Journal of Number Theory 131 (2011) 1390–1396

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Journal of Number Theory

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Hessenberg matrices and the Pell and Perrin numbers

Fatih Yilmaz ∗, Durmus Bozkurt

Selcuk University, Science Faculty Department of Mathematics, 42250 Campus Konya, Turkey

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 January 2011Revised 15 February 2011Accepted 16 February 2011Available online 29 March 2011Communicated by David Goss

Keywords:Hessenberg matrixPell numberPerrin number

In this paper, we investigate the Pell sequence and the Perrin se-quence and we derive some relationships between these sequencesand permanents and determinants of one type of Hessenberg ma-trices.

© 2011 Elsevier Inc. All rights reserved.

1. Introduction

The Pell and Perrin sequences [1] are defined by the following recurrence relations, respectively:

Pn = 2Pn−1 + Pn−2, where P1 = 1, P2 = 2,

Rn = Rn−2 + Rn−3, where R0 = 3, R1 = 0, R2 = 2

for n > 2. The first few values of the sequences are

n 1 2 3 4 5 6 7 8 9Pn 1 2 5 12 29 70 169 408 985Rn 0 2 3 1 2 5 5 7 10

* Corresponding author.E-mail addresses: fyilmaz@selcuk.edu.tr (F. Yilmaz), dbozkurt@selcuk.edu.tr (D. Bozkurt).

0022-314X/$ – see front matter © 2011 Elsevier Inc. All rights reserved.doi:10.1016/j.jnt.2011.02.002

F. Yilmaz, D. Bozkurt / Journal of Number Theory 131 (2011) 1390–1396 1391

The permanent of a matrix is similar to the determinant but all the signs used in the Laplace expan-sion of minors are positive. The permanent of an n-square matrix is defined by

perA =∑σ∈Sn

n∏i=1

aiσ (i)

where the summation extends over all permutations σ of the symmetric group Sn [6].Let A = [aij] be an m×n matrix with row vectors r1, r2, . . . , rm . We call A contractible on column k,

if column k contains exactly two nonzero elements. Suppose that A is contractible on column k withaik �= 0 �= a jk and i �= j. Then the (m − 1) × (n − 1) matrix Aij:k obtained from A replacing row i witha jkri + aikr j and deleting row j and column k is called the contraction of A on column k relative torows i and j. If A is contractible on row k with aki �= 0 �= akj and i �= j, then the matrix Ak:i j = [AT

ij:k]T

is called the contraction of A on row k relative to columns i and j. We know that if A is a nonnegativematrix and B is a contraction of A [2], then

per A = perB. (1)

It is known that there are a lot of relations between determinants or permanents of matrices andwell-known number sequences. For example, in [2], the authors consider the relationships betweenthe sums of Fibonacci and Lucas numbers by Hessenberg matrices.

In [4], Lee defined the matrix

£n =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 1 0 · · · 01 1 1 0 · · · 0

0 1 1 1...

0 0 1 1. . . 0

......

. . .. . . 1

0 0 · · · 0 1 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

and showed that

per(£n) = Ln−1

where Ln is the nth Lucas number.In [5], the author investigated general tridiagonal matrix determinants and permanents. Also he

showed that the permanent of the tridiagonal matrix based on {ai}, {bi}, {ci} is equal to the deter-minant of the matrix based on {−ai}, {bi}, {ci}.

In [3], the authors found (0, 1,−1) tridiagonal matrices whose determinants and permanents arenegatively subscripted Fibonacci and Lucas numbers. Also, they give an n × n (1,−1) matrix S , suchthat perA = det(A ◦ S), where A ◦ S denotes Hadamard product of A and S . Let S be a (1,−1) matrixof order n, defined with

S =

⎡⎢⎢⎢⎢⎣

1 1 · · · 1 1−1 1 · · · 1 11 −1 · · · 1 1...

.... . .

......

1 1 · · · −1 1

⎤⎥⎥⎥⎥⎦ . (2)

In the present paper, we consider the Pell and Perrin numbers as determinants and permanents ofupper Hessenberg matrices, A = (aij) is an upper Hessenberg matrix if aij = 0 for i > j + 1.

1392 F. Yilmaz, D. Bozkurt / Journal of Number Theory 131 (2011) 1390–1396

2. Determinantal representations of the Pell and Perrin numbers

In this section, we define one type of upper Hessenberg matrix of odd order and show that thepermanents of these type of matrices are the Pell numbers. Let Hn = [hij]n×n be an n-square matrixwith ht,t+2 = 1, hs,s+2 = −1 for t = 2, 4, . . . , n−3

2 and s = 1, 3, . . . , n−12 and hi, j = 1 for |i − j| � 1 and

otherwise 0. Namely:

Hn =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 1 −11 1 1 1 0

1 1 1 −1. . .

. . .. . .

. . .

1 1 1 11 1 1 −1

0 1 1 11 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (3)

Theorem 1. Let Hn be an n-square matrix as in (3), then

perHn = perH (n−2)n = Pn

where Pn is the nth Pell number.

Proof. By definition of the matrix Hn , it can be contracted on column 1. Let Hrn be the rth contraction

of Hn . If r = 1, then

H1n =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

2 0 1 01 1 1 −10 1 1 1 1

. . .. . .

. . .. . .

1 1 1 −11 1 1

0 1 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

Since H1n also can be contracted according to the first column,

H2n =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

2 3 −2 0 01 1 1 1 0 0

1 1 1 −11 1 1 1

. . .. . .

. . .. . .

1 1 1 −10 1 1 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

1 1

F. Yilmaz, D. Bozkurt / Journal of Number Theory 131 (2011) 1390–1396 1393

Going with this process, we have

H3n =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

5 0 2 0 01 1 1 1 0 0

1 1 1 −11 1 1 1

. . .. . .

. . .. . .

1 1 1 −10 1 1 1

1 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

and contracting H3n according to the first column

H4n =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

5 7 −5 0 01 1 1 1 0 0

1 1 1 −11 1 1 1

. . .. . .

. . .. . .

1 1 1 −10 1 1 1

1 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

Continuing this method, we obtain the rth contraction

Hrn =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Pr+1 0 Pr 0 01 1 1 −1 0 0

1 1 1 11 1 1 −1

. . .. . .

. . .. . .

1 1 1 −10 1 1 1

1 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, if r is odd

Hrn =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Pr Pr−1 + Pr −Pr 0 01 1 1 1 0 0

1 1 1 −11 1 1 1

. . .. . .

. . .. . .

1 1 1 −10 1 1 1

1 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, if r is even

where 2 � r � n − 4. Hence

Hn−3n =

[ Pk Pk−1 + Pk −Pk1 1 1

]

0 1 1

1394 F. Yilmaz, D. Bozkurt / Journal of Number Theory 131 (2011) 1390–1396

which, by contraction of Hn−3n on column 1,

Hn−2n =

[Pn 01 1

].

By (1), we have perHn = perH(n−2)n = Pn . �

Let K (n) = [kij] be n × n matrix with k11 = 1, k12 = 2, k13 = 3, k21 = 1, k23 = 1 and km,m+1 =km+1,m = 1 for m = 3, 4, 5, . . . ,n − 1 and kp,p+2 = 1 for p = 3, 4, . . . ,n − 2. Clearly:

Kn =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 2 3 01 0 0 0 0

1 0 1 1. . .

. . .. . .

. . .

1 0 1 10 1 0 1

1 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (4)

Theorem 2. Let Kn be an n-square matrix as in (3), then

perKn = perK (n−2)n = Rn

where Rn is the nth Perrin number.

Proof. By definition of the matrix Kn , it can be contracted on column 1. Namely,

K 1n =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

2 3 0 0 01 0 1 1 00 1 0 1 1

. . .. . .

. . .. . . 0

1 0 1 11 0 1

0 1 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

K 1n also can be contracted on the first column,

K 2n =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

3 2 2 0 0 01 0 1 1 00 1 0 1 1

. . .. . .

. . .. . . 0

1 0 1 11 0 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

0 1 0

F. Yilmaz, D. Bozkurt / Journal of Number Theory 131 (2011) 1390–1396 1395

Continuing this process, we have

K rn =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

Rr+1 Rr+2 Rr 0 0 01 0 1 1 00 1 0 1 1

. . .. . .

. . .. . . 0

1 0 1 11 0 1

0 1 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

for 1 � r � n − 4. Hence

K n−3n =

[ Rn−2 Rn−1 Rn−31 0 10 1 0

]

which by contraction of K n−3n on column 1, gives

K n−2n =

[Rn−1 Rn

1 0

].

By applying (1) we have perKn = perK (n−2)n = Rn, which is desired. �

Let S be a matrix as in (2) and denote the matrices Hn ◦ S and Kn ◦ S by An and Bn , respectively.Thus

An =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 1 −1−1 1 1 1 0

−1 1 1 −1. . .

. . .. . .

. . .

−1 1 1 1−1 1 1 −1

0 −1 1 1−1 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

and

Bn =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 2 3 0−1 0 0 0 0

−1 0 1 1. . .

. . .. . .

. . .

−1 0 1 10 −1 0 1

−1 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

Then, we have

det(An) = perHn = Pn

and

det(Bn) = perKn = Rn.

1396 F. Yilmaz, D. Bozkurt / Journal of Number Theory 131 (2011) 1390–1396

Acknowledgment

This research is supported by TUBITAK and Selcuk University Scientific Research Project Coordina-torship (BAP). This study is a part of corresponding author’s PhD Thesis.

References

[1] K. Kaygısız, D. Bozkurt, k-generalized order-k Perrin number representation by matrix method, Ars Combin., in press.[2] E. Kılıç, Dursun Tasçı, On families of bipartite graphs associated with sums of Fibonacci and Lucas numbers, Ars Combin. 89

(2008) 31–40.[3] E. Kılıç, D. Tasçı, Negatively subscripted Fibonacci and Lucas numbers and their complex factorizations, Ars Combin. 96

(2010) 275–288.[4] G.Y. Lee, k-Lucas numbers and associated bipartite graphs, Linear Algebra Appl. 320 (2000) 51.[5] D.H. Lehmer, Fibonacci and related sequences in periodic tridiagonal matrices, Fibonacci Quart. 12 (1975) 150–158.[6] H. Minc, Permanents, Encyclopedia Math. Appl., vol. 6, Addison–Wesley Publishing Company, London, 1978.