C o m m u n . Fa c . S c i. U n iv . A n k . S é r. A 1 M a th . S ta t. Vo lu m e 6 6 , N u m b e r 1 , P a g e s 2 4 2 –2 5 3 (2 0 1 7 ) D O I: 1 0 .1 5 0 1 / C o m m u a 1 _ 0 0 0 0 0 0 0 7 9 3 IS S N 1 3 0 3 –5 9 9 1
ON THE GENERALIZED PERRIN AND CORDONNIER MATRICES ADEM S ¸ AH I·N Abstract. In the present paper, we study the associated polynomials of Perrin and Cordonnier numbers. We de…ne generalized Perrin and Cordonnier matrices using these polynomials. We obtain the inverse of generalized Cordonnier matrices and give some relationships between generalized Perrin and Cordonnier matrices. In addition, we give a factorization of generalized Cordonnier matrices. Finally, we give some determinantal representation of associated polynomials Cordonnier numbers.
1. Introduction There are several hundreds of papers on Fibonacci numbers and other recurrence related sequences published during the last 30 years. Perrin numbers and Cordonnier numbers are some of them. Perrin numbers and Cordonnier numbers are Pn = Pn 2 + Pn 3 for n > 3 and P1 = 0; P2 = 2; P3 = 3; Cn = Cn
2
+ Cn
3
for n > 3 and C1 = 1; C2 = 1; C3 = 1;
respectively. The characteristic equation associated with the Perrin and Cordonnier sequence is x3 x 1 = 0 with roots ; ; ; in which = 1; 324718, is called plastic number and Cn+1 Pn+1 lim = lim = : n!1 Cn n!1 Pn The plastic number is used in art and architecture. Richard Padovan studied on plastic number in Architecture and Mathematics in [20, 21]. Christopher Bartlett found a signi…cant number of paintings with canvas sizes that have the aspect ratio of approximately 1.35. This ratio reminds Plastic number[1]. In [17] authors constructed the Plastic number in a heuristic way, explaining its relation to human Received by the editors: May 15, 2016, Accepted: Sept. 25, 2016. 2010 Mathematics Subject Classi…cation. Primary 5A24, 11B68; Secondary 15A09, 15A15. Key words and phrases. Associated polynomials of Perrin and Cordonnier numbers, generalized Perrin and Cordonnier matrices. c 2 0 1 7 A n ka ra U n ive rsity C o m m u n ic a tio n s d e la Fa c u lté d e s S c ie n c e s d e l’U n ive rsité d ’A n ka ra . S é rie s A 1 . M a th e m a tic s a n d S ta tistic s.
242
ON THE GENERALIZED PERRIN AND CORDONNIER M ATRICES
243
perception in three-dimensional space through architectural style of Dom Hans van der Laan. In [22], authors de…ned associated polynomials of Perrin and Cordonnier sequences as; Pn (x) = x2 Pn
2 (x)
3 (x)
for n > 3 and P1 (x) = 0; P2 (x) = 2; P3 (x) = 3x;
Cn (x) = x2 Cn
2 (x) + Cn 3 (x)
for n > 3 and C1 (x) = 1; C2 (x) = x; C3 (x) = x2 ;
+ Pn
respectively. If we take x = 1 we obtain Pn (x) = Pn and Cn (x) = Cn . Yilmaz and Taskara [26] developed the matrix sequences that represent Padovan and Perrin numbers. Kaygisiz and Bozkurt [5] de…ned k sequences of generalized order-k Perrrin numbers. Kaygisiz and Sahin [9] de…ned generalized Van der Laan and Perrin Polynomials, and generalizations of Van der Laan and Perrin Numbers. Many researchers have studied Linear Algebra of the some matrices. In [2] authors discussed the Linear Algebra of the Pascal Matrix, in [14] authors examined the linear algebra of the k-Fibonacci matrix and the symmetric k-Fibonacci matrix. In [12] authors studied on the Pell Matrix. Sahin [23] gave the (q; x; s)-Fibonacci and Lucas matrices, obtained the inverse of these matrices and give some factorization of these matrices. Lee et al. [13] de…ned Fibonacci matrices and gave the factorization of Fibonacci matrix and obtained inverse of this matrix. In addition many researchers have studied matrix representations of number sequences. Yilmaz and Bozkurt [25] gave matrix representation of Perrin sequences. Kaygisiz and Sahin [10] calculated terms of associated polynomials of Perrin and Cordonnier numbers by using determinants and permanents of various Hessenberg matrices. More examples can be found in [3, 6, 7, 8, 11, 15, 16, 19, 23]. Lemma 1.1. (Cf. Theorem of [3]) Let An be an n n lower Hessenberg matrix for all n 1 and de…ne det(A0 ) = 1. Then, det(A1 ) = a11 and for n 2 det(An ) = an;n det(An
1)
+
n X1 r=1
[( 1)n
r
n Y1
an;r (
aj;j+1 ) det(Ar
1 )]:
j=r
In this paper, …rst we de…ne generalized Perrin and Cordonnier matrices using associated polynomials of Perrin and Cordonnier numbers. We obtain the inverse of generalized Cordonnier matrices with the aid of determinants of some Hessenberg matrices which obtained from a part of these matrices. We also give some relationships between generalized Perrin and Cordonnier matrices in this section. Secondly, we give a factorization of generalized Cordonnier matrices. In the last section we give some determinantal representation of associated polynomials Cordonnier numbers.
ADEM S ¸ AH I·N
244
2. Generalized Perrin and Cordonnier matrices De…nition 2.1. Let n be any positive integer, the n n lower triangular generalized Cordonnier matrix Cn;x = [ci;j ]i;j=1;2;:::;n are de…ned by Ci 0;
ci;j = For example, C4;x
if i j > 0 otherwise.
j+1 (x);
2
1 6 x =6 4 x2 1 + x3
0 1 x x2
0 0 1 x
(1)
3 0 0 7 7: 0 5 1
De…nition 2.2. Let n be any positive integer, the n n lower triangular generalized Perrin matrix Pn;x = [pi;j ]i;j=1;2;:::;n are de…ned by Pi 0;
pi;j = For example, P4;x
j+2 (x);
2
2 6 3x =6 4 2x2 2 + 3x3
if i j > 0 otherwise.
0 2 3x 2x2
0 0 2 3x
De…nition 2.3. Let n be any positive integer, the n sequence H Cn;x = [ai;j ]i;j=1;2;:::;n are de…ned by ai;j =
Ci 0;
j+2 (x);
(2)
3 0 0 7 7: 0 5 2
if i j + 1 otherwise.
n lower Hessenberg matrix 0
Pn Lemma 2.4. Let c0 (x) = 1, c1 (x) = x; cn+1 (x) = xcn (x)+ k=1 ( 1)n Then, det(H Cn;x ) = cn (x) for any positive integer n 1:
(3) k+1
Cn
k+3 (x)ck 1 (x):
Proof. We proceed by induction on n. The result clearly holds for n = 1. Now suppose that the result is true for all positive integers less than or equal to n. We prove it for n + 1. In fact, by using Lemma 1.1 we have 2 3 n n X Y 4( 1)n+1 i an+1;i aj;j+1 det(H Ci 1;x )5 det(H Cn+1;x ) = x det(H Cn;x ) + i=1
=
x det(H Cn;x ) +
n X
j=i
( 1)n+1 i Cn
i+3 (x) det(H Ci 1;x )
i=1
From the hypothesis of induction, we obtain n X det(H Cn+1;x ) = xcn (x) + ( 1)n+1 i Cn i=1
i+3 (x)ci 1 (x)
:
:
ON THE GENERALIZED PERRIN AND CORDONNIER M ATRICES
245
Therefore, det(H Cn;x ) = cn (x) holds for all positive integers n. Example 2.5. We obtain c3 (x); c4 (x) by using 2 Lemma 2.4. 2 3 x 1 x 1 0 6 x2 x 2 x 1 5 = 1 = c3 (x);det6 det4 x 4 1 + x3 x2 1 + x3 x2 x 4 x + x 1 + x3 c4 (x):
0 1 x x2
3 0 0 7 7 = x = 1 5 x
Corollary 2.6. Let (H Cn;x ) be the n n Hessenberg matrix in (3). Then, det(H Cn;x ) = cn (x) = xn 3 for any positive integer n 3: Proof. We proceed by induction on n. The result clearly holds for n = 3. Now suppose that the result is true for all positive integers less than or equal to n. We prove it for n + 1. It is clear by the Laplace expansion of the last column that, det(H Cn+1;x )
=
x det(Mn;n )
=
x det(H Cn;x )
=
n 3
xx
det(Mn det(Mn
det(Mn
1;n ) 1;n )
1;n )
2
and since nth row of Mn 1;n is equal x ((n 1)th row of Mn 1;n ) + ((n 2)th row of Mn 1;n ); det(Mn 1;n ) = 0, where Mi;j is the (i; j) minor matrix of H Cn;x . So we obtain det(H Cn+1;x ) = xn 2 : Theorem 2.7. Let n be any positive integer, Cn;x is n n lower triangular generalized Cordonnier matrix in (1) and (H Cn;x ) be the n n Hessenberg matrix in (3). Then (Cn;x ) 1 = [cpi;j ] is obtained by 8 < ( 1)i j det(H Ci j;x ); if i j > 0 p 1; if i j = 0 [ci;j ] = : 0; otherwise.
Proof. Note that it su¢ ces to prove that Cn;x (Cn;x ) 1 = In . We take Cn;x (Cn;x ) 1 = Pn [ai;j ]1 i;j n . It is obvious that ai;j = k=0 ci;k cpk;j = 0 for i j < 0 and ai;j = Pn p p 1 we have k=0 ci;k ck;j = ci;i ci;i = 1 for i = j: For i > j ai;j =
n X
k=0
ci;k cpk;j
=
i X
ci;k cpk;j
k=j
= Ci j+1 (x) Ci j (x)c1 (x) + + C1 (x)( 1)i j ci j (x) Pi j and we know Ci j+1 (x) = s=1 ( 1)s+1 cs (x)Ci j+1 s (x) from de…nition of cn (x). Pn p 1 which implies that Thus, we obtain ai;j = k=0 ci;k ck;j = 0 for i > j 1 Cn;x (Cn;x ) = In :
ADEM S ¸ AH I·N
246
Now, we show a relation between the generalized Perrin and Cordonnier matrices. The n n lower triangular matrix Tn;x := [ri;j ], (1 i; j n) is de…ned by (P i j k k=0 ( 1) Pi j k+2 (x)ck (x); if i > j; ri;j = 0; otherwise: Theorem 2.8. Let n be any positive integer, Pn;x and Cn;x are n gular generalized Perrin and Cordonnier matrices, then
n lower trian-
Tn;x Cn;x = Pn;x : Proof. Note that it su¢ ces to prove that Pn;x (Cn;x ) ri;j = 0 for i j > 0: For i j 1 we have n X
pi;k cpk;j
=
k=0
n X
Pi
k+2 (x)(
1)k
j
ck
j (x)
Pi
k k+2 (x)( 1)
j
ck
j (x) =
1
= Tn;x . It is obvious that
k=0
=
i X k=j
which implies that Pn;x (Cn;x )
i j X
Pi+j
1)k ck (x) = ri;j
k+2 (x)(
k=0 1
= Tn;x ; as desired.
Example 2.9. We obtain relation between the generalized Perrin and Cordonnier matrices for n=5 by using Theorem 2.8. T5;x C5;x
=
P5;x : 2 2 0 0 6 x 2 0 6 6 x2 x 2 6 3 4 x x2 x x4 x3 x2 2 2 0 6 3x 2 6 2 2x 3x = 6 6 4 3x3 + 2 2x2 4 3x + 2x 3x3 + 2
32 1 0 0 6 x 0 0 7 76 2 6 0 0 7 76 x 3 5 4 1+x 2 0 x + x4 x 2 3 0 0 0 0 0 0 7 7 2 0 0 7 7: 3x 2 0 5 2x2 3x 2
Now we give a companion matrix Qn;x as follows: 2 0 1 6 .. 6 . 0 0 6 6 0 0 Qn;x = 6 6 .. 6 . 0 0 6 4 0 0 ( 1)n+1 cn (x) ( 1)n cn 1 (x)
0 1 x x2 1 + x3
0
0
0 1
0 0
0 1 0 0 c3 (x) 0
0 0 1 x x2
0
0 0 0 0 1
3
7 7 7 7 7 7 0 7 7 1 5 x n 0 0
0 0 0 1 x
:
n
3 7 7 7 7 5
ON THE GENERALIZED PERRIN AND CORDONNIER M ATRICES
Theorem 2.10. Let m is
247
n be the integers, then the last column of matrix (Qn;x )m 2 6 6 6 4
Cm
n+2 (x)
.. . Cm (x) Cm+1 (x)
3
7 7 7: 5
Proof. We proceed by induction on m. The result clearly holds for m = 1. Now suppose that the result is true for all positive integers less than or equal to m. We prove it for m + 1. The last column of matrix (Qn;x )m+1 is 2
1
0 0
0 0
0
0
0
0 1
0 0
0 0
3
3 72 7 Cm n+2 (x) 76 7 .. 76 7 . 76 7 74 .. 5 C (x) 7 m . 0 0 0 1 0 7 C (x) 5 m+1 0 0 0 0 1 ( 1)n+1 cn (x) ( 1)n cn 1 (x) c3 (x) 0 x 3 2 3 2 Cm n+3 (x) Cm n+3 (x) 7 6 7 6 .. .. 7 6 7 6 . . 7: 6 7=6 4 4 5 Cm+1 (x) 5 Cm+1 (x) Cm+2 (x) ( 1)n+1 cn (x)Cm n+2 (x) + + c1 (x)Cm+1 (x)
6 6 6 6 6 6 6 6 4 =
0
..
.
2.1. Factorizations of Generalized Cordonnier matrices. The set of all nsquare matrices is denoted by Hn : A matrix H 2 Hn of the form 2
6 6 H=6 6 4
H11
0
0 .. .
H22 .. .
0
in which Hii 2 Hni (i = 1; 2; :::; k) and
k P
.
0 .. .
. 0
0 Hkk
.. ..
3 7 7 7 7 5
ni = n, is called block diagonal. No-
i=1
tationally, such a matrix is often indicated as H = H11 H22 Hkk ; this is called the direct sum of the matrices H11 ; H22 ; ; Hkk : Lee et al. [13, 14] and Sahin[23] gave some factorization. Like these we consider factorization of Cn;x . Let In be the identity matrix of order n. We de…ne the
ADEM S ¸ AH I·N
248
matrices C n;x = [1]
(Cn
1;x )
and
2
6 6 Dn = 6 4
Lemma 2.11. (C k;x )(Dk
1 c1 (x) .. .
0
0
7 7 7: 5
In+2
( 1)n+1 cn+2 (x)
3)
= Ck;x for k
3
3:
(4)
Proof. We take (C k;x ) = [ci;j ], (Dk 3 ) = [di;j ] and Ck;x = [ci;j ] and obtain k P ci;s ds;j for i; j = 1; 2; :::; k: It is obvious from matrix product and de…nition
s=1
of In+2 that c11 = 1,
k P
ci;s ds;j = ci;j for i = 1; 2; :::; k and j = 2; :::; k: For j = 1;
s=1
ci;1 =
k X
ci;s ds;1 = Ci
( 1)k
1 (x)c1 (x)
1
C1 (x)ci
1 (x):
s=1
Using ck 1 (x) = xck 2 (x) C3 (x)ck 3 (x) + + ( 1)k 2 Ck (x)c0 (x) and c0 (x) = C1 (x) = 1; we obtain Ck (x) = Ck 1 (x)c1 (x) + ( 1)k 1 ck 1 (x). So using these last two equation the equation ci;1 = Ci 1 (x)c1 (x) + ( 1)i 1 C1 (x)ci 1 (x) = Ci (x) is obtained. Theorem 2.12. Let n (Dn 4 ))(Dn 3 ).
3 be any positive integer. Then Cn;x = (In
2
(D
1 ))
(I1
Proof. From Lemma 2.11 and matrix product we obtain =
(C n;x )(Dn
=
[(I2 .. .
=
(In
3
C 3;x )(In
C 3;x ) = (In
2
(D
Cn;x
and (In
3
Cn;x = (In
2
Cn
(D
3)
= [(I1
2;x )(I2
1 )).
Cn
(Dn
3
1;x )(I1
5 ))](I1
(D0 ))
(Dn (Dn
(I1
4 ))](Dn 3 )
4 ))(Dn 3 )
(Dn
4 ))(Dn 3 )
Thus, we obtain
1 ))(In 3
(D0 ))
(I1
(Dn
4 ))(Dn 3 ):
Example 2.13. We give a factorization for C5;x by using Theorem 2.12: (I3 D 1 )(I2 2 1 0 0 0 6 0 1 0 0 6 = 6 6 0 0 1 0 4 0 0 0 1 0 0 0 x
D0 )(I1 D1 )D2 32 0 1 0 0 6 0 1 0 0 7 76 6 0 7 76 0 0 1 0 54 0 0 x 1 0 0 0
0 0 0 0 0 0 1 0 0 1
32 76 76 76 76 54
1 0 0 0 0
0 1 x 0 1
32 0 0 0 6 0 0 0 7 76 6 1 0 0 7 76 0 1 0 54 0 0 1
1 x 0 1 x
3 0 0 0 0 1 0 0 0 7 7 0 1 0 0 7 7 0 0 1 0 5 0 0 0 1
ON THE GENERALIZED PERRIN AND CORDONNIER M ATRICES
2
1 x x2 3 x +1 x + x4
6 6 = 6 6 4
Lemma 2.14. Dn are the (n + 3)
(Dn )
1
2
6 6 =6 4
0 1 x x2 3 x +1
0 0 1 x x2
0 0 0 1 x
0 0 0 0 1
249
3
7 7 7: 7 5
(n + 3) Hessenberg matrices in (4). Then 1 c1 (x) .. .
0
0 In+2
( 1)n+2 cn+2 (x)
Proof. The proof is obvious that matrix product.
3
7 7 7: 5
Corollary 2.15. Let n 3 be any positive integer. Then (Cn;x ) (Dn 4 )) 1 (In 2 (D 1 )) 1 : Proof. Proof is obvious that previous lemma and equations (Ik Ik (Dn k 3 ) 1 :
1
= (Dn
(Dn
3)
1
k 3 ))
(I1 1
=
3. Determinantal representation of associated polynomials Cordonnier numbers Sahin and Ramirez gave a method for determinantal representation of Convolved Lucas polynomials in [24]. Using similar method, we give determinantal representation of Cn (x). Theorem 3.1. Let n 1 be an integer; Cn (x) be the nth associated polynomials (x) Cordonnier numbers and An = [ai;j ]i;j=1;2;:::;n be an n n Hessenberg matrix de…ned as 8 > if i j = 1; < 1; (5) ai;j = ( 1)i j ci j+1 (x); if i j > : 0; otherwise: Then
det( An(x) ) = Cn+1 (x): (x)
Proof. We proceed by induction on m. The result clearly holds for n = 1; det( A1 ) = x = C2 (x). Now suppose that the result is true for all positive integers less than or equal to n 1. We prove it for n. A(x) n [C1 (x) C2 (x)
T
Cn (x)] = [0 0
T
0 Cn+1 (x)]
ADEM S ¸ AH I·N
250
In fact, using Cramers rule we have (x)
Cn (x)
=
det( An
det(
1 )Cn+1 (x) (x) An ) (x)
) Cn+1 (x) =
det( An )Cn (x) (x)
det( An
1)
From the hypothesis of induction we obtain det( A(x) n ) = Cn+1 (x): (x)
Therefore, det( An ) = Cn+1 (x) holds for all positive integers n. Example 3.2. We obtain the 2 x 1 6 0 x 6 6 1 0 det 6 6 x 1 6 2 4 x x x3 x2 Corollary 3.3. Let m Then
polynomial C7 (x) by using Theorem 3.1. 3 0 0 0 0 1 0 0 0 7 7 x 1 0 0 7 7 = 2x3 + x6 + 1: 0 x 1 0 7 7 1 0 x 1 5 x 1 0 x
n be the integers, en is nth row of the identity matrix In .
m T en ( A(x) n )(Qn;x ) en = Cm+2 (x):
Proof. Proof is obvious from matrix product, Theorem 2.10 and equation A(x) n [Cm
n+2 (x)
T
Cm+1 (x)] = [0 0
T
0 Cn+2 (x)] :
Example 3.4. We obtain the polynomial C7 (x) by using Corollary 3.3. (x)
e6 ( A6 )(Q6;x )5 eT6 = 2x3 + x6 + 1 = C7 (x) Theorem 3.5. Let n 1 be an integer; Cn (x) be the nth associated polynomials (x) Cordonnier numbers and + Bn = [bs;t ]s;t=1;2;:::;n be an n n Hessenberg matrix de…ned as 8 >i; if s t = 1; < bs;t = (i)s t cs t+1 ; if s t > : 0; otherwise: Then det(+ Bn(x) ) = Cn+1 (x): Proof. If we multiply the kth column by ( 1)( i)k and the jth row by ( 1)ij of p (x) 1, then the determinant is not altered. Therefore the matrix An , where i = we get the desired result.
ON THE GENERALIZED PERRIN AND CORDONNIER M ATRICES
251
Example 3.6. We obtain the polynomial C6 (x) by using Theorem 3.5. 3 2 x i 0 0 0 6 0 x i 0 0 7 7 6 2 5 6 x i 0 7 det 6 1 0 7 = 2x + x 5 4 ix 1 0 x i x2 ix 1 0 x P Qn The permanent of a n-square matrix is de…ned by perA = 2Sn i=1 ai (i) ; where the summation extends over all permutations of the symmetric group Sn (cf. [18]). There is a relation between permanent and determinant of a Hessenberg matrix (cf. [4, 7]). Then it is clear the following corollary. Corollary 3.7. Let n 1 be an integer; Cn (x) be the nth associated polynomials (x) (x) Cordonnier numbers, + An = [us;t ]s;t=1;2;:::;n and Bn = [vs;t ]s;t=1;2;:::;n be the n n Hessenberg matrices de…ned as 8 8 > > if s t = 1; if s t = 1; < i; > : : 0; otherwise 0; otherwise p (x) (x) Where i = 1. Then per(+ An ) = per( Bn ) = Cn+1 (x): (x)
Corollary 3.8. Let n 1 be an integer, An be the n n Hessenberg matrix in (5), Cn (x) is the nth associated polynomials Cordonnier numbers and 3 2 1 0 0 .. 7 6 \ 6 (x) . 7 An = 6 7: (x) 4 ( A ) 0 5 n 1
1
Then,
\ (x) ( An )
1
= Cn;x :
Proof. Proof is obvious from Theorem 2.7. Acknowledgements. The author would like to express their pleasure to the anonymous reviewer for his/her careful reading and making some useful comments which improved the presentation of the paper. References [1] Bartlett, C. and Huylebrouck D., Art and math of the 1:35 ratio rectangle, Symmetry: Culture and Science. 24(2013). [2] Brawer, R. and Pirovino M., The linear algebra of the Pascal matrix, Linear Algebra Appl. 174(1992), 13–23.
252
ADEM S ¸ AH I·N
[3] Cahill, N.D., D’Errico J.R., Narayan D.A. and Narayan J.Y., Fibonacci determinants, College Math. J. 33(2002), 221–225. [4] Gibson, P. M., An identity between permanents and determinants, Amer. Math. Monthly. 76(1969), 270–271. [5] Kaygisiz, K., Bozkurt D., k-Generalized Order-k Perrin Number Presentation by Matrix Method, Ars Combinatoria, 105(2012), 95-101. [6] Kaygisiz, K. and Sahin A., Generalized bivariate Lucas p-Polynomials and Hessenberg Matrices, J. Integer Seq. 15 Article 12.3.4.(2012). [7] Kaygisiz, K. and Sahin A., Determinant and permanent of Hessenberg matrix and generalized Lucas polynomials, Bull. Iranian Math. Soc. 39(6)(2013), 1065–1078. [8] Kaygisiz, K. and Sahin A., A new method to compute the terms of generalized order-k Fibonacci numbers, J. Number Theory. 133(2013), 3119–3126. [9] Kaygisiz, K. and Sahin A., Generalized Van der Laan and Perrin Polynomials, and Generalizations of Van der Laan and Perrin Numbers, Selçuk Journal of Applied Math., 14(1)(2013),89103. [10] Kaygisiz, K. and Sahin A., Calculating terms of associated polynomials of Perrin and Cordonnier numbers, Notes on Number Theory and Discrete Mathematics, 20(1)(2014),10-18. [11] Kilic, E. and Stakhov A.P., On the Fibonacci and Lucas p-numbers, their sums, families of bipartite graphs and permanents of certain matrices, Chaos Solitons Fractals. 40(2009), 2210–2221. [12] Kilic, E. and Tasci D., The linear algebra of the Pell matrix, Bol. Soc. Mat. Mexicana. 2(11)(2005), 163-174. [13] Lee, G.Y., Kim J.S. and Lee S.G., Factorizations and eigenvalues of Fibonacci and symmetric Fibonacci matrices, Fibonacci Quart. 40(3)(2002), 203–211. [14] Lee, G.Y. and Kim J.S., The linear algebra of the k-Fibonacci matrix, Linear Algebra Appl. 373(2003), 75–87. [15] Li, H-C., On Fibonacci-Hessenberg matrices and the Pell and Perrin numbers, Appl. Math. Comput. 218(17)(2012), 8353–8358. [16] Li, H. and MacHenry T., Permanents and Determinants, Weighted Isobaric Polynomials, and Integer Sequences, Journal of Integer Sequences. 16 (2013)Article 13.3.5 [17] Marohni´c, L. and Strmeµcki T., Plastic Number: Construction and Applications, Advanced Research in Scienti…c Areas. (3)7(2012), 1523-1528. [18] Minc, H. Encyclopedia of Mathematics and its Applications, Permanents, Vol.6, AddisonWesley Publishing Company, 1978. [19] Ocal, A.A., Tuglu N. and Altinisik E., On the representation of k-generalized Fibonacci and Lucas numbers, Appl. Math. Comput. 170(1)(2005), 584–596. [20] Padovan, R., Dom Hans Van Der Laan and the Plastic Number, Nexus IV: Architecture and Mathematics, eds. Kim Williamsand Jose Francisco Rodrigues, Fucecchio (Florence): Kim Williams Books, 2002. [21] Padovan, R., Dom Hans van der Laan: Modern Primitive, Architectura Natura Press, 1994. [22] Shannon, A.G., Anderson P.G. and Horadam A. F., Properties of Cordonnier, Perrin and Van der Laan numbers. International Journal of Mathematical Education in Science and Technology. 37(2006), 825-831. [23] Sahin, A., On the Q analogue of …bonacci and lucas matrices and …bonacci polynomials, Utilitas Mathematica, 100(2016), 113–125. [24] Sahin, A. and Ramírez, J. R., Determinantal and permanental representations of convolved Lucas polynomials, Appl. Math. Comput., 281(2016), 314–322. [25] Yilmaz, F. and Bozkurt D., Hessenberg matrices and the Pell and Perrin numbers, J. Number Theory. 131(8)(2011),1390–1396. [26] Yilmaz, N. and Taskara N., Matrix Sequences in terms of Padovan and Perrin Numbers, Journal of Applied Mathematics (2013), Article ID 941673.
ON THE GENERALIZED PERRIN AND CORDONNIER M ATRICES
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Current address : Faculty of Education, Gaziosmanpa¸sa University, 60250 Tokat, TURKEY E-mail address :
[email protected],
[email protected] 