J. Ana. Num. Theor. 4, No. 2, 153-158 (2016) 153 Journal of Analysis & Number Theory An International Journal

http://dx.doi.org/10.18576/jant/040211

A Note on Generalized k-Pell Numbers and Their Determinantal Representation

Ahmet Oteles¸¨ 1,∗ and Mehmet Akbulak2 1 Department of , Faculty of Education, Dicle University, TR-21280, Diyarbakır, Turkey 2 Department of Mathematics, Art and Science Faculty, Siirt University, TR-56100, Siirt, Turkey

Received: 3 Jan. 2016, Revised: 9 Apr. 2016, Accepted: 11 Apr. 2016 Published online: 1 Jul. 2016

Abstract: In this paper, we investigate permanents of an n × n (0,1,2)- by contraction method. We show that the permanent of the matrix is equal to the generalized k-Pell numbers.

Keywords: Pell , Permanent, Contraction of a matrix

1 Introduction where the summation extends over all σ of the symmetric group Sn. The well-known Pell sequence {Pn} is defined by the Let A = [ai j] be an m × n real matrix with row vectors , for n > 2 α1,α2,...,αm. We say A is contractible on column (resp. P = 2P + P row) k if column (resp. row) k contains exactly two n n−1 n−2 nonzero entries. Suppose A is contractible on column k where P1 = 1 and P2 = 2. with aik 6= 0 6= a jk and i 6= j. Then the (m − 1) × (n − 1) In [1], the authors defined k of the matrix Ai j:k obtained from A by replacing row i with k α α generalized order-k Pell numbers Pn as shown: a jk i + aik j and deleting row j and column k is called the contraction of A on column k relative to rows i and j. i i i  i Pn = 2Pn−1 + Pn−2 + ... + Pn−k (1) If A is contractible on row k with aki 6= 0 6= ak j and T for n > 0 and1 ≤ i ≤ k, with initial conditions T i 6= j, then the matrix Ak:i j = Ai j:k is called the h i 1 if n = 1 − i, contraction of A on row k relative to columns i and j. We Pi = for 1 − k ≤ n ≤ 0, n  0 otherwise, say that A can be contracted to a matrix B if either B = A or there exist matrices A0,A1,...,At (t ≥ 1) such that i where Pn is the nth term of the ith sequence. The sequence A0 = A, At = B, and Ar is a contraction of Ar−1 for k Pn is reduced to the usual Pell sequence {Pn} for k = 2. r = 1,...,t. 4 4 In [2], Lee defined the following matrix  For example, for i = 4, they obtained P−3 = 1, P−2 = P4 = P4 = 0 and gave first few terms of generalized order −1 0 10 ... 0 10 ... 00 4-Pell sequence as:  11 1 ... 1 0 ... 00  1,2,5,13,34,88,228,.... L (n,k) = 011 1 ... 10 ... 00  ......  k  ......  The authors called Pn the generalized k-    000 0 ... 00 ... 11  for i = k in (1)[1].   The permanent of an n-square matrix A = [ai j] is (n,2) defined by and showed that perL = Ln−1 and also n (n,k) (k) (k) perL = l , where Ln and ln are respectively the perA = ∑ ∏aiσ(i) n−1 σεSn i=1 nth and nth k-Lucas number.

∗ Corresponding author e-mail: [email protected]

c 2016 NSP Natural Sciences Publishing Cor. 154 A. Oteles¸,¨ M. Akbulak: A note on generalized k-pell numbers

In [3], Minc defined generalized Fibonacci numbers of as follows: order r as 1 1 −1 0 ········· 0 0, if n < 0, . .  11 11 .. .   1, if n = 0, f (n,r)=  .. .   r  01 11 −1 . .   ∑ f (n − k,r), if n > 0.    ......  k=1  ......   Hn =     . . . . .  Minc also defined the n × n super-diagonal (0,1)-matrix  ...... 1 0    F (n,r) = [ f ] as in the following:  . . . . .  i j  ...... −1     . . . .  1 1 0 0  ...... 1  ······ ···   . . .  0 ············ 01 1   1 .. .. .     . .   0 .. .. 0  and    . . .  12 3 0 ········· 0 F (n,r)=  . .. .. 1  (2)    .. .   . . . .  10 0 0 . .  ......     .. .   . . . .   01 0 1 1 . .   ......       . . . .   0 ········· 0 11   . .. 1 01 1 .. .    Kn =  .  ......   ...... 0  where    . . . . .   ...... 1    1, if i = j + 1and 1 ≤ j ≤ r − 1,  . . . .   ...... 1   1, if 1 ≤ i ≤ n − r − 2 and i ≤ j ≤ i + r − 2,   fi j =  0 ············ 0 10   1, if n − r + 3 ≤ i ≤ n and i ≤ j ≤ n,   0, otherwise. They showed that  Then he proved that (n−2) perHn = perHn = Pn perF (n,r)= f (n,r − 1). and (n−2) In [4], the authors denoted the matrix F (n,r) in (2) perKn = perKn = Rn as F (n,k) and obtained permanent of this matrix, the same where Pn is nth Pell number, Rn is nth , and result in [3, Theorem 2], by applying contraction to the (n−2) (n−2) matrix F (n,k). Hn and Kn is (n − 2)th contraction of the matrix Hn and K , respectively. In [5], Kilic defined the n × n super-diagonal (0,1,2)- n In this paper, we show that the permanent of the n × n matrix S(k,n) as: (0,1,2)-matrix S(k,n) equals to the (n + 1)th generalized k-Pell number by the contraction method. 2 1 ··· 1 0 ··· 0 . .  12 1 ... 1 .. .   ..   01 2 1 ... . 0  2 Generalized k-Pell numbers by contraction    . . . . .  method S(k,n)=  ...... 1  (3)    ......   ......    In this section, we use generalized order-k Pell sequence  ......  k  . . . . 1  Pn given in (1) for i = k. For this sequence, it can be    0 ········· 0 12  written for n > k ≥ 2   k k k k k and proved that the permanent of S(k,n) equals to the P1−k = 1, P2−k = P3−k = ... = P−1 = P0 = 0 (n + 1)th generalized k-Pell number. In [6], Yilmaz and Bozkurt defined the n × n upper and k k k k Hessenberg matrix Hn = [hi j] and n × n matrix Kn = [ki j] Pn = 2Pn−1 + Pn−2 + ... + Pn−k.

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First few terms of the generalized k-Pell numbers are column 1 such that follows in [1] as: 52 0 ······ 0 k k k k . . P1 = 2P0 + P−1 + ... + P1−k = 1,  12 1 .. .  k k k k . P2 = 2P1 + P0 + ... + P2−k = 2 × 1 = 2,  .. .. .  (n)  01 2 . . .  k k k k T1 =  , P3 = 2P2 + P1 + ... + P3 k = 2 × 2 + 1 = 5,  . . . . .  −  ...... 0  k k k k   P4 = 2P3 + P2 + ... + P4−k = 2 × 5 + 2 + 1 = 13,  . . . .   ...... 1  Pk 2Pk Pk ... Pk 2 13 5 2 1 34,....   5 = 4 + 3 + + 5−k = × + + + =  0 ······ 0 12    Then, some terms of this sequence can be given for some (1) (1) (n) k and n values in the following table: where t11 = 5 = P3 and t12 = 2 = P2. Also the matrix T1 can be contracted on column 1 such that n k 1 2 3 4 5 6 7 8 9 10 2 1 2 5 12 29 70 169 408 985 2378 12 5 0 ······ 0 . . 3 1 2 5 13 33 84 214 545 1388 3535  .. .  4 1 2 5 13 34 88 228 591 1532 3971 1 2 1 . 5 1 2 5 13 34 89 232 605 1578 4116  .. .  (n)  01 2 1 . .  T =  , 2  . . . . .  It is easily seen that from the above table, Pk is the  ...... 0  n   usual Pell sequence {P } for k = 2.  . . . .  n   ...... 1  The following Lemma is well-known from [7].    0 ······ 0 12    Lemma 2.1. Let A be a nonnegative integral matrix of order n for n > 1 and let B be a contraction of A. Then (2) (2) where t11 = 12 = P4 and t12 = 5 = P3. Continuing this process, we obtain the rth contraction of T (n) as follows perA = perB. (4) Pr+2 Pr+1 0 ······ 0 . . Let S(k,n) be the (k + 1)st (0,1,2)-matrix of order n  1 2 1 .. .  (n,k) given in (3). For convenience, we use the notation S  .. .  F (n,k) (n)  0 1 21 . .  instead of S(k,n). If be the matrix as in (2) and In Tr =  ,  . . . . .  be the identity matrix of order n, then it is immediately  ...... 0  (n,k) F (n,k) (n,k)   seen that S = + In. Then the matrix S is  . . . .   ...... 1  contractible on column 1 relative to rows 1 and 2. In   particular, if k = 2, the matrix S(n,k) reduced to the  0 ······ 0 12  (n)   tridiagonal Toeplitz matrix T = tridiagn (1,2,1) of order n as follows: for 2 ≤ r ≤ n − 2. So, we have

21 0 0 (n) Pn Pn−1 ······ Tn−2 = . (6) 12 1 0 0  1 2   ···  . . 01 2 1 .. . By combining (4)and (6), we get (n)   T =  . . . . . . (5)  ......  (n) (n)  . 0  perT = perT  .  n−2  ......  = 2P + P  . . . . 1  n n−1    0 ······ 0 12  = Pn+1, and the proof is completed. Lemma 2.2. Let the matrix T (n) beasin (5). Then (n,k) (t) Lemma 2.3. Let St = si j be the tth contraction of the (n,k) h i (n) matrix S given in (3)and 1 ≤ t ≤ n − 3. Then perT = Pn+1, k Pt 2, if j = 1, where P be the n 1 st Pell number. + n+1 ( + )  t+1 (t) ∑ Pk, if 2 j k t, (n) (r) s =  r ≤ ≤ − for k > t +2 1, j  r 1 Proof. Let Tr = ti j be the rth contraction of the matrix  = (t) (n) h i (n) s − Pk , if k −t + 1 ≤ j ≤ n −t, T given in (5). The matrix T can be contracted on  1, j−1 t+ j−k  

c 2016 NSP Natural Sciences Publishing Cor. 156 A. Oteles¸,¨ M. Akbulak: A note on generalized k-pell numbers

and relative to rows 1 and 2, we write

Pk , if j = 1, (t+1) (t) (t) t+2 s11 = 2s11 + s12  t+1 t+1 (t) ∑ k s =  Pr , if j = 2, for k ≤ t + 2. = 2Pk + ∑ Pk 1, j  r=t+3−k t+2 r  r=1 (t) k s1, j−1 − Pt+ j−k, if 3 ≤ j ≤ n −t, k k k  = 2Pt+2 + Pt+1 + ... + P1 .   k k k (t) k Also since P = P = ... = P = 0 for k > t + 2, we In any case, if s −P < 0, we assume that si j = 0. 0 −1 t−k+3 1, j−1 t+ j−k k k k write P0 + P 1 + ... + P = 0. Consequently, we get Proof. We will prove the lemma by induction on t. − t−k+3 (n,k) If we apply the contraction to the matrix S s(t+1) = 2Pk + Pk + ... + Pk + Pk + Pk + ... + Pk according to column 1, we get 11 t+2 t+1 1 0 −1 t+3−k k = Pt+3, 53 3 3 ... 32 0 ... 0 12 1 1 ... 11 1 ... 0 and   . . . 0 ...... (t+1) (t) (t) (n,k)   s = s + s , q = 2,3,...,k −t − 1 S =   1,q 11 1,q+1 1  ..   .  t+1   = Pk + ∑ Pk  .. ..  t+2 r  0 . . 1  r=1  0 12  t+2   k Pk Pk + Pk Pk + Pk ... Pk + Pk Pk 0 ... 0 = ∑ Pr . 3 2 1 2 1 2 1 2 r=1  12 1 ... 1 11 ... 0  ...... (t+1)+1  0 . . .  (t+1)   Thus, s = ∑ Pk for q = 2,3,...,k −(t +1), finally =  . , 1,q r  ..  r=1    . .   0 .. .. 1  s(t+1) = s(t) + s(t)   1,k−t 11 1,k−t+1  0 12    (t) (t) k = s11 + s1,k−t − Pt+(k−t+1)−k (1) (1) (1) (1) k k k k (t+1) k where s11 = P3 , s12 = ... = s , = P2 +P1 , s , = P2 = 1 k−1 1 k = s1,k−t−1 − P(t+1)+(k−t)−k. k k k (1) k (P2 +P1 )−P1 = s1,k−1 −P1 . So, it is easy to show that the assumption is true for t = 1. So by the recurrence relation, we reach We now assume the assumption is true for t. Then, there are two cases need to prove. (t+1) (t+1) k s1, j = s1, j−1 − P(t+1)+ j−k, (t) k (t) Case 1. We consider k > t +2. Since s11 = Pt+2, s12 = (t) ∑t+1 k (t) (t) k for k −t ≤ j ≤ n − (t + 1) and proof of Case 1 completes. ... = s1,k−t = r=1 Pr , and s1, j = s1, j−1 −Pt+ j−k for k−t + (n,k) (t) Case 2. In this case, we will consider k ≤ t + 2. For 1 ≤ j ≤ n−t, we clearly write the matrix St = si j as: t = 1 and k = 2, Lemma 2 is obtained. We assume that h i (t) k (t) ∑t+1 k (t) (t) k s11 = Pt+2, s12 = r=t+3−k Pr and s1, j = s1, j−1 − Pt+ j−k t+1 t+1 t+1 t+1 k ∑ k ∑ k ∑ k ∑ k k k k (n,k) (t) Pt+2 Pr Pr ... Pr Pr ... Pt+1 + Pt Pt+1 0 ... 0 for 3 ≤ j ≤ n − t. Then, the matrix St = si j follows  r=1 r=1 r=1 r=2  h i 1 2 11 ... 1 1 ... 1 10 as:  . . . . .   0 ......  t+1 t+1 t+1   Pk ∑ Pk ∑ Pk ∑ Pk ... Pk + Pk Pk 0 ... 0  .  t+2 r r r t+1 t t+1  .. 1   r=t+3−k r=t+4−k r=t+5−k   . 12 1 11 ... 1 110  .  . . . . .  .   0 ......       ..   ..   . 1   . 1 .  . . .   .   0 ...... 1   .     .   0 12   ..     1  (7)  . . .   0 ...... 1  Now, we must prove our assumption for t + 1. Since the  0 12  (n,k)   matrix St given by (7) is contractible on column 1 (8)

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(n,k) If we apply the contraction to the matrix St given in (8) 3 Conclusion according to its first column, we get The famous sequences (e.g. Fibonacci, Pell) s(t+1) 2s(t) s(t) 11 = 11 + 12 provide invaluable opportunities for exploration, and t+1 (k) (k) contribute handsomely to the beauty of mathematics, = 2Pt+2 + ∑ Pr especially number theory. Among these sequences, Pell r=t+3−k numbers have achieved a kind of celebrity status. There is (k) (k) (k) = 2Pt+2 + Pt+1 + ... + Pt−k+3 a vast literature concerned with this sequence. This study (k) is a different application which consider the connection = Pt+3, between the permanents of a matrix and Pell numbers. (t+1) (t) (t) s12 = s11 + s13 k (t) k = Pt+2 + s12 − Pt+3−k Acknowledgement t+1 k k k The authors are grateful to the anonymous referee for a = Pt+2 + ∑ Pr − Pt+3−k r=t+3−k careful checking of the details and for helpful comments k k k k k that improved this paper. = Pt+2 + Pt+1 + ... + Pt+4−k + Pt+3−k − Pt+3−k t+2 k = ∑ Pr r=t+4−k References (t+1)+1 = ∑ Pk [1] E. Kilic, D. Tasci, The generalized Binet formula, r representation and sums of the generalized order-k Pell r=(t+1)+3−k numbers, Taiwanese Journal of Mathematics 10 (6), 1661- and 1670 (2006). (t+1) (t) (t) [2] G.Y. Lee, k-Lucas numbers and associated bipartite graphs, s13 = s11 + s14 Linear Algebra and Its Application 320, 51-61 (2000). (t) (t) k = s11 + s13 − Pt+4−k [3] H. Minc, Permanents of (0,1)-Circulants, Canadian (t+1) Mathematical Bulletin 7 (2), 253-263 (1964). = s − Pk . 12 (t+1)+3−k [4]G. Y. Lee, S. G. Lee and H. G. Shin, On the k-generalized Thus, we get the recurrence relation Fibonacci matrix Qk, Linear Algebra and Its Application 251, 73-88 (1997). (t+1) (t+1) k s1, j = s1, j−1 − P(t+1)+ j−k, [5] E. Kilic, On the usual Fibonacci and generalized order-k Pell numbers, Ars Combinatoria 88, 33-45 (2008).  and the proof of Case 2 completes. [6] F. Yilmaz, D. Bozkurt, Hessenberg matrices and the Pell and k Theorem 2.4. Let Pn+1 be the (n + 1)st generalized k-Pell Perrin numbers, Journal of Number Theory 131, 1390-1396 number for n ≥ k. Then (2011). (n,k) k [7] R. A. Brualdi, P. M. Gibson, Convex polyhedra of doubly perS = Pn+1. stochastic matrices I: applications of the permanent function, Journal of Combinatorial Theory, A 22, 194-230 (n,k) Proof. Since the matrix S is contractible, by Lemma 3 (1977). we obtain n−2 n−2 k k k k P ∑ Pr ∑ Pr − P (n,k)  n−1 n−k  ¨ S = r=n−k r=n−k , Ahmet Oteles¸ n−3 12 1   received the PhD degree  01 2  in Mathematics at the   Selc¸uk University. His ∑n−2 k k ∑n−2 k. where r=n−k Pr − Pn−k = r=n−k+1 Pr So we get research interests are: Linear k k k k algebra and its applications, (n,k) P P + P + ... + P S = n n−1 n−2 n−k+1 . matrix theory, number theory n−2 1 2   and graph theory. He has By combining the last equality and (4), we reach published research articles in (n,k) (n,k) reputed international journals perS = perSn−2 of mathematics. He is a referee of mathematical journals. k k k = 2Pn + Pn−1 + ... + Pn−k+1 k = Pn+1, and the proof is completed.

c 2016 NSP Natural Sciences Publishing Cor. 158 A. Oteles¸,¨ M. Akbulak: A note on generalized k-pell numbers

Mehmet Akbulak is Associate Professor of Mathematics at Siirt University. He started his working life as a mathematics teacher in a high school in Konya at 2000. He started working at the Selc¸uk University as a Research Assistant at 2005. He received his PhD degree in Mathematics at the Selc¸uk University at 2009. He has been working in the Siirt University for 7 years. He is a referee of many international journals in the frame of pure and applied mathematics. His main research interests are: matrix theory, number theory and graph theory.

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