Theoretical Mathematics & Applications, vol.2, no.2, 2012, 115-124
ISSN: 1792- 9687 (print), 1792-9709 (online)
International Scientific Press, 2012
Generalized Fibonacci Sequences
V.K. Gupta1, Yashwant K. Panwar2 and Omprakash Sikhwal3
Abstract
The Fibonacci sequence is famous for possessing wonderful and amazing
properties. In this paper, we introduce generalized Fibonacci sequences and
related identities consisting even and odd terms. Also we present connection
formulas for generalized Fibonacci sequences, Jacobsthal sequence and
Jacobsthal-Lucas sequence.
Mathematics Subject Classification: 11B39, 11B37
Keywords:
Generalized
Fibonacci
sequence,
Jacobsthal-Lucas
sequence,
Jacobsthal sequence, Binet’s formula
1
2
3
Department of Mathematics, Govt. Madhav Science College, Ujjain, India,
e-mail: dr_vkg61@yahoo.com
Department of Mathematics, Mandsaur Institute of Technology, Mandsaur, India,
e-mail: yashwantpanwar@gmail.com
Department of Mathematics, Mandsaur Institute of Technology, Mandsaur, India,
e-mail: opbhsikhwal@rediffmail.com ; opsikhwal@gmail.com
Article Info: Received : March 7, 2012. Revised : April 21, 2012
Published online : June 30, 2012
116
1
Generalized Fibonacci Sequences
Introduction
It is well-known that the Fibonacci sequence, Lucas sequence, Pell sequence,
Pell-Lucas sequence, Jacobsthal sequence and Jacobsthal-Lucas sequence are
most prominent examples of recursive sequences.
The Fibonacci sequence [8] is defined by the recurrence relation
Fk  Fk 1  Fk 2 , k  2 with F0  0, F1  1
(1.1)
The Lucas sequence [8] is defined by the recurrence relation
Lk  Lk 1  Lk 2 , k  2
with L0  2, L1  1
(1.2)
The Jacobsthal sequence [2] is defined by the recurrence relation
J k  J k 1  2 J k 2 , k  2 with J 0  0, J1  1
(1.3)
The Jacobsthal-Lucas sequence [2] is defined by the recurrence relation
jk  jk 1  2 jk 2 , k  2 with j0  2, j1  1
(1.4)
B. Singh, O. Sikhwal and S. Bhatnagar [5] defined Fibonacci-Like Sequence
Sk  Sk 1  Sk  2 , k  2
with S0  2, S1  2
(1.5)
The second order recurrence sequence has been generalized in two ways mainly,
first by preserving the initial conditions and second by preserving the recurrence
relation.
Kalman and Mena [6] generalize the Fibonacci sequence by
Fn  aFn 1  bFn  2 , n  2
with F0  0, F1  1
(1.6)
Horadam [3] defined generalized Fibonacci sequence { H n } by
H n  H n 1  H n  2 , n  3 with H1  p , H 2  p  q
(1.7)
where p and q are arbitrary integers.
In this paper, we introduce generalized Fibonacci sequence and present
identities consisting even and odd terms.
Further we defined connection
formulas for generalized Fibonacci sequence, Jacobsthal sequence and
Jacobsthal-Lucas sequence.
V. K. Gupta, Y. K. Panwar and O. Sikhwal
2
117
Generalized Fibonacci Sequence
We define generalized Fibonacci sequence as
Fk  p Fk 1  q Fk  2 , k  2
with F0  a , F1  b ,
(2.1)
where p , q , a and b are positive integers
For different values of p , q , a and b many sequences can be determined.
We focus two cases of sequences Vk k 0 and U k k 0 which generated in (2.1).
If p  1 , q  a  b  2 , we get
with V0  2 , V1  2
V k  V k 1  2 V k  2 , for k  2
(2.2)
The first few terms of Vk k 0 are 2, 2, 6, 10, 22, 42 and so on. Its Generating
function is defined by
Vk 
2
1  x  2 x2
(2.3)
Its Binet’s formula is defined by
Vk  2
1k 1  2k 1
1  2
(2.4)
where 1 and 2 are the roots of the characteristic equation 2 x 2  x  1  0 .
If p  1 , q  a  2 , b  0 we get
U k  U k 1  2 U k  2 for k  2 with U 0  2 , U 1  0
The first few terms of
U k k 0
(2.5)
are 2, 0, 4, 4, 12, 20 and so on. Its
Generating function is defined by
Uk 
2 (1  x)
1  x  2x2
(2.6)
Its Binet’s formula is defined by
1k 1  2k 1
Uk  4
1  2
(2.7)
where 1 and 2 are the roots of the characteristic equation 2 x 2  x  1  0 .
118
3
Generalized Fibonacci Sequences
Identities of Generalized Fibonacci Sequence
Now we present identities consisting even and odd terms.
Theorem 3.1. If Vk k 0 and U k k 0 are the generalized Fibonacci sequences,
then
8
8
j2 k  2k 1 ,

8
8 9
3
Vk U k  j2 k  (2) k 1  
8
9
3 8
j2 k  2k 1 ,
 9
3
k even
k odd
Proof. By Binet’s formulas (2.4) and (2.7), we have
8
(1k 1  2k 1 ) (1k 1  2k 1 )
9
8
 (12 k  22 k )  1k 2k (121  211 )
9
  
8
  j2 k  (2) k ( 1  2 ) 
9
2 1 
Vk U k 
8
j2 k  (2) k 1 (12  22 )

9
8
8
 j2 k  (2) k 1
9
3

8
8
j2 k  2k 1 , k even

8
8 9
3
Vk U k  j2 k  (2) k 1  
8
9
3 8
j2 k  2k 1 , k odd
 9
3
(3.6)

Theorem 3.2.
(i ) Vk21  Vk2 
If Vk k 0 is the generalized Fibonacci sequence, then
4k 1 8
  j2 k  (2) k 
3
9
V. K. Gupta, Y. K. Panwar and O. Sikhwal


(ii ) Vk21  Vk21  


119
5 k 1 1 k 3
(4)  (2) , k even
3
3
5 k 1 1 k 3
(4)  (2) , k odd
3
3
Proof. (i): By Binet’s formula (2.4), we have
4
(1k  2k ) 2  (1k 1  2k 1 ) 2 

9
4
 12 k  12 k  2  2k  22 k  2  2(12 ) k (12  1)
9
4
 312 k  2(12 k  22 k )  2(2) k 
9
4
 312 k  2 j2 k  2(2) k 
9
Vk21  Vk2 
Vk21  Vk2 
4k 1 8
  j2 k  (2) k 
3
9
(3.11)
(ii): By Binet’s formula (2.4), we have
4
(1k 2  2k 2 )2  (1k  2k )2 
9
4
 12 k (14  1)  22 k (42  1)  2(12 ) k (12 ) k  1
9
Vk21  Vk21 
4
1512k  6(2)k 
9
5
(1) k
(2) k 3
 (4) k 1 
3
3
Vk21  Vk21 


Vk21  Vk21  


5 k 1 1 k 3
(4)  (2) , k even
3
3
5 k 1 1 k 3
(4)  (2) , k odd
3
3

Following theorems can be solved by Binet’s formula (2.4) and (2.7)
120
Generalized Fibonacci Sequences
Theorem 3.3. If U k k 0 is the generalized Fibonacci sequence, then
(i ) U
2
k 1
U
2
k 2
4k  2 32

  j2 k  (2) k 
3
9


(ii ) U k23  U k21  


(3.17)
5 k  2 1 k 5
(4)  (2) , k even
3
3
5 k  2 1 k 5
(4)  (2) , k odd
3
3
(3.18)
Theorem 3.4. Prove that
k
V0  V3  V6  ...  V3k 3   V3n 3
n 1
Corollary 3.5.
Theorem 3.6.
4
, k even
 7 J 3k

 2 (2 J  1) , k odd
3k
 7
V0  V3  V6  ...  V3k 3
4 k
 21 (8  1),

 2 (23k 1  5),
 21
(3.19)
k even
(3.20)
k odd
Prove that
8
 7 J 3k ,
U 2  U 5  U 8  ...  U 3k 1  
 2 (4 J  1),
3k
 7
(3.21)
k even
k odd
V. K. Gupta, Y. K. Panwar and O. Sikhwal
Corollary 3.7.
121
8 k
 21 (8  1) , k even
U 2  U 5  U 8  ...  U 3k 1   k 1
 8  20 , k odd
 21
(22)
Theorem 3.8. Prove that
k
V1  V4  V7  ...  V3k  2   V3n  2 
n 1
1
1
5
j3k  (8) k 
3
21
7
Corollary 3.9. V1  V4  V7  ...  V3k  2
8 k
 21 (8  1),
  k 1
 (8  20) ,

21
(3.23)
k even
(3.24)
k odd
Theorem 3.10. Prove that
k
U 3  U 6  U 9  ...  U 3k   U 3k 
n 1
Corollary 3.11.
2
2
10
j3k  (8) k 
3
21
7
U 3  U 6  U 9  ...  U 3k
16 k
 (8  1),
  21
 2 (8k 1  22),
 21
Theorem 3.12. Prove that
U1  U 4  U 7  ...  U 3k  2
4
 21 ( j3k  2),

 4 ( j  7),
 21 3k
k even
k odd
(3.25)
k even
(3.26)
k odd
122
Generalized Fibonacci Sequences
Corollary 3.13. U1  U 4  U 7  ...  U 3 k  2
4
4 k
 (8  1) , k even
  21
 4 (8k  8) , k odd
 21
Connection Formulas
Finally we present connection formulas for generalized Fibonacci Sequences,
Jacobsthal sequence and Jacobsthal-Lucas sequence.
Theorem 4.1. Prove that
(i )
2Vk 1  Vk  2 jk 1
(4.1)
(ii )
Vk  4Vk 1  2 jk 1
(4.2)
Proof. (i): For k  0 ,
2V01 V0  2  2  2  2  2 1  2 j1 ,
which is true for k  0 .
For k  1,
2V11  V1  2  6  2  10  2  5  2 j2 ,
which is also true for k  1 .
If result is true for k  n , then 2Vn1 Vn  2 jn1 . Now
2V(n1)1  V(n1)  2Vn2  Vn1
 2 jn 1  2(2 jn )
(By hypothesis)
 2( jn 1  2 jn )  2 jn  2
Therefore, 2V(n1)1  V(n1)  2 jn2 , which is true for k  n  1 .
This completes the proof.
V. K. Gupta, Y. K. Panwar and O. Sikhwal
123
(ii): By Binet’s formula (2.4), we have
Vk  4Vk 1  2
1k 1  2k 1
k  2k
8 1
 2 jk 1
1  2
1  2
Following theorems can be solved with the help of Binet’s formulas and as well as
induction method.

Theorem 4.2. Prove that
(i )
U k 1  4U k  4 jk
(4.3)
(ii ) 2 U k  2  U k 1  4 jk
(4.4)
Theorem 4.3. Prove that


jk 1  4 jk  


(i )


2 jk 1  jk  


(ii )
5
9
Vk ,
2
9
U k 2 ,
4
9
Vk 1 ,
2
9
U k 2 ,
4
k 0
(4.5)
k 0
k 1
(4.6)
k 0
Conclusion
In this paper we have stated and derived many identities of generalized
Fibonacci sequences consisting even and odd terms through Binet’s formulas.
Finally we presented some connection formulas and defined through induction
method.
124
Generalized Fibonacci Sequences
Acknowledgements. We are thankful to referees for their valuable comments.
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with
Applications,
A