#A83 INTEGERS 18 (2018)

THE ADJACENCY-PELL-HURWITZ NUMBERS

Josh Hiller Department of and Computer Science, Adelpi University, New York [email protected]

Ye¸sim Akuz¨ um¨ Faculty of Science and Letters, Kafkas University, Turkey yesim [email protected]

Om¨ ur¨ Deveci Faculty of Science and Letters, Kafkas University, Turkey yesim [email protected]

Received: 9/25/17, Revised: 8/22/18, Accepted: 10/11/18, Published: 10/26/18

Abstract In this paper, we define the k-adjacency-Pell-Hurwitz numbers by using the Hurwitz of order 4m which is obtained by the aid of the characteristic polynomial of the adjacency-Pell sequence. Firstly, we give relationships between the k-adjacency- Pell-Hurwitz numbers and the generating matrices for these sequences. Further, we obtain the Binet formula for the (2m 1)-adjacency-Pell-Hurwitz numbers. Also, we derive relationships between the k-adjacency-P ell-Hurwitz numbers and perma- nents and determinants of certain matrices. Finally, we give the combinatorial and exponential representations of the k-adjacency-Pell-Hurwitz numbers.

1. Introduction

It is well-known that the Pell sequence is defined by the following equation:

Pn+1 = 2Pn + Pn 1 for n > 0, where P0 = 0, P1 = 1. The adjacency-type sequence is defined in [3] by an mn-order recurrence equation:

n,m n,m n,m xnm+k = xnm n+1+k + xk n,m n,m n,m n,m for k 1, where x1 = = xnm n+1 = 0, xnm n+2 = 1, xnm n+3 = = · · · · · · xn,m = 0 and n, m 2. nm INTEGERS: 18 (2018) 2

Karaduman and Deveci defined the adjacency-Pell sequence as follows:

a (mn + k) = 2a (mn n + k + 1) + a (k) m,n m,n m,n for the integers k 1, m 2 and n 4, with initial constants a (1) = = m,n · · · a (mn 1) = 0 and a (mn) = 1 [5]. m,n m,n Consider the k-step recurrence sequence:

an+k = c0an + c1an+1 + + ck 1an+k 1 · · · where c0, c1, . . . , ck 1 are real constants. Earlier, Kalman [8] derived a number of closed-form formulas for some generalized sequences via the method as follows: If the companion matrix A is defined by

0 1 0 0 0 0 0 1 · · · 0 0 2 ·.· · 3 0 0 0 .. 0 0 A = [a ] = 6 7 i,j k k 6 . . . . . 7 ⇥ 6 . . . . . 7 6 7 6 0 0 0 0 1 7 6 · · · 7 6 c0 c1 c2 ck 2 ck 1 7 6 7, 4 5 then a0 an a1 an+1 An 2 . 3 = 2 . 3 . . 6 7 6 7 6 ak 1 7 6 an+k 1 7 6 7 6 7 4 5 4 5 for n > 0. Consider f a real polynomial of degree n given by

n n 1 f (x) = a0x + a1x + + an 1x + an. · · · INTEGERS: 18 (2018) 3

Hurwitz [7] introduced the matrix Hn = [hi,j]n n associated to f as follows: ⇥ a a a 0 0 0 1 3 5 · · · · · · · · · . . . 2 a0 a2 a4 . . . 3 . . . 6 0 a1 a3 . . . 7 6 7 6 . .. . . 7 6 . a a . 0 . . 7 6 0 2 7 6 . . . . 7 Hn = 6 . 0 a .. a . . 7 6 1 n 7 6 . . .. . 7 6 . . a0 . an 1 0 . 7 6 7 6 . . . 7 6 . . 0 an 2 an . 7 6 7 6 . . . 7 6 . . . a a 0 7 6 n 3 n 1 7 6 0 0 0 a a a 7 6 n 4 n 2 n 7. 4 · · · · · · · · · 5 Several suthors have used homogeneous linear recurrance relations to deduce mis- cellaneous properties for a plethora of sequences; see for example [4, 6, 9, 10, 13, 14, 15, 16, 17, 18, 19]. In particular, Deveci and Shannon defined the adjacency- type numbers and examined their structural properties [3]. The adjacency-Pell numbers, their miscellaneous properties and applications in groups were studied by Deveci and Karaduman in [5]. In the present paper, we define the k-adjacency- Pell-Hurwitz numbers by a recurrence relations of order 4m, (m 2) and give their generating matrices, Binet formulas, permanental, determinantal, combina- torial, exponential representations, and we derive a formula for the sums of the k-adjacency-Pell-Hurwitz numbers.

2. The Main Results

For m 2 and n = 4 it is clear that the characteristic polynomial of the adjacency- Pell sequence is 4m 4m 3 p (x) = x 2x 1. (2.1) Then by (2.1), we see that the Hurwitz matrix H4m = [hi,j]4m 4m associated to a polynomial p is ⇥

2 if i = 2k 1 and j = k + 1 for 1 k 2m, 1 if i = 2k and j = k + 2m for 1 k 2m, [h ] = 8 i,j 4m 4m 1 if i = 2k and j = k for 1 k 2  ⇥ > < 0 otherwise.   > We define the:>k-adjacency-Pell-Hurwitz numbers by using the Hurwitz matrix H4m as shown: (k,m) (k,m) (k,m) x4m+u = 2x4m 2k+1+u + x4m 2k 2+u (2.2) INTEGERS: 18 (2018) 4

(k,m) for the integers u 1, m 2 and 1 k 2m 1, with initial constants x1 = (k,m) (k,m)  (k,m) = x4m 1 = 0 and x4m = 1. Here xi is the ith term of the kth sequence · · · according to the constant m. By (2.2), we may write

2 0 0 1 0 0 0 1 0 0 0 · 0· · 0 0 2 0 1 0 · 0· · 0 0 0 3 · · · 6 0 0 1 0 0 0 0 7 (1) (1) 6 · · · 7 Mm = mi,j = 6 0 0 0 0 1 0 0 7 4m 4m 6 · · · 7 h i ⇥ 6 ...... 7 6 ...... 7 6 7 6 0 0 0 0 0 1 0 7 6 · · · 7 6 0 0 0 0 0 0 1 7 6 · · · 7, 4 5 () () Mm = mi,j = 4m 4m h i ⇥ (2 + 2) th # 0 0 0 0 2 0 0 1 0 0 0 0 0 1 0 0 · 0· · · · · 0 0 2 0 1 0 · · · 0 0 3 · · · 6 0 0 1 0 0 0 7 6 · · · 7 6 0 0 0 1 0 0 0 7 6 · · · 7 6 ...... 7 6 ...... 7 6 7 6 ...... 7 6 ...... 7 6 7 6 0 0 0 1 0 0 0 0 0 7 6 · · · · · · 7 6 1 0 0 0 0 0 7 6 · · · 7 6 0 0 0 0 1 0 0 0 7 6 · · · · · · 7 6 0 0 0 0 1 0 0 0 7 6 · · · · · · 7 6 . . . . . 7 6 ...... 7 6 7 6 . . . . . 7 6 ...... 7 6 7 6 0 0 0 0 0 1 0 7 6 7 6 0 0 · · · 0 0 · · · 0 0 1 7 6 7 , 4 · · · · · · 5 INTEGERS: 18 (2018) 5 for 2 2m 2 and   0 0 0 2 0 0 1 1 0 · 0· · 0 0 2 0 1 0 · · · 0 0 3 · · · 6 0 0 1 0 0 0 7 (2m 1) (2m 1) 6 . . · · · . 7 Mm = mi,j = 6 ...... 7 4m 4m 6 . . . . . 7 ⇥ 6 7 h i 6 . . . . 7 6 ...... 7 6 7 6 0 0 1 0 0 7 6 · · · 7 6 0 0 0 1 0 7 6 · · · 7 . 4 5 (↵) We call matrix Mm the ↵-adjacency-Pell-Hurwitz matrix of size 4m 4m. ⇥ By an inductive argument on ⌧, we obtain

(1,m) (1,m) (1,m) (1,m) x4m+⌧ x4m+⌧ 3 x4m+⌧ 2 x4m+⌧ 1 0 0 0 0 (1,m) (1,m) (1,m) (1,m) · · · 2 x4m+⌧ 1 x4m+⌧ 4 x4m+⌧ 3 x4m+⌧ 2 0 0 0 0 3 (1,m) (1,m) (1,m) (1,m) · · · x4m+⌧ 2 x4m+⌧ 5 x4m+⌧ 4 x4m+⌧ 3 0 0 0 0 6 · · · 7 6 x(1,m) x(1,m) x(1,m) x(1,m) 0 0 0 0 7 ⌧ 6 4m+⌧ 3 4m+⌧ 6 4m+⌧ 5 4m+⌧ 4 7 M (1) = 6 0 0 0 0 1 0 0 · · · 0 7 m 6 7 6 0 0 0 0 0 1 0 · · · 0 7 ⇣ ⌘ 6 7 6 ...... · · · . 7 6 ...... 7 6 7 6 0 0 0 0 0 0 1 0 7 6 7 6 0 0 0 0 0 · · · 0 0 1 7 6 7, 4 · · · 5 for ⌧ 1,

(,m) (,m) (,m) x4m+⌧ x4m+⌧+1 x4m+⌧+2 2 (,m) (,m) · · · (,m) 2 x4m+⌧ 1 x4m+⌧ x4m+⌧+2 3 3 (,m) (,m) · · · (,m) x4m+⌧ 2 x4m+⌧ 1 x4m+⌧+2 4 6 · · · 7 6 . . . 7 ⌧ 6 . . . 7 () 6 (,m) (,m) (,m) 7 Mm = 6 7 x4m+⌧ 2 x4m+⌧ 2+1 x4m+⌧ 2 E 6 · · · 7 ⇣ ⌘ 6 (,m) (,m) (,m) 7 6 x4m+⌧ 2 1 x4m+⌧ 2 x4m+⌧ 3 7 6 · · · 7 6 0 0 0 7 6 · · · 7 6 . . . 7 6 . . . 7 6 7 6 0 0 0 7 6 · · · 7, 4 5 INTEGERS: 18 (2018) 6 for ⌧ 1 and 2 2m 2, where E is the following 4m (4m 2 + 1) matrix:   ⇥ (,m) (,m) (,m) x4m+⌧ 3 x4m+⌧ 2 x4m+⌧ 1 0 0 0 0 (,m) (,m) (,m) · · · x4m+⌧ 4 x4m+⌧ 3 x4m+⌧ 2 0 0 0 0 2 · · · 3 . . . 6 . . . 0 0 0 0 7 6 (,m) (,m) (,m) · · · 7 6 x4m+⌧ 2 3 x4m+⌧ 2 2 x4m+⌧ 2 1 0 0 0 0 7 6 (,m) (,m) (,m) · · · 7 E = 6 x x x 0 0 0 0 7 6 4m+⌧ 2 4 4m+⌧ 2 3 4m+⌧ 2 2 · · · 7 6 0 0 0 1 0 0 0 7 6 7 6 0 0 0 0 1 0 · · · 0 7 6 7 6 . . . . . · · · . 7 6 ...... 7 6 . . . . . 7 6 7 6 0 0 0 0 0 1 0 7 6 · · · 7 6 0 0 0 0 0 0 1 7 4 · · · 5 and ⌧ (2m 1) Mm = ⇣ ⌘ (2m 1,m) (2m 1,m) (2m 1,m) (2m 1,m) (2m 1,m) (2m 1,m) x4m+⌧ x4m+⌧+1 x8m+⌧ 4 x4m+⌧ 3 x4m+⌧ 2 x4m+⌧ 1 (2m 1,m) (2m 1,m) · · · (2m 1,m) (2m 1,m) (2m 1,m) (2m 1,m) 2 x4m+⌧ 1 x4m+⌧ x8m+⌧ 5 x4m+⌧ 4 x4m+⌧ 3 x4m+⌧ 2 3 (2m 1,m) (2m 1,m) · · · (2m 1,m) (2m 1,m) (2m 1,m) (2m 1,m) x4m+⌧ 2 x4m+⌧ 1 x8m+⌧ 6 x4m+⌧ 5 x4m+⌧ 4 x4m+⌧ 3 6 · · · 7 6 ...... 7 6 ...... 7 6 7 6 (2m 1,m) (2m 1,m) (2m 1,m) (2m 1,m) (2m 1,m) (2m 1,m) 7 6 x⌧+3 x⌧+4 x4m+⌧ 1 x⌧ x⌧+1 x⌧+2 7 6 (2m 1,m) (2m 1,m) · · · (2m 1,m) (2m 1,m) (2m 1,m) (2m 1,m) 7 6 x⌧+2 x⌧+3 x4m+⌧ 2 x⌧ 1 x⌧ x⌧+1 7 6 (2m 1,m) (2m 1,m) · · · (2m 1,m) (2m 1,m) (2m 1,m) (2m 1,m) 7 6 x⌧+1 x⌧+2 x4m+⌧ 3 x⌧ 2 x⌧ 1 x⌧ 7 6 · · · 7, 4 5 t (k,m) (k,m) for ⌧ 3. Let m 2 and let St = xi such that 1 k 2m 1. We i=1   introduce matrix H (k, m) by P

1 0 0 0 1 · · · 2 (k) 3 H (k, m) = 0 Mm 6 . 7 6 . 7 6 7 6 0 7 6 7 4 5 for 1 k 2m 1. Note that H (k, m) is a of size (4m + 1)   ⇥ INTEGERS: 18 (2018) 7

(4m + 1), and it can be shown by induction that:

1 0 0 0 (1,m) · · · S⌧+4m 1 2 (1,m) 3 S⌧+4m 2 6 (1,m) 7 6 S⌧+4m 3 7 6 ⌧ 7 ⌧ 6 (1,m) (1) 7 (H (1, m)) = 6 S⌧+4m 4 Mm 7 for ⌧ 1, 6 7 6 0 ⇣ ⌘ 7 6 7 6 0 7 6 7 6 . 7 6 . 7 6 7 6 0 7 6 7 4 5 1 0 0 0 (,m) · · · S⌧+4m 1 2 (,m) 3 S⌧+4m 2 6 7 6 . 7 6 . 7 ⌧ 6 ⌧ 7 (H (, m)) = 6 S(,m) M () 7 for ⌧ 1 and 2 2m 2 6 ⌧+4m 2 2 m 7   6 7 6 0 ⇣ ⌘ 7 6 7 6 0 7 6 7 6 . 7 6 . 7 6 7 6 0 7 6 7 and 4 5 1 0 0 0 (2m 1,m) · · · S⌧+4m 1 2 (2m 1,m) 3 S⌧+4m 2 ⌧ ⌧ 6 . (2m 1) 7 (H (2m 1, m)) = 6 . M 7 (⌧ 3) . 6 m 7 6 (2m 1,m) 7 6 ⇣ ⌘ 7 6 S⌧+2 7 6 (2m 1,m) 7 6 S⌧+1 7 6 (2m 1,m) 7 6 S 7 6 ⌧ 7 4 5 Lemma 2.1. The equation x4m + 2x3 1 = 0 does not have multiple roots for any integer m 2. Proof. Let q (x) = x4m + 2x3 1 and suppose v is a multiple root of q (x). Since q (0) = 0, it follows that v = 0. Then, the hypotheses q (v) = 0 and q0 (v) = 0 imply 4m 63 3 3 26 m 3 4m 3 v = and v = , respectively. It follows that v > 0 and v < 0, 2m 4m 3 inequalities that cannot hold simultaneously for m 2. This is a contradiction resulting from our assumption that v is a multiple root, which concludes the proof of the lemma. INTEGERS: 18 (2018) 8

(2m 1) Let q (v) be the characteristic polynomial of matrix Mm . Then, q (v) = 4m 3 (2m 1) v + 2v 1, a clear fact because Mm is a companion matrix. (2m 1) Let v1, v2, . . .,v4m be the eigenvalues of Mm . By Lemma 2.1, we know that (2m 1) these are 4m distinct numbers. Let Vm be the following : 4m 1 4m 1 4m 1 (v1) (v2) (v4m) 4m 2 4m 2 · · · 4m 2 (v ) (v ) (v ) 2 1 2 m 3 (2m 1) . . · · · . Vm = . . . 6 7 6 v v v 7 6 1 2 4m 7 6 1 1 1 7 6 7. 4 · · · 5 (2m 1) (2m 1) Denote by Vm (i, j) the matrix obtained from Vm by replacing the jth column by ⌧+4m i v1 ⌧+4m i v (2m 1) 2 2 3 Cm (i, j) = . . 6 ⌧+4m i 7 6 v 7 6 4m 7. We can give the generalized Binet formula 4for the (2m 5 1)-adjacency-Pell-Hurwitz numbers with the following theorem. ⌧ (2m 1) (2m 1,⌧) Theorem 2.1. For the matrix Mm = mi,j , for ⌧ 3, 4m 4m ⇣ ⌘ h i ⇥ (2m 1) (2m 1,⌧) det Vm (i, j) mi,j = (2m 1) . (2.3) det Vm

Proof. Consider the integer ⌧ 3 to be fixed. Since v1, v2, . . . , v4m are distinct, the (2m 1) (2m 1) (2m 1) (2m 1) matrix Mm is diagonalizable. Then, Mm Vm = Vm Dm, where (2m 1) D = (v , v , . . . , v ). Since det Vm = 0, we can write m 1 2 4m 6 1 (2m 1) (2m 1) (2m 1) Vm Mm Vm = Dm.

(2m⇣ 1) ⌘ Then, the matrix Mm is similar to Dm and so ⌧ (2m 1) (2m 1) (2m 1) ⌧ Mm Vm = Vm (Dm) . ⇣ ⌘ We can now easily establish the following linear system of equations:

(2m 1,⌧) 4m 1 (2m 1,⌧) 4m 2 (2m 1,⌧) n+4m i mi,1 (v1) + mi,2 (v1) + + mi,4m = (v1) (2m 1,⌧) 4m 1 (2m 1,⌧) 4m 2 · · · (2m 1,⌧) n+4m i m (v ) + m (v ) + + m = (v ) 8 i,1 2 i,2 2 · · · i,4m 2 > . > . < (2m 1,⌧) 4m 1 (2m 1,⌧) 4m 2 (2m 1,⌧) n+4m i mi,1 (v4m) + mi,2 (v4m) + + mi,4m = (v4m) > · · · . > :The> numbers in formula (2.3) are solutions of the last linear system. INTEGERS: 18 (2018) 9

Theorem 2.1 gives immediately: (2m 1,m) Corollary 2.2. Let x⌧ be the ⌧th element of the (2m 1)-adjacency-Pell- Hurwitz sequence, then

(2m 1) (2m 1) (2m 1,m) det Vm (4m, 4m) det Vm (4m 1, 4m 1) x⌧ = (2m 1) = (2m 1) det Vm det Vm (2m 1) det Vm (4m 2, 4m 2) = (2m 1) . det Vm Now we consider the permanental representations of the k-adjacency-Pell-Hurwitz numbers. Definition 2.1. Let M = [m ] be u v real matrix and let r1, r2, . . . , ru and i,j ⇥ c1, c2, . . . , cv be respectively, the row and column vectors of M. If r↵ contains exactly two non-zero entries, then M is contractible on row ↵ . Similarly, M is contractible on column provided c contains exactly two non-zero entries.

Let x1, x2, . . . , xu be row vectors of the matrix M and let M be contractible in th the ↵ column with mi,↵ = 0, mj,↵ = 0 and i = j. Then the (u 1) (v 1) 6 6 6 th ⇥ matrix Mij:↵ obtained from M by replacing the i row with mi,↵xj + mj,↵xi and deleting the jth row and the ↵th column is called the contraction in the ↵th column relative to the ith row and the jth row.

The permanent of a u-square matrix A = [ai,j] is defined by

u

per(A) = ai,(i), S i=1 X2 u Y where the summation extends over all permutations of the symmetric group Su. In [1], Brualdi and Gibson showed that per (A) = per (B) if A is a real matrix of order u > 1 and B is a contraction of A. Let n 4m and let X (k, (m, n)) = xm,n,k , 1 k 2m 1, be the n n i,j   ⇥ super-diagonal matrices defined using theh followingi cases: 1. i = and j = + 2k 2 for 1 n 2k 1,   2. i = , j = + 2k + 1 for 1 n 2k 1, and j = 1 for 2 n,     3. otherwise.

2 if case (1) applies m,n,k 8 xi,j > 1 if case (2) applies > <> 0 if case(3) applies > , > :> INTEGERS: 18 (2018) 10 where m is as in the definition of the k-adjacency-Pell-Hurwitz numbers. Then we have the following theorem.

Theorem 2.3. For 1 k 2m 1, we have   (k,m) per(X (k, (m, n))) = x4m+n.

Proof. Consider the matrix X (1, (m, n)). We will use induction on n. Assume the equation holds for n 4m. Then we must show that the equation per(X (1, (m, n))) = (1,m) x4m+n holds for n + 1. If we expand per(X (1, (m, n))) by the Laplace expansion of permanent according to the first row, then we obtain

per(X (1, (m, n + 1))) = 2 per(X (1, (m, n))) + per(X (1, (m, n 3))). (1,m) Since per(X (1, (m, n 3))) = x4m+n 3, we obtain (1,m) (1,m) (1,m) per(X (1, (m, n + 1))) = 2x4m+n + x4m+n 3 = x4m+n+1. The proofs for 2 k 2m 1 are similar to the above and are omitted.   Let n 4m. Now we define the n n matrices Y (k, (m, n)) = ym,n,k , 1 k ⇥ i,j   2m 1, using the following cases: h i 1. i = and j = + 2k 2 for 1 n 2k 1,   2. i = , j = + 2k 2 for 1 n 2k 1, and j = 1 for 2 n,     3. otherwise,

With these three cases in mind we now define the desired matrix: 2 if case (1) applies m,n,k 8 yi,j = > 1 if case (2) applies > <> 0 if case(3) applies , > > where m is as in the definition: of the k-adjacency-Pell-Hurwitz numbers. In the next theorem we obtain another permanental representation.

Theorem 2.4. For 1 k 2m 1, we have   (k,m) per(Y (k, (m, n))) = x4m 2k 2+n. Proof. Consider matrices Y (, (m, n)) = ym,n, , 2 2m 2. We will use i,j   induction on n. Suppose that the equationh holds ifor n 4m. Then we must show INTEGERS: 18 (2018) 11 that the equation holds for n + 1. If we expand the perY (, (m, n)) by the Laplace expansion of permanent according to the first row, then we obtain per(Y (, (m, n + 1))) = 2 per(Y (, (m, n 2 + 2)))+per(Y (, (m, n 2 1))) (,m) (,m) (,m) = 2x4m 4+n + x4m 4+n 3 = x4m 2+n 1 for 2 2m 2. Thus, the conclusion is obtained.   The proofs for the matrices Y (1, (m, n)) and Y (2m 1, (m, n)) are similar. We now consider the sums of the k-adjacency-Pell-Hurwitz numbers by using their permanental representations. Let n > 4m and suppose that Z (1, (m, n)), Z (, (m, n)), (2 2m 2) and Z (2m 1, (m, n)) are the n n matrices de-   ⇥ fined by

(n 4) th # 1 1 0 0 1 · · · · · · 2 0 Y (1, (m, n 1)) 3 Z (1, (m, n)) = (2.4) 6 . 7 6 . 7 6 7 6 0 7 6 7, 4 5 (n 4) th # 1 1 0 0 1 · · · · · · 2 0 Y (, (m, n 1)) 3 Z (, (m, n)) = (2.5) 6 . 7 6 . 7 6 7 6 0 7 6 7 and 4 5

(n 6) th # 1 1 0 0 1 · · · · · · 2 0 Y (2m 1, (m, n 1)) 3 Z (2m 1, (m, n)) = (2.6) 6 . 7 6 . 7 6 7 6 0 7 6 7. 4 5 Then we have the following theorem. INTEGERS: 18 (2018) 12

Theorem 2.5. For 1 k 2m 1, let the n n matrices Z (k, (m, n)) be as in   ⇥ (2.4),(2.5) and (2.6). Then

n+4m 1 (1,m) per(Z (1, (m, n))) = x" , "=1 X n+4m 2 3 per(Z (, (m, n))) = x(,m), (2 2m 2) , "   "=1 X and n 1 (2m 1,m) per(Z (2m 1, (m, n))) = x . " "=1 X Proof. Consider the matrices Z(2m 1, (m, n)). Expanding per(Z(2m 1, (m, n))) with respect to the first row, we have per(Z (2m 1, (m, n))) = per(Z (2m 1, (m, n 1)))+per(Y (2m 1, (m, n 1))). By Theorem 2.3 and the inductive argument on n, we easily complete the proof. The proofs for 1 k 2m 2 are similar to the above and are omitted.   A matrix M is called convertible if there is an n n (1, 1)-matrix K such that ⇥ perM = det (M K), where M K denotes the Hadamard product of M and K. Let n > 4m and let W be the n n matrix defined by ⇥ 1 1 1 1 1 1 1 1 · · · 1 1 2 1 1 1 · · · 1 1 3 · · · W = 6 ...... 7 . 6 ...... 7 6 7 6 1 1 1 1 1 7 6 · · · 7 6 1 1 1 1 1 7 6 · · · 7 4 5 It is easy to see that per(X (k, (m, n)) = det (X (k, (m, n)) W )), per(Y (k, (m, n)) = det (Y (k, (m, n)) W )) and per(Z (k, (m, n)) = det (Z (k, (m, n)) W )) for n > 4m and 1 k 2m 1.   Consider the n n matrix ⇥ c1 c2 cn 1 0 · · · 0 C = C (c1, c2, . . . , cn) = 2 . . . 3 . .. . 6 7 6 0 1 0 7 6 · · · 7. 4 5 INTEGERS: 18 (2018) 13

For detailed information about the companion matrix, see [11, p.69] and [12, p.284].

(⌧) Theorem 2.6. (Chen and Louck [2]).The (i, j) entry ci,j (c1, c2, . . . , cn) in the ⌧ matrix C (c1, c2, . . . , cn) is given by the following formula:

(⌧) tj + tj+1 + + tn t1 + + tn t1 tn ci,j (c1, c2, . . . , cn) = · · · · · · c1 cn t1 + t2 + + tn ⇥ t1, . . . , tn · · · (t1,tX2,...,tn) · · · ✓ ◆ (2.7) where the summation is over nonnegative integers satisfying t1 + 2t2 + + ntn = t + +t (t1+ +tn)! · · · ⌧ i + j, 1 ··· n = ··· is a multinomial coecient, and the coecients t1,...,tn t1!...tn! in (2.7) are defined to be 1 if ⌧ = i j. Now we give the combinatorial representations for the (2m 1)-adjacency-Pell- Hurwitz sequence by the following Corollary. (2m 1,m) Corollary 2.7. Let x⌧ be the ⌧th element of the (2m 1)-adjacency-Pell- Hurwitz sequence such that ⌧ 3 and m 2. Then

(2m 1,m) t4m 2 + t4m 1 + t4m t1 + + t4m t4m 3 x⌧ = · · · ( 2) t1 + t2 + + t4m ⇥ t1, . . . , t4m (t1,tX2...,t4m) · · · ✓ ◆

t4m 1 + t4m t1 + + t4m t4m 3 = · · · ( 2) t1 + t2 + + t4m ⇥ t1, . . . , t4m (t1,tX2...,t4m) · · · ✓ ◆

t4m t1 + + t4m t4m 3 = · · · ( 2) , t1 + t2 + + t4m ⇥ t1, . . . , t4m (t1,tX2...,t4m) · · · ✓ ◆ where the summation is over nonnegative integers satisfying t +2t + +(4m) t = 1 2 · · · 4m ⌧.

Proof. In Theorem 2.6, if we choose n = 4m and i = j such that 4m 2 i, j 4m,   then the proof follows from (2).

Now we give the generating function of the k-adjacency-Pell-Hurwitz numbers.

Let

(k,m) (k,m) (k,m) (k,m) 2 (k,m) u (k,m) u+1 g (y) = x4m + x4m+1y + x4m+2y + + x4m+u 1y + x4m+uy + . · · · · · · Then 2k 1 (k,m) (k,m) 2k 1 (k,m) 2k (k,m) 2k+1 2y g (y) = x4m 2y + x4m+12y + x4m+22y (k,m) 2k 1+u (k,m) 2k+u + + x4m+u 12y + x4m+u2y + · · · · · · and 2k+2 (k,m) (k,m) 2k+2 (k,m) 2k+3 (k,m) 2k+4 y g (y) = x4m y + x4m+1y + x4m+2y INTEGERS: 18 (2018) 14

(k,m) 2k+2+u (k,m) 2k+2+u+1 + + x4m+u 1y + x4m+uy + . · · · · · · Thus, we have

(k,m) 2k 1 (k,m) 2k+2 (k,m) (k,m) 4m 1 g (y) + 2y g (y) y g (y) = x y . 4m By the definition of the k-adjacency-Pell-Hurwitz numbers, we obtain y4m 1 g(k,m) (y) = , 1 + 2y2k 1 y2k+2 2k+2 2k 1 where 1 k 2m 1 and 0 y 2y < 1.   6 Now we give an exponential representation for the k-adjacency-Pell-Hurwitz numbers.

2k+2 2k 1 Theorem 2.8. For 1 k 2m 1 and 0 y 2y < 1, the k-adjacency-   6 Pell-Hurwitz numbers have the following exponential representation:

2k 1 i (k,m) 4m 1 1 y 3 i g (y) = y exp y 2 . i i=1 ! X Proof. Since

4m 1 (k,m) y 4m 1 2k 1 2k+2 ln g (y) = ln = ln y ln 1 + 2y y 1 + 2y2k 1 y2k+2 and

2k 1 2k+2 2k+2 2k 1 ln 1 + 2y y = ln 1 y 2y 2k 1 3 1 2k 1 2 3 2 = [y y 2 + y y 2 2 1 2k 1 i 3 i + + y y 2 + ] · · · i · · · 2k 1 i 1 y i = y3 2 , i i=1 ! X we have 2k 1 i (k,m) 4m 1 1 y 3 i ln g (y) ln y = y 2 . i i=1 X Therefore, we obtain

2k 1 i g (x) 1 y i ln = y3 2 . y4m 1 i i=1 X The last formula implies the one in the text of the theorem, thus concluding the proof. INTEGERS: 18 (2018) 15

Acknowledgement. This work was supported by the Commission for the Scientific Research Projects of Kafkas University, Project number 2016-FM-23. The authors would like to thank the anonymous referee for a careful reading and many helpful comments that improved the final version of this manuscript.

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