AbstractIn this paper, we investigate the Pell sequence and the Perrin sequence and we derive some relationships between these sequences and permanents and determinants of one type of Hessenberg matrices

Bozkurt, Durmus

Yilmaz, Fatih

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Journal of Number Theory 131 (2011) 1390–1396Contents lists available at ScienceDirect
Journal of Number Theory
www.elsevier.com/locate/jnt
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 2011
Revised 15 February 2011
Accepted 16 February 2011
Available online 29 March 2011
Communicated by David Goss
Keywords:
Hessenberg matrix
Pell number
Perrin number
In this paper, we investigate the Pell sequence and the Perrin se-
quence and we derive some relationships between these sequences
and 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 9
Pn 1 2 5 12 29 70 169 408 985
Rn 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 1391The 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 with
aik = 0 = a jk and i = j. Then the (m − 1) × (n − 1) matrix Aij:k obtained from A replacing row i with
a jkri + aikr 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 k with aki = 0 = akj and i = j, then the matrix Ak:i j = [ATij:k]T
is called the contraction of A on row k relative to columns i and j. We know that if A is a nonnegative
matrix and B is a contraction of A [2], then
perA = perB. (1)
It is known that there are a lot of relations between determinants or permanents of matrices and
well-known number sequences. For example, in [2], the authors consider the relationships between
the sums of Fibonacci and Lucas numbers by Hessenberg matrices.
In [4], Lee deﬁned the matrix
£n =
⎡
⎢⎢⎢⎢⎢⎢⎢⎢⎣
1 0 1 0 · · · 0
1 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 are
negatively subscripted Fibonacci and Lucas numbers. Also, they give an n × n (1,−1) matrix S , such
that perA = det(A ◦ S), where A ◦ S denotes Hadamard product of A and S . Let S be a (1,−1) matrix
of order n, deﬁned with
S =
⎡
⎢⎢⎢⎢⎣
1 1 · · · 1 1
−1 1 · · · 1 1
1 −1 · · · 1 1
...
...
. . .
...
...
1 1 · · · −1 1
⎤
⎥⎥⎥⎥⎦ . (2)
In the present paper, we consider the Pell and Perrin numbers as determinants and permanents of
upper 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) 1390–13962. 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 the
permanents of these type of matrices are the Pell numbers. Let Hn = [hij]n×n be an n-square matrix
with ht,t+2 = 1, hs,s+2 = −1 for t = 2, 4, . . . , n−32 and s = 1, 3, . . . , n−12 and hi, j = 1 for |i − j| 1 and
otherwise 0. Namely:
Hn =
⎡
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
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
⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
. (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 0
1 1 1 −1
0 1 1 1 1
. . .
. . .
. . .
. . .
1 1 1 −1
1 1 1
0 1 1
⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎦
.
Since H1n also can be contracted according to the ﬁrst column,
H2n =
⎡
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
2 3 −2 0 0
1 1 1 1 0 0
1 1 1 −1
1 1 1 1
. . .
. . .
. . .
. . .
1 1 1 −1
0 1 1 1
⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
.1 1
F. Yilmaz, D. Bozkurt / Journal of Number Theory 131 (2011) 1390–1396 1393Going with this process, we have
H3n =
⎡
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
5 0 2 0 0
1 1 1 1 0 0
1 1 1 −1
1 1 1 1
. . .
. . .
. . .
. . .
1 1 1 −1
0 1 1 1
1 1
⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
and contracting H3n according to the ﬁrst column
H4n =
⎡
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
5 7 −5 0 0
1 1 1 1 0 0
1 1 1 −1
1 1 1 1
. . .
. . .
. . .
. . .
1 1 1 −1
0 1 1 1
1 1
⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
.
Continuing this method, we obtain the rth contraction
Hrn =
⎡
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
Pr+1 0 Pr 0 0
1 1 1 −1 0 0
1 1 1 1
1 1 1 −1
. . .
. . .
. . .
. . .
1 1 1 −1
0 1 1 1
1 1
⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
, if r is odd
Hrn =
⎡
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
Pr Pr−1 + Pr −Pr 0 0
1 1 1 1 0 0
1 1 1 −1
1 1 1 1
. . .
. . .
. . .
. . .
1 1 1 −1
0 1 1 1
1 1
⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
, if r is even
where 2 r  n − 4. Hence
Hn−3n =
[ Pk Pk−1 + Pk −Pk
1 1 1
]
0 1 1
1394 F. Yilmaz, D. Bozkurt / Journal of Number Theory 131 (2011) 1390–1396which, by contraction of Hn−3n on column 1,
Hn−2n =
[
Pn 0
1 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 0
1 0 0 0 0
1 0 1 1
. . .
. . .
. . .
. . .
1 0 1 1
0 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 0
1 0 1 1 0
0 1 0 1 1
. . .
. . .
. . .
. . . 0
1 0 1 1
1 0 1
0 1 0
⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎦
.
K 1n also can be contracted on the ﬁrst column,
K 2n =
⎡
⎢⎢⎢⎢⎢⎢⎢⎢⎣
3 2 2 0 0 0
1 0 1 1 0
0 1 0 1 1
. . .
. . .
. . .
. . . 0
1 0 1 1
1 0 1
⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎦
.0 1 0
F. Yilmaz, D. Bozkurt / Journal of Number Theory 131 (2011) 1390–1396 1395Continuing this process, we have
Krn =
⎡
⎢⎢⎢⎢⎢⎢⎢⎢⎣
Rr+1 Rr+2 Rr 0 0 0
1 0 1 1 0
0 1 0 1 1
. . .
. . .
. . .
. . . 0
1 0 1 1
1 0 1
0 1 0
⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎦
for 1 r  n − 4. Hence
Kn−3n =
[ Rn−2 Rn−1 Rn−3
1 0 1
0 1 0
]
which by contraction of Kn−3n on column 1, gives
Kn−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 1
0 −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) 1390–1396Acknowledgment
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 Tas¸çı, On families of bipartite graphs associated with sums of Fibonacci and Lucas numbers, Ars Combin. 89
(2008) 31–40.
[3] E. Kılıç, D. Tas¸çı, 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.


Hessenberg matrices and the Pell and Perrin numbers 

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Hessenberg matrices and the Pell and Perrin numbers

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