is the n × n inverse matrix of _{− h i ′} n

  1 E e

n defined with entries E = for every i, j = 1, 2, 3, · · · , n, where

n i,j _{′ X} i

  i − 1

  

i +k

  e = (−1) E . (9)

  k−j +3 i,j

  − k

  1

  k =j _{′ ′}

  From the equation (9), we see that ∀i = j, e = 1 dan ∀j ≥ 2, e = _{′ ′} i,j i,j e + e .

  i− i− 1,j−1 1,j ₋

  1 Using the definition of P n as in 1.1, M n as in 1.2 and E as in (9), n

  we can derive the following theorem. ₋

1 Theorem 1.2

  Given an E matrix, so for every natural number n with

  n

  the Pascal P matrix defined by equation (1) and the M tetranacci matrix

  n n ₋

  1 E defined in equation (2), it can be express is that M n = P n . _{− −} n

  1

  1 M

  Proof. It is enough to prove that P n = E , in similar way as in

  n n theorem 1.1.

  4. CONCLUDING REMARKS In this paper the authors discuss only the relationship between the lower triangular tetranacci matrix and the Pascal matrix. Then from the relationship of two matrices we obtain the formula for a new matrix i.e. the matrix E . In addition, using the new matrix obtain the factorization of the

  n

  relationship between the tetranacci matrix and the Pascal matrix. There- fore, further discussing could be focused on the upper triangular tetranacci matrix, symmetric tetranacci matrix, and n-nacci matrix.

  

REFERENCES

1.

  R. Brawer and M. Pirovino, The linear algebra of the Pascal matrix, Linear Algebra and Its Applications, 174 (1992), 13–23.

  2. G. S. Call and D. J. Velleman, Pascal matrices, The American Mathe- matical Monthly, 100 (1993), 372–376.

  

3. T. Koshy , Fibonacci and Lucas Numbers with Applications, Wiley In-

terscience, New York, (2001).

  4. G. Y. Lee, J. S. Kim and S. G. Lee, Factorizations and eigenvalues of

  Fibonacci and symmetric Fibonacci matrices, Fibonacci Quarterly, 40 (2002), 203–211.

  

Mirfaturiqa, Sri Gemawati & M. D. H.Gamal – Tetranacci matrix via pascal’s matrix

5.

  G. Y. Lee, J. S. Kim and S. H. Cho, Some combinatorial identities via Fibonacci numbers, Discrete Applied Mathematics, 130(2003), 527–534.

  

6. N. Sabeth, S. Gemawati and H. Saleh, A factorization of the tribonacci

  matrix and the Pascal matrix, Applied Mathematical Sciences, 11 (2017), 489–497.

  7. J. Varnadore, Pascal’s triangle and Fibonacci numbers, The Mathematics Theacher, 84 (1991), 314–316.

  

8. Z. Zhang and X. Wang, A factorization of the symmetric Pascal matrix in-

  volving the Fibonacci matrix, Discrete Applied Mathematics, 155 (2007), 2371–2376.

  Mirfaturiqa : Graduate Student, Department of Mathematics, Faculty of Mathe-

matics and Natural Sciences, University of Riau, Bina Widya Campus, Pekanbaru

  28293, Indonesia.

  E-mail: turiqamirfa@gmail.com

  Sri Gemawati : Senior Lecturer, Department of Mathematics, Faculty of Mathe-

matics and Natural Sciences, University of Riau, Bina Widya Campus, Pekanbaru

  28293, Indonesia.

  E-mail: gemawati.sri@gmail.com

  

M. D. H. Gamal : Associate Professor, Department of Mathematics, Faculty of

Mathematics and Natural Sciences, University of Riau, Bina Widya Campus, Pekan-

baru 28293, Indonesia.

  E-mail: mdhgamal@unri.ac.id