hessenberg matrices and the pell and perrin numbers
TRANSCRIPT
Journal of Number Theory 131 (2011) 1390–1396
Contents 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 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: [email protected] (F. Yilmaz), [email protected] (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 11394 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.