Padovan Numbers by the Permanents of a Certain Complex Pentadiagonal Matrix

In this paper, we consider a certain type of complex pentadiagonal matrices. Then we show that the permanents of this matrix generate Padovan numbers. Finally, we give a Maple procedure in order to verify our result.

The permanent of a n × n matrixA = (a ij ) is defined by acadj@garmian.edu.krdVol.5, No.2 (June, 2018) where the summation extends over all permutations σ of the symmetric group S n .Thepermanent of a matrix is analogous to the determinant, where all of the signs used in theLaplace expansion of minors are positive.Permanents have many applications in physics, chemistry, graph theory, electrical engineering, and so on [5,6,7,8,9].One of the most important applications of permanents is the relationship between some special types of matrices and the wellknown number sequences.There are many papers in relation to that applications.[10,11,12,13,14,15,16,17,18,19,20,21,22,23,24] are some of them.In this paper, we consider a certain type of complex pentadiagonal matrices.Then we show that the permanents of this matrix generate Padovan numbers.Finally, we give a Maple procedure in order to verify our result.

Main Results
Let A = [a ij ]be an m × n real matrix with row vectors a 1 , a 2 , … , a m .We say A iscontractible on column(resp.row) k if column (resp.row) kcontains exactly twononzero entries.Suppose A is contractibleon column k with a ik ≠ 0 ≠ a jk and i ≠ j.Then the (m − 1) × (n − 1)matrix A ij:k obtained fromA by replacing row i with a jk α i + a ik α j and deleting row j and column k is called the contraction of A on column k relative to rows i and j.If A is contractible on row a ki ≠ 0 ≠ a kj and i ≠ j, then the matrix T is called the contraction of A on row k relative to columns i and j.
We say that A can be contracted to a matrix B if either B = A or there exist matrices A 0 , A 1 , . . ., A t (t ≥ 1) such that A 0 = A, A t = B, and A r is a contraction of A r−1 for r = 1, . . ., t [6].
Brualdi and Gibson [6] proved the following result about the permanent of a matrix.Lemma 1 LetA be a nonnegative integral matrix of order n for n > 1 and let B be a contraction of A. Then perA = perB. ( Let H n = h ij be an n × n pentadiagonal matrix as the following where i = √−1.If n = 5, then we obtain the permanent of H 5 by using Laplace expansion as the following By the contraction method introduced by Brualdi in [6], we now present the following theorem that gives the relationship between the permanent of the pentadiagonal matrix H n and the Padovan number P (n).
Theorem 2 Let H n be the n × n pentadiagonal matrix given by (2).Then the permanent of the matrix is equal to the n th Padovan number P (n).

Proof.Let H
Since the definition of the matrix H n ; thematrix H n can be contracted on column 1 so that acadj@garmian.edu.krdVol.5, No.2 (June, 2018) Since the matrix H n 1 can be contracted on column 1 Furthermore, the matrix H n 2 can be contracted on column 1 and P (3) = P (4) = 2, P (5) = 3 so that acadj@garmian.edu.krdVol.5, No.2 (June, 2018) The following Maple procedure calculates the permanent of the pentadiagonal matrix H n given by (2).