An application of Fibonacci numbers
in matrices
Erdal Karaduman
Fen Edebiyat Fak€ultesi Matematik B€ol€um€u, Atat€urk University, 25240 Erzurum, Turkey
Abstract
In this paper, we investigate the determinants of the matrices obtained by k
sequences of the generalized order-k Fibonacci numbers.
2002 Elsevier Inc. All rights reserved.
Keywords: Fibonacci numbers; The generalized order-k Fibonacci numbers; Fibonacci sequences;
Matrix; Determinant
In the recent years, there has been much interest in applications of Fibonacci
numbers and sequences. Karaduman and Yavuz proved that the periods of the
2-step Fibonacci recurrences in finite nilpotent groups of nilpotency class 5 and
exponent p are pkðpÞ, for 2 < p 6 2927, where p is prime and kðpÞ is the periods
of ordinary 2-step Fibonacci sequences [3]. The theory has been generalized
in [4] to the 2-step general Fibonacci sequences in finite nilpotent groups of
nilpotency class 4 and exponent p. Also, it is shown in [5] that the period of 2-
step general Fibonacci sequence is equal to the length of fundamental period of
the 2-step general recurrence constructed by two generating elements of the
group of exponent p and nilpotency class 2. CAYLEY has been used for calculations
in [5].
Lee has investigated the relationship between gðkÞ
n Fibonacci numbers and
lðkÞ
n Lucas numbers [6]. Takahashi gives a fast algorithm which is based on the
product of Lucas numbers to compute large Fibonacci numbers [7]. Fibonacci
sequences have been an interesting subject in applied mathematics. West has
E-mail address: eduman@atauni.edu.tr (E. Karaduman).
0096-3003/$ - see front matter 2002 Elsevier Inc. All rights reserved.
doi:10.1016/S0096-3003(02)00827-5
Applied Mathematics and Computation 147 (2004) 903–908
www.elsevier.com/locate/amc
shown using transfer matrices that the number jSnð123; 3214Þj of permutations
avoiding the patterns 123 and 3214 is the Fibonacci number F2n [8].
Kalman [1] derives a number of closed-form formulas for the generalized
Fibonacci sequence by matrix methods. In [2], Er has extended the matrix
representation and showed that the sums of the generalized Fibonacci numbers
could be derived directly using this representation.
Define k sequences of the generalized order-k Fibonacci numbers as shown:
gi
n ¼ X
k
j¼1
gi
nj; for n > 0 and 1 6 i 6 k; ð1Þ
with boundary conditions
gi
n ¼ 1; i ¼ 1 n;
0; otherwise
for 1 k 6 n 6 0;
where gi
n is the nth term of the ith sequence. When k ¼ 2, generalized order-k
Fibonacci sequence is reduced to the conventional Fibonacci sequence.
Following the approach taken by Kalman [1], a k k square matrix A can be
defined as follows:
A ¼
111 1 1
100 0 0
010 0 0
.
.
. .
.
. .
.
. .
.
. .
.
.
000 0 0
000 1 0
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
:
Then, by a property of matrix multiplication, we have
½gi
nþ1gi
n gi
nkþ2
T ¼ A½gi
ngi
n1 gi
nkþ1
T
: ð2Þ
To deal with the k sequences of the generalized order-k Fibonacci series
simultaneously, a k k square matrix Gn is defined as follows [2]:
Gn ¼
g1
n g2
n gk
n
g1
n1 g2
n1 g2
n1
.
.
. .
.
. .
.
.
g1
nkþ1 g2
nkþ1 gk
nkþ1
2
6
6
6
4
3
7
7
7
5:
Generalizing Eq. (2), we have
Gnþ1 ¼ AGn ð3Þ
Then, by an inductive argument, one may rewrite it as
Gnþ1 ¼ An
G1: ð4Þ
904 E. Karaduman / Appl. Math. Comput. 147 (2004) 903–908
Now, it can be readily seen that, by Definition (1), G1 ¼ A; therefore, Gn ¼
An. Thus, Eqs. (3) and (4) may be rewritten as shown:
Gnþ1 ¼ GnG1 ¼ G1Gn ð5Þ
or
Gnþ1 ¼ Anþ1
:
In other words, G1 is commutative under matrix multiplication.
Theorem 1. If the Gn has the form
Gn ¼
g1
n g2
n gk
n
g1
n1 g2
n1 gk
n1
.
.
. .
.
. .
.
.
g1
nkþ1 g2
nkþ1 gk
nkþ1
2
6
6
6
4
3
7
7
7
5
then,
det Gn ¼ ð1Þ
n
; if k is even
1; if k is odd
;
where, gi
n ¼ Pk
j¼1 gi
nj, for n > 0 and 1 6 i 6 k, with boundary conditions
gi
n ¼ 1; i ¼ 1 n;
0; otherwise;
for 1 k 6 n 6 0:
Proof. Since for 8k,
G1 ¼ A ¼
111 1 1
100 0 0
010 0 0
.
.
. .
.
. .
.
. .
.
. .
.
.
000 0 0
000 1 0
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
after ðk 1Þ-step elementary rows operations G1 is reduced to the triangular
matrix form as shown:
11 1 1 1
0 1 1 1 1
0 0 1 1 1
.
.
. .
.
. .
.
. .
.
. .
.
.
00 0 1 1
00 0 0 1
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
:
E. Karaduman / Appl. Math. Comput. 147 (2004) 903–908 905
So, det G1 ¼ det A ¼ 1 ð1Þ
k1
. On the other hand, since Gn ¼ An we have
det Gn ¼ ðdet AÞ
n
det Gn ¼ ðð1Þ
k1
Þ
n
: ð6Þ
Therefore, if k is even, ðk 1Þ will be odd. So, ð1Þ
k1 ¼ 1. Thus, from (6),
we have det Gn ¼ ð1Þ
n
. If k is odd, ðk 1Þ will be even. So, ð1Þ
k1 ¼ 1 and
thus, from (6), we have det Gn ¼ 1. We are done.
Now, if we ignore the condition n > 0 in Definition (1), we can define k
sequences of the generalized order-k Fibonacci numbers as shown:
gi
n ¼ X
k
j¼1
gi
nj; 1 6 i 6 k ð7Þ
with boundary conditions
gi
n ¼ 1; i ¼ 1 n;
0; otherwise;
for 1 k 6 n 6 0;
where gi
n is the nth term of the ith sequence. When k ¼ 2, generalized order-k
Fibonacci sequence is reduced to the conventional Fibonacci sequence with
initial data g1
0 ¼ 1, g1
1 ¼ 1.
We define k k square matrices Gn, B and C as follows:
Gn ¼
g1
n g2
n gk
n
g1
n1 g2
n1 gk
n1
.
.
. .
.
. .
.
.
g1
nkþ1 g2
nkþ1 gk
nkþ1
2
6
6
6
4
3
7
7
7
5;
B ¼
20 21 22 2k2 2k1
20 20 21 2k3 2k2
0 20 20 2k4 2k3
.
.
. .
.
. .
.
. .. . .
.
. .
.
.
000 ..
. 20 21
000 20 20
2
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
5
;
C ¼
000 0 1
100 0 1
010 0 1
.
.
. .
.
. .
.
. .
. . .
.
. .
.
.
000 .. . 0 1
000 1 1
2
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
5
:
Now, it can be readily seen that, by Definition (7), G1 ¼ B. Then, by an
inductive argument, we write
906 E. Karaduman / Appl. Math. Comput. 147 (2004) 903–908
Gnþ1 ¼ BCn
: ð8Þ
But, B is not commutative under matrix multiplication.
Theorem 2. If Gn has the form as shown:
Gn ¼
g1
n g2
n gk
n
g1
n1 g2
n1 gk
n1
.
.
. .
.
. .
.
.
g1
nkþ1 g2
nkþ1 gk
nkþ1
2
6
6
6
4
3
7
7
7
5
then,
det Gn ¼ 1; if k is odd
ð1Þ
n
; if k is even
:
Proof. Since Gn ¼ G1Cn1 ¼ BCn1, we have
det Gn ¼ det G1 ðdet CÞ
n1
: ð9Þ
So, after ðk 1Þ-step elementary rows operations G1 is reduced to the triangular
matrix form as shown:
20 21 22 2k2 2k1
0 1 2 2k3 2k2
0 0 1 2k4 2k3
.
.
. .
.
. .
.
. .
.
. .
.
.
00 0 1 20
00 0 0 1
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
Therefore, we have det G1 ¼ 1 ð1Þ
k1
. Simultaneously, replacing first row of
C with the other rows of C, C is reduced to the triangular matrix form as
shown:
100 0 1
010 0 1
001 0 1
.
.
. .
.
. .
.
. .
.
. .
.
.
000 1 1
000 0 1
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
:
So, we have det C ¼ ð1Þ
k1
. Thus, we have det Gn ¼ ð1Þ
k1
ðð1Þ
k1
Þ
n1
,
from (9). On the other hand, if k is even then we have ð1Þ
k1 ¼ 1, and if k is
odd then we have ð1Þ
k1 ¼ 1. Therefore, we have
E. Karaduman / Appl. Math. Comput. 147 (2004) 903–908 907
det Gn ¼ 1; if k is odd
ð1Þ
n
; if k is even
:
We are done.
From Eqs. (1) and (7), we see that the condition n > 0 does not change the
determinants of Gn. But, it yields that G1 is commutative under matrix multiplication
or not. i.e. if we do not ignore the condition n > 0 then G1 is
commutative under matrix multiplication, otherwise it is not.
References
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(1982) 73–76.
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204–207.
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p and nilpotency class 4, Applied Mathematics and Computation (AMC 7418), in press.
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(3) (1998) 216–221.
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908 E. Karaduman / Appl. Math. Comput. 147 (2004) 903–908