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 Mathematics, 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)-matrix by contraction method. We show that the permanent of the matrix is equal to the generalized k-Pell numbers. Keywords: Pell sequence, Permanent, Contraction of a matrix 1 Introduction where the summation extends over all permutations 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 recurrence relation, 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 sequences 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-Pell number 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 Lucas number 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 Perrin number, 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. c 2016 NSP Natural Sciences Publishing Cor. J. Ana. Num. Theor. 4, No. 2, 153-158 (2016) / www.naturalspublishing.com/Journals.asp 155 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.