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On the order-k generalized Lucas numbers
Applied Mathematics and Computation 155 (2004) 637–641 www.elsevier.com/locate/amc
On the order-k generalized Lucas numbers Dursun Tasci *, Emrah Kilic Department of Mathematics, Gazi University, 06500 Teknikokullar, Ankara, Turkey
Abstract In this paper we give a new generalization of the Lucas numbers in matrix representation. Also we present a relation between the generalized order-k Lucas sequences and Fibonacci sequences. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Fibonacci numbers; Lucas numbers; The generalized order-k Lucas numbers
1. Introduction It is well known that fFn g, the Fibonacci sequence is defined by a recurrence relation, that is, Fn ¼ Fn1 þ Fn2 for n P 3 such that F1 ¼ 1 and F2 ¼ 1. Also, one can obtain the Fibonacci sequence by matrix methods. Indeed, it is clear that Fn 0 ; ¼ An 1 Fnþ1 where A¼
0 1
1 : 1
Kalman [1] mentioned that this is a special case of a sequence which is defined recursively as a linear combination of the preceding k terms:
*
Corresponding author. E-mail address: [email protected] (D. Tasci).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00804-X
638
D. Tasci, E. Kilic / Appl. Math. Comput. 155 (2004) 637–641
anþk ¼ c0 an þ c1 anþ1 þ þ ck1 anþk1 ; where c0 ; c1 ; . . . ; ck1 are real constants. In [1], Kalman obtained a number of closed-form formulas for the generalized sequence by matrix method. In [2] Er defined k sequences of the generalized order-k Fibonacci numbers as shown: for n > 0 and 1 6 i 6 k gni ¼
k X
i cj gnj
ð1Þ
j¼1
with boundary conditions for 1 k 6 n 6 0; 1 if i ¼ 1 n; gni ¼ 0 otherwise;
ð2Þ
where cj , 1 6 j 6 k; are constant coefficients, gni is the nth term of ith sequence. Er showed that 2 i 3 2 3 gnþ1 gni i i 6 gn 7 6 gn1 7 6 7 6 7 ð3Þ 6 .. 7 ¼ A6 .. 7; 4 . 5 4 . 5 i i gnkþ2 gnkþ1 where 2
c1 61 6 60 6 0 A¼6 6 . 6 .. 6 40 0
c2 0 1 0 .. .
c3 0 0 1 .. .
.. .
ck1 0 0 0 .. .
0 0
0 0
0 1
3 ck 07 7 07 7 0 7; .. 7 . 7 7 05 0
ð4Þ
being a k k companion matrix. Then he derived Gnþ1 ¼ AGn ;
ð5Þ
where 2 6 6 Gn ¼ 6 4
gn1 1 gn1 .. .
gn2 2 gn1 .. .
.. .
gnk k gn1 .. .
1 gnkþ1
2 gnkþ1
k gnkþ1
3 7 7 7; 5
ð6Þ
generalizing the matrix equation in (3). Recently, Karaduman [3] showed that G1 ¼ A, and therefore, Gn ¼ An . Also he proved that
D. Tasci, E. Kilic / Appl. Math. Comput. 155 (2004) 637–641
det Gn ¼
n
ð1Þ 1
639
if k is even; if k is odd:
In this paper we give an order-k generalization of the usual Lucas sequence fLn g, which is defined recursively as Ln ¼ Ln1 þ Ln2 for n P 3 with initial condition L1 ¼ 1 and L2 ¼ 3: We present a matrix representation of the order-k generalization of the Lucas sequence. Then we calculate the determinant of the matrix obtained by our sequence. Furthermore, we find a relation between Gn obtained by the sequence of the generalized order-k Fibonacci numbers and the matrix obtained by the sequence of the generalized order-k Lucas numbers.
2. The main results Define k sequences of the generalized order-k Lucas numbers as shown: lin ¼
k X
linj ;
ð7Þ
j¼1
for n > 0 and 1 6 i 6 k, with boundary (initial) conditions 8 if i ¼ 2 n; <2 lin ¼ 1 if i ¼ 1 n; : 0 otherwise;
ð8Þ
for 1 k 6 n 6 0; where lin is the nth term of the ith sequence. When i ¼ 1 and k ¼ 2, the generalized order-k Lucas sequence reduces to negative usual Fibonacci sequence, i.e., l1n ¼ Fnþ1 for all n 2 Zþ . When we choose k ¼ 4 and i ¼ 3, the generalized order-k Lucas sequence is as follows: . . . ; l32 ¼ 1; l31 ¼ 2; l30 ¼ 0; l31 ¼ 1; l32 ¼ 2; l33 ¼ 5; l34 ¼ 8; l35 ¼ 16; l36 ¼ 31; . . . By (7), we can write 2 6 6 6 6 6 4
linþ1 lin i ln1 .. . linkþ2
3
2
1 7 61 7 6 7 60 7¼6 7 4 .. 5 .
1 0 1 .. .
1 0 0 .. .
.. .
1 0 0 .. .
0
0
0
1
3 2 li 3 1 n lin1 7 0 76 7 6 7 li 7 0 76 n2 7; 6 .. 7 6 .. 7 5 . 4 . 5 0 linkþ1
ð9Þ
640
D. Tasci, E. Kilic / Appl. Math. Comput. 155 (2004) 637–641
for the generalized order-k 2 1 1 1 ... 61 0 0 ... 6 0 1 0 ... A¼6 6. . . 4 .. .. .. 0 0 0 ...
Lucas sequences. Letting 3 1 1 0 07 7 0 0 7: .. .. 7 . .5 1 0
ð10Þ
To deal with the k sequences of the generalized order-k Lucas sequences simultaneously, we define a k k matrix Hn as follows: 2 1 3 l2n lkn ln 1 2 k 6 ln1 ln1 ln1 7 6 7 ð11Þ Hn ¼ 6 . .. 7: . . .. .. 4 .. . 5 l1nkþ1
l2nkþ1
lknkþ1
Generalizing Eq. (9), we have Hnþ1 ¼ A Hn :
ð12Þ
Lemma 1. Let A and Hn be as in (10) and (11), respectively. Then Hnþ1 ¼ An H1 , where 3 2 1 1 1 1 1 6 1 2 0 0 07 7 6 6 0 1 2 0 07 6 : H1 ¼ 6 .. .. 7 .. .. . . .. .7 . . . . 7 6 . 4 0 0 0 2 05 0
0
0
1
2
Proof. By (12), we have Hnþ1 ¼ AHn . Then by induction and a property of matrix multiplication, we have Hnþ1 ¼ An H1 : Also H1 ¼ A K, where 2
1 6 0 6 0 K¼6 6 . 4 .. 0
2 1 0 .. .
0 2 1 .. .
.. .
0 0 0 .. .
0
0
1
Thus, Hnþ1 ¼ Anþ1 K.
3 7 7 7: 7 5
ð13Þ
D. Tasci, E. Kilic / Appl. Math. Comput. 155 (2004) 637–641
641
Theorem 2. Let Hn be as in (11). Then 1 if k is odd; det Hnþ1 ¼ nþ1 ð1Þ if k is even: Proof. From Lemma 1, we have Hnþ1 ¼ Anþ1 K. Then det Hnþ1 ¼ ðdet AÞ
nþ1
det K;
where det A ¼ ð1Þkþ1 Thus
det Hnþ1 ¼
and
1 ð1Þnþ1
detK ¼ ð1Þk : if k is odd; if k even:
The following theorem presents a relation between the generalized order-k Lucas sequence and Fibonacci sequence, which was given by Er in [2]. Theorem 3. Let Gn and Hn be as in (6) and (11), respectively. Then Hn ¼ Gn K, where K is the k k matrix in (13). Proof. In [2] Er, showed that Gn ¼ An . Also we obtain Hn ¼ An K. Thus we have Hn ¼ Gn K: In Theorem 3, letting k ¼ 2. Then we have 1 1 1 2 ln l2n gn2 gn ¼ 1 : 2 0 1 l1n1 l2n2 gn1 gn2 2 2 for all n 2 Z, we have l2n ¼ 2gnþ1 gn2 , Therefore, l2n ¼ 2gn1 gn2 . Since gn1 ¼ gnþ1 2 2 where ln and gn is the usual Lucas and Fibonacci numbers, respectively. Indeed, we generalize a relation Lucas and Fibonacci numbers, i.e., Ln ¼ 2Fnþ1 Fn (see p. 176, [4]).
References [1] D. Kalman, Generalized Fibonacci numbers by matrix methods, Fibonacci Quart. 20 (1) (1982) 73–76. [2] M.C. Er, Sums of Fibonacci numbers by matrix methods, Fibonacci Quart. 22 (3) (1984) 204– 207. [3] E. Karaduman, An application of Fibonacci numbers in matrices, Appl. Math. Comp., in press. [4] S. Vajda, Fibonacci and Lucas Numbers and the Golden Section, Chichester, Brisbane, Toronto, New York, 1989.
On the order-k generalized Lucas numbers Dursun Tasci *, Emrah Kilic Department of Mathematics, Gazi University, 06500 Teknikokullar, Ankara, Turkey
Abstract In this paper we give a new generalization of the Lucas numbers in matrix representation. Also we present a relation between the generalized order-k Lucas sequences and Fibonacci sequences. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Fibonacci numbers; Lucas numbers; The generalized order-k Lucas numbers
1. Introduction It is well known that fFn g, the Fibonacci sequence is defined by a recurrence relation, that is, Fn ¼ Fn1 þ Fn2 for n P 3 such that F1 ¼ 1 and F2 ¼ 1. Also, one can obtain the Fibonacci sequence by matrix methods. Indeed, it is clear that Fn 0 ; ¼ An 1 Fnþ1 where A¼
0 1
1 : 1
Kalman [1] mentioned that this is a special case of a sequence which is defined recursively as a linear combination of the preceding k terms:
*
Corresponding author. E-mail address: [email protected] (D. Tasci).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00804-X
638
D. Tasci, E. Kilic / Appl. Math. Comput. 155 (2004) 637–641
anþk ¼ c0 an þ c1 anþ1 þ þ ck1 anþk1 ; where c0 ; c1 ; . . . ; ck1 are real constants. In [1], Kalman obtained a number of closed-form formulas for the generalized sequence by matrix method. In [2] Er defined k sequences of the generalized order-k Fibonacci numbers as shown: for n > 0 and 1 6 i 6 k gni ¼
k X
i cj gnj
ð1Þ
j¼1
with boundary conditions for 1 k 6 n 6 0; 1 if i ¼ 1 n; gni ¼ 0 otherwise;
ð2Þ
where cj , 1 6 j 6 k; are constant coefficients, gni is the nth term of ith sequence. Er showed that 2 i 3 2 3 gnþ1 gni i i 6 gn 7 6 gn1 7 6 7 6 7 ð3Þ 6 .. 7 ¼ A6 .. 7; 4 . 5 4 . 5 i i gnkþ2 gnkþ1 where 2
c1 61 6 60 6 0 A¼6 6 . 6 .. 6 40 0
c2 0 1 0 .. .
c3 0 0 1 .. .
.. .
ck1 0 0 0 .. .
0 0
0 0
0 1
3 ck 07 7 07 7 0 7; .. 7 . 7 7 05 0
ð4Þ
being a k k companion matrix. Then he derived Gnþ1 ¼ AGn ;
ð5Þ
where 2 6 6 Gn ¼ 6 4
gn1 1 gn1 .. .
gn2 2 gn1 .. .
.. .
gnk k gn1 .. .
1 gnkþ1
2 gnkþ1
k gnkþ1
3 7 7 7; 5
ð6Þ
generalizing the matrix equation in (3). Recently, Karaduman [3] showed that G1 ¼ A, and therefore, Gn ¼ An . Also he proved that
D. Tasci, E. Kilic / Appl. Math. Comput. 155 (2004) 637–641
det Gn ¼
n
ð1Þ 1
639
if k is even; if k is odd:
In this paper we give an order-k generalization of the usual Lucas sequence fLn g, which is defined recursively as Ln ¼ Ln1 þ Ln2 for n P 3 with initial condition L1 ¼ 1 and L2 ¼ 3: We present a matrix representation of the order-k generalization of the Lucas sequence. Then we calculate the determinant of the matrix obtained by our sequence. Furthermore, we find a relation between Gn obtained by the sequence of the generalized order-k Fibonacci numbers and the matrix obtained by the sequence of the generalized order-k Lucas numbers.
2. The main results Define k sequences of the generalized order-k Lucas numbers as shown: lin ¼
k X
linj ;
ð7Þ
j¼1
for n > 0 and 1 6 i 6 k, with boundary (initial) conditions 8 if i ¼ 2 n; <2 lin ¼ 1 if i ¼ 1 n; : 0 otherwise;
ð8Þ
for 1 k 6 n 6 0; where lin is the nth term of the ith sequence. When i ¼ 1 and k ¼ 2, the generalized order-k Lucas sequence reduces to negative usual Fibonacci sequence, i.e., l1n ¼ Fnþ1 for all n 2 Zþ . When we choose k ¼ 4 and i ¼ 3, the generalized order-k Lucas sequence is as follows: . . . ; l32 ¼ 1; l31 ¼ 2; l30 ¼ 0; l31 ¼ 1; l32 ¼ 2; l33 ¼ 5; l34 ¼ 8; l35 ¼ 16; l36 ¼ 31; . . . By (7), we can write 2 6 6 6 6 6 4
linþ1 lin i ln1 .. . linkþ2
3
2
1 7 61 7 6 7 60 7¼6 7 4 .. 5 .
1 0 1 .. .
1 0 0 .. .
.. .
1 0 0 .. .
0
0
0
1
3 2 li 3 1 n lin1 7 0 76 7 6 7 li 7 0 76 n2 7; 6 .. 7 6 .. 7 5 . 4 . 5 0 linkþ1
ð9Þ
640
D. Tasci, E. Kilic / Appl. Math. Comput. 155 (2004) 637–641
for the generalized order-k 2 1 1 1 ... 61 0 0 ... 6 0 1 0 ... A¼6 6. . . 4 .. .. .. 0 0 0 ...
Lucas sequences. Letting 3 1 1 0 07 7 0 0 7: .. .. 7 . .5 1 0
ð10Þ
To deal with the k sequences of the generalized order-k Lucas sequences simultaneously, we define a k k matrix Hn as follows: 2 1 3 l2n lkn ln 1 2 k 6 ln1 ln1 ln1 7 6 7 ð11Þ Hn ¼ 6 . .. 7: . . .. .. 4 .. . 5 l1nkþ1
l2nkþ1
lknkþ1
Generalizing Eq. (9), we have Hnþ1 ¼ A Hn :
ð12Þ
Lemma 1. Let A and Hn be as in (10) and (11), respectively. Then Hnþ1 ¼ An H1 , where 3 2 1 1 1 1 1 6 1 2 0 0 07 7 6 6 0 1 2 0 07 6 : H1 ¼ 6 .. .. 7 .. .. . . .. .7 . . . . 7 6 . 4 0 0 0 2 05 0
0
0
1
2
Proof. By (12), we have Hnþ1 ¼ AHn . Then by induction and a property of matrix multiplication, we have Hnþ1 ¼ An H1 : Also H1 ¼ A K, where 2
1 6 0 6 0 K¼6 6 . 4 .. 0
2 1 0 .. .
0 2 1 .. .
.. .
0 0 0 .. .
0
0
1
Thus, Hnþ1 ¼ Anþ1 K.
3 7 7 7: 7 5
ð13Þ
D. Tasci, E. Kilic / Appl. Math. Comput. 155 (2004) 637–641
641
Theorem 2. Let Hn be as in (11). Then 1 if k is odd; det Hnþ1 ¼ nþ1 ð1Þ if k is even: Proof. From Lemma 1, we have Hnþ1 ¼ Anþ1 K. Then det Hnþ1 ¼ ðdet AÞ
nþ1
det K;
where det A ¼ ð1Þkþ1 Thus
det Hnþ1 ¼
and
1 ð1Þnþ1
detK ¼ ð1Þk : if k is odd; if k even:
The following theorem presents a relation between the generalized order-k Lucas sequence and Fibonacci sequence, which was given by Er in [2]. Theorem 3. Let Gn and Hn be as in (6) and (11), respectively. Then Hn ¼ Gn K, where K is the k k matrix in (13). Proof. In [2] Er, showed that Gn ¼ An . Also we obtain Hn ¼ An K. Thus we have Hn ¼ Gn K: In Theorem 3, letting k ¼ 2. Then we have 1 1 1 2 ln l2n gn2 gn ¼ 1 : 2 0 1 l1n1 l2n2 gn1 gn2 2 2 for all n 2 Z, we have l2n ¼ 2gnþ1 gn2 , Therefore, l2n ¼ 2gn1 gn2 . Since gn1 ¼ gnþ1 2 2 where ln and gn is the usual Lucas and Fibonacci numbers, respectively. Indeed, we generalize a relation Lucas and Fibonacci numbers, i.e., Ln ¼ 2Fnþ1 Fn (see p. 176, [4]).
References [1] D. Kalman, Generalized Fibonacci numbers by matrix methods, Fibonacci Quart. 20 (1) (1982) 73–76. [2] M.C. Er, Sums of Fibonacci numbers by matrix methods, Fibonacci Quart. 22 (3) (1984) 204– 207. [3] E. Karaduman, An application of Fibonacci numbers in matrices, Appl. Math. Comp., in press. [4] S. Vajda, Fibonacci and Lucas Numbers and the Golden Section, Chichester, Brisbane, Toronto, New York, 1989.