Selçuk J. Appl. Math. Selçuk Journal of Special Issue. pp. 27-32, 2012 Applied Mathematics
On the Pell Sequence and Hessenberg Matrices Fatih Yılmaz, Durmus Bozkurt
Selcuk University, Science Faculty Department of Mathematics, 42250, Konya, Turkiye e-mail: fyilm az@ selcuk.edu.tr,db ozkurt@ selcuk.edu.tr
Presented in 3 National Communication Days of Konya Eregli Kemal Akman Vocational School, 28-29 April 2011.
Abstract. In this paper, we investigate the relationship between permanents of one type of upper-Hessenberg matrices (of odd order) and the Pell numbers. At the final part of this paper, we give a Maple 13 source code to strength our calculations.
Key words: Hessenberg matrix; Permanent; Pell number.
2000 Mathematics Subject Classification: 05B20; 15A15; 11B39. 1. Introduction and Preliminaries
Among numerical sequences, Pell numbers have achieved a kind of celebrity status. Although the sequence has been studied extensively for a long time, it remains to fascinating and there always seems to be some amazing properties aspects that are revealed by looking at it closely.
The Pell sequence is defined by the recurrence relation for 2
(1) = 2−1+ −2
with initial conditions 1 = 1 and 2 = 2 (see [3]. The first few values of the
sequence in (1) are given as the following form:
1 2 3 4 5 6 7 8 9 10
1 2 5 12 29 70 169 408 985 2378
Furthermore the permanent of an -square matrix is defined by
= X ∈ Y =1 ()
where the summation extends over all permutations of the symmetric group
of the same matrix but the minor difference is that all of the signs used in the Laplace expansion of minors are positive.
Let = [] be an × matrix with row vectors 1 2 . We call
is contractible on column , if the th column contains exactly two non zero elements. Suppose that is contractible on column with 6= 0 6= 0
and 6= . Then the ( − 1) × ( − 1) matrix : = [:] obtained from
replacing row with + and deleting row and column This
progress is called the contraction of on column relative to rows and . If is contractible on row with 6= 0 6= 0 and 6= , then the matrix
: = [: ] is called the contraction of on row relative to columns
and . This contraction method(s) can be found in [1]. We know that if is a nonnegative integer matrix and is a contraction of (by [1]), then
=
In literature, there are a lot of papers which investigate relationships between number sequences and permanents of matrices. For example, Lee (see [4]) de-fined the following matrix
£= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 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
£= −1
where is the th Lucas number. Ocal et. al. (see [8]) gave some
determi-nantal and permanental representations of -generalized Fibonacci and Lucas numbers. In [6], the authors derived some relations between permanents of some tridiagonal matrices with applications to the negatively and positively subscripted usual Fibonacci and Lucas numbers. In [9], the authors defined × upper Hessenberg matrix as below:
(2) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 11 12 13 · · · 1 21 22 23 · · · 2 0 32 33 · · · 3 .. . ... ... . .. ... 0 0 0 · · · ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Then they derived some relationships between the sums of second order linear recurrences and permanents or determinants of certain Hessenberg matrices. In [5], the authors showed that there are relationships between a generalized
Lucas sequence and the permanent (and so the determinant) of some Hessen-berg matrices. Kılıç and Ta¸sçı (see [1]) investigated similar relationships among Fibonacci, Lucas numbers (and their sums) and permanents of Hessenberg ma-trices. Moreover, the authors (see [7]) investigated the Pell and Perrin sequence and derived some relations between these sequences and permanents and deter-minants of one type of Hessenberg matrices.
In this paper, we consider one type of upper-Hessenberg matrix of odd order. We also investigate relationships between permanents of the upper Hessenberg matrix and sums of consecutive five Pell numbers.
2. Determinantal representations of the Pell numbers
In this section, we consider a special form of upper Hessenberg matrix given by (2). Let us define -square ( = 2 + 1, = 2 3 ) upper-Hessenberg matrix of odd order, as follows:
(3) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 3 1 −1 1 1 0 1 1 1 1 1 1 1 1 1 −1 0 0 . .. ... ... ... ... ... . .. ... ... ... ... 0 1 1 1 1 0 1 1 1 −1 1 1 1 0 1 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
Theorem 1. Let be an -square matrix as in (3), then
= (−2)= +2
X
=−2
where is the th Pell number.
Proof. Let us compute first few permanents of the matrix given by (3). In other words: For = 2, = 5; per5= 4+ 3+ 2+ 1+ 0= 4 X =0 = 20 For = 3, = 7; per7= 5+ 4+ 3+ 2+ 1= 5 X =1 = 49
For = 4, = 9; per9= 6+ 5+ 4+ 3+ 2= 6 X =2 = 118 For = 5, = 11; per11= 7+ 6+ 5+ 4+ 3= 7 X =3 = 285
More generally; we obtain:
For = , = 2 + 1; per = per2+1= +2
X
=−2
As it can be seen that the permanent of the matrix given by (3) is equal to sum of five consecutive terms of the Pell sequence.
By the definition of matrix, it can be contracted according to the first column as (1) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 4 2 4 4 3 0 · · · 0 1 1 1 −1 0 0 1 1 1 1 . .. ... ... ... 1 1 1 1 0 1 1 1 −1 1 1 1 0 1 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
In here, (1) can also be contracted by first column,
(2) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 6 8 0 3 0 0 · · · 0 1 1 1 1 0 0 1 1 1 −1 . .. ... ... ... 1 1 1 1 0 1 1 1 −1 1 1 1 0 1 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
Hence by iterating this procedure, we obtain the th contraction as:
(4) ()= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ +3P =−1 −6 6 +2P =−2 0 0 · · · 0 1 1 1 −1 0 0 1 1 1 1 . .. . .. ... ... 1 1 1 1 0 1 1 1 −1 1 1 1 0 1 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
where = 2 + 1 and = 1 2 . In fact we can rewrite (4) for even numbers as follows: ()= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ +2P =−2 +3P =−1 − µ +2 P =−2 +6 ¶ − µ +2 P =−2 −6 ¶ 0 · · · 0 1 1 1 1 1 1 1 −1 . .. . .. . .. . .. 1 1 1 −1 1 1 1 0 1 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ,
where = 2 ( = 2 3 ) and 3 ≤ ≤ − 3 Hence
(−3)= ⎛ ⎜ ⎜ ⎜ ⎝ +1 X =−3 +2 X =−2 − Ã +1 X =−3 + 6 ! − Ã +1 X =−3 − 6 ! 1 1 1 0 1 1 ⎞ ⎟ ⎟ ⎟ ⎠
By contraction (−3) according to first column, we obtain:
(−3)= ⎛ ⎜ ⎝ +2 X =−2 − 6 6 1 1 ⎞ ⎟ ⎠ That is: = (−2)= +2 X =−2 which is desired. 3. An Application
In this section we give a Maple 13 source code to check the results easily. restart:
with(LinearAlgebra): permanent:=proc(n) local i,j,k,c,C;
c:=(i,j)-piecewise(i=j+1,1,j=i+1,1,j=i+2,(-1)^i,j=4 and i=1,1,j=5 and i=1,1,j=6 and i=2,1,j=5 and i=2,1,j=1 and i=1,3,i=j,1);
C:=Matrix(n,n,c): for k from 0 to n-3 do print(k,C):
for j from 2 to n-k do
od: C:=DeleteRow(DeleteColumn(Matrix(n-k,n-k,C),1),2): od: print(k,eval(C)): end proc: References
1. Kılıç, E. and Ta¸sçı, D., On families of bipartite graphs associated with sums of Fibonacci and Lucas numbers, Ars Combinatoria, 89 (2008) 31-40.
2. Minc, H., Encyclopedia of Mathematics and Its Applications, Permanents, Vol. 6, Addison-Wesley Publishing Company, London, 1978.
3. Yılmaz, F., “Generalization of Some Number Sequences”, Ms. Thesis, Selçuk University, Institute of Natural and Applied Sciences, 2009 (In Turkish).
4. G. Y. Lee,-Lucas numbers and associated bipartite graphs, Linear Algebra and Its Applications, 320 (2000) 51.
5. Kılıç, E., Ta¸sçı, D. And Haukkanen, P., On the genaralized Lucas sequences by hessenberg matrices, Ars Combinatoria 95 (2010) 383-395.
6. Kılıç, E. and Ta¸sçı, D., On the permanents of some tridiagonal matrices with appli-cations to the Fibonacci and Lucas numbers, Rocky Mountain Journal of Mathematics, V. 37, No:6, 2007.
7. Yılmaz, F. and Bozkurt, D., Hessenberg matrices and the Pell and Perrin numbers, Journal of Number Theory, 131 (2011) 1390-1396.
8. Ocal, A., Tuglu, N. and Altinisik, E., On the representation of -generalized Fi-bonacci and Lucas numbers, Applied Mathematics and Computation, 170 (2005) 584-596.
9. Kılıç, E. and Ta¸sçı, D., On sums of second order linear recurrences by Hessenberg matrices, Rocky Mountain Journal of Mathematics, V:38, Number 2, 2008.
