Generalized Fibonacci Sequences

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: [email protected] Department of Mathematics, Mandsaur Institute of Technology, Mandsaur, India, e-mail: [email protected] Department of Mathematics, Mandsaur Institute of Technology, Mandsaur, India, e-mail: [email protected] ; [email protected]

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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Koshy,

Fibonacci

and

Lucas

Numbers

Wiley-Interscience Publication, New York, 2001.

with

Applications,

A