Download A Generalization of Recursive Integer Sequences of Order 2

A Generalization of Recursive
Integer Sequences of Order 2
Stephen A. Parry
Missouri State REU
August 1, 2007
Synopsis
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A ring extension from Ω to Ωα.
The definition of Period-1 and Period-2
orbits, along with an algorithm for finding
Period-2 matrices.
Discuss isomorphisms
Generalized Relations
Powers of Period-1 and Period-2 2x2
Matrices
Multiplying NxM Recursive Matrices
Generalizations
Problems that need additional research
Definitions
Definition: A recursive integer sequence of order 2 is a sequence in the
form An   An1   An2 ,  ,  Z.
Definition: The Fibonacci sequence is a recursive sequence
Fn  Fn  1  Fn  2, F 1  1, F 2  1.
{...1,1, 2,3,5,8,...}
Definition: The Lucas sequence is a recursive sequence
Ln  Ln  1  Ln  2, L1  1, L 2  3.
{...1,3, 4, 7,11,18,...}
Definition:   a  b b   M (2, Z) 



b
a



Recursive Matrix

We will use this definition of a recursive matrix
interchangeably with the recursive sequence
representation.
Definition: A recursive matrix is an nxm matrix in the form
 An
A
 n 1


 An r
An 1
An c 




An r c 
Note: The recursive matrix need not be square.
Recursive Matrices
Example:
An  2 A n 1 4 A n  2 A1  1, A2  2
{...1, 2,8, 24,...}
 24 8 
 8 2


Example:
{...1, 2,8, 24,...}
 24 8
8 2

2 
1 
An  A n 1  A n  2 A1  1, A2  1
{...1,1, 2,3,5,8,...}
 2 1
1 1


{...1,1, 2,3,5,8,13,...}
13
8

5

3
8
5
3
2
5
3
2
1
3
2 
1

1
More Definitions
Definition: F is the set of recursive matrices with integer multiples of
Fibonacci entries.
Definition: L is the set of recursive matrices with integer multiples of
Fibonacci entries.
Examples:
 2
3 1   F
1 
18 11 7
7 4 3

 4 3 1
4
1
2

 L


 2 1
1 1  F, 


About Ω
Theorem:
Ω forms an integral domain.
Definition: We define σ as the shift map.
Theorem:
Theorem:
F
n   2
 F1
a
n
F1 
1

1
F0 

n
1
F  nZ

0
 4 3
A
L
3
1


F4 k 
F
A2 k  5k  4 k 1
F
F
F
4 k 1 
 4k
L4 k  2 
L
A2 k 1  5k  4 k 3
L

 L4 k  2 L4 k 1 
Examples
Fibonacci * Fibonacci
 2 1  3 2
1 1  2 1 



8 5

F

 5 3
Fibonacci * Lucas
 2 1  4 3
1 1  3 1



11 7 

L

 7 4
Lucas * Lucas
7 4  4 3
 4 3   3 1



8 5 
 5
F

 5 3
l L
f
f F
f F
F
L
F
L
Ring Extension
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
Now we have a result for the Fibonacci and Lucas
numbers, we would like to make a generalization for
all recursive sequences,
An   An1   An2 ,  ,  Z.
The matrix we would require in the 2x2 case is
 b   a b 
 b
.
a




Since the identity is in Ω, then we must have the
identity in the ring extension. Thus, β=1.
Therefore, we will concentrate on
An   An1  An2 ,  Z.
Ring Extension
 b  a b 
  
a 
 b
b  a b 


b
a


More Definitions
Definition:
Definition:
Pw1
A Period-1 sequence is any sequence when expressed in
matrix form will be closed under multiplication. Define
this set as Pw1 .
A Period-2 sequence is any sequence when expressed in matrix
form will be closed under multiplication in the union of the
Period-2 sequence and its complimentary Period-1 sequence.
Define this set as Pw2 .
Pw2
R
Cw
Isomorphism
Theorem:
Theorem:

  Z[  ]  Z[ ],    
1
1
0
(1   )2 n /  2    1  A2 n 1  A2 n
such that
An   An 1  An 2 , A1  1, A2   .
Example:
(1   )3 /  2    1 
(1  3  3 2   3 ) /  2    1
 5  8  F5  F6
Period-2 2x2 Matrices
Theorem: We express a Period-1 sequence as
{...1,  ,  2  1,  3  2 ...}.

 2  1

 

.
1
Theorem: We express a Period-2 sequence for odd α as
{... ,  2  2,  3  3 ...}.
 3  3  2  2 
   2
.
 
  2
Theorem: We express a Period-2 sequence for even α as
1
{... ,  2  2,  3  3 ...}.
2
1  3  3  2  2 
   2

2  2
 
Proof for odd α
Definition: Given a recursive sequence in the form, An   An1  An2
we define the characteristic polynomial,
x

Proof:
  2  4
2
  2  4
2
x 2   x  1.
.
.
 2    1
 2  4 2   2  4(  1)
 ( 2  2)  
Note: The discriminant of the characteristic polynomial plays an
important role.
Proof for odd α
 3  3  2  2 
   2

 
  2
 6  7 4  13 2  4  5  6 3  8 
  

5
3
4
2


6


8



5


4


2
4
2
3



3


1

 2 
2
2
2
 (  4) 

(


4)



3
 2 1 
   2
2 n  n  Pw1
2 n 1  Pw2
α=5
Example:
Period-1
Period-2
{...1,5, 26,...}.
{...5, 27,140...}.
 26 5
5  
.

 5 1
140 27 
5  
.

 27 5 
 26 5 5 1 
 5 1  1 0 



135 26 


26
5


140 27   27 5 
 27 5   5 2 



135 26 
 29 

 26 5 
 26 5 5 1  5 1 
 5 1  1 0  1 0 




701 135


135 26 
140 27   27 5   27 5 
 27 5   5 2   5 2 




3775 727 
 29 

 727 140 
Generalized Recursive Relations
 A2
Definition:    
 A1
Theorem:
Theorem:
A1 
A0 
is the shift map  Z..
 Fn 1 Fn 
 

F
F
n 1 
 n
 An 1 An 
n
  

 A n An 1 
n
1
Theorem: An m  An1 Am  An Am1
Proof:
Theorem:
Proof:
 m n 1   m n1.
A22k 1   A2 k A2 k 1  A22k  1
det(n  )  1.
The Shift Map
Example:   2.
2 1
2  

1 0 
Period-1
{...0,1, 2,5,12...}
Period-2
{...0,1,3, 7,17...}
7 3
 3 1


5 2
2 1


5 2  2 1  12 5 
 2 1  1 0    5 2 


 

7 3  2 1  17 7 
 3 1  1 0    7 3 


 

{...0,1, 2,5,12...}
{...0,1,3, 7,17...}
 n
 n
Pw1,2
Pw2,2
A Characteristic of Period-1 Matrices

Period-1 Matrices act as units to their correlating
Period-2 Matrices
n
Theorem: Let   E   , then    E    Z
Proof:
 b  a b   B3
 
  B
b
a

  2
B2
B1 
 2b   a  b  b  a   B4
   

b   B3
 b  a
B
 n   n 3
 Bn  2
Bn  2 
.

Bn 1 
 n
E
B3 
B2 
Powers of Λα and ηα
Theorem:
Theorem:
 A2 n1
  
 A2 n
n
For α even
A2 n 
A2 n1 

2 n 1
  
 1 

4


2
n 1
 2 n 2
n
2 n
Theorem:
For α odd

  2  2n
 1 
 
4 

2 n 1
   4 
2
n 1
 2 n2
    4  2 n
2n
2
n
Periodicity
2
n
n
n
Theorem: If C   , then C   , or C 
 b  a b 
Proof: C  
a 
 b
( b  a ) 2  b 2
2
C 
  b  2ab
X  ( b  a ) 2  b 2
Y   b  2ab
Z  b2  a 2
X  0, Z  0,
let  b  2ab  0
b( b  2a )  0
2a
b  0, b 

 b  2ab 

b2  a 2 
If b  0,
1 0 
C  a
 n

0 1 
2a
If b 
,

 2
2  2
  3



C  a  2
  2

1
 

a  k 
 ,  odd
ak

2

 ,  even
4x4 Period-1 and Period-2 Matrices
Example:
Note: Similar to Period-1 and Period-2 2x2 matrices, it is possible to create a
general formula for every α.
Higher Degree Periods
Theorem: For n  1, 2  a, b, c  Z such that the primitive case of
(a  b )n  c n is true.
Proof:
The proof is dependent on the fact the degree of the ring extension
is 2.
Q=
( )  n c
Ωα

Ω
2
Q=
nc
Thus, there fails to exist periods of degree greater than 2.
The NxM case
Definition: A complete orbit is an orbit closed under Pw1  Pw2 .
11 7 
C1  7 4   L
 4 3 
Example:
34 21 13
C1C1T  5  21 13 8   F
13 8 5 
573 361 
C1C1T C1  5 354 223  F, L
 219 138 
F
L
This complete one orbit, not two; this fails to be a complete orbit.
Resulting Generalizations
Theorem: We are guaranteed a complete orbit when we are given nxm
matrix where n  Z., m  2Z..
Theorem: Every nxn recursive matrix, Mn, where n is even , forms a ring.
Theorem: The set of nxm matrices that form a complete orbit is a
semigroup.
Theorem:
If C 2  P1 , then C  P1 , or C  P2
Pw1
Pw2
Problems of Interest


Relations between Period-1 and Period-2
Sequences:
( Bn  Bn 1 ) 2 k   k ( A1  A2 ) k ( n 1) .
Finding more isomorphisms





Continued fraction maps
Eigenvalue maps
Determinant maps
Forming relationships for any power of every nxm
recursive matrix in the Period-1 and Period-2 sets.
Studying recursive relations of greater order.
References









[1] George E. Andrews. Number Theory. Dover, 1994.
[2] Lin Dazheng. Fibonacci matrices. Fibonacci Quarterly, 37(1):1420, 1999.
[3] Michele Elia. A note on fibonacci matrices of even degree.
International Journal of Mathematics and Mathematical Sciences,
27(8):457-469, 2001.
[4] John B. Fraleigh. A First Course in Abstract Algebra. AddisonWesley, seventh edition edition, 2003.
[5] Ross Honsberger. Mathematical Gems III. MAA, 1985.
[6] T.F. Mulcrone. Semigroup examples in introductory modern
algebra. The American Mathematical Monthly, 69(4):296{301, Apri.,
1962.
[7] N.N. Vorobyov. The Fibonacci Numbers. The University of Chicago,
1966.
[8] Lawrence C. Washington. Some remarks on fibonacci matrices.
Fibonacci Quarterly, 37(4):333-341, 1999.
[9] Kung-Wei Yang. Fibonacci with a golden ring. Mathematics
Magazine, 70(2):131-135, Apr.,1997.