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Linear recurrences
and Fibonacci numbers
A rabbit problem
In a rabbit farm, we want to know the number of
does (female rabbits) we will have after a certain
number of months if
• A doe take one month to mature
• A doe gives birth to a doe every month after that.
• Rabbits never die.
• In the first month, we have only one newborn
doe.
n Fn Bn
1 1 0
2
1
1
3
2
1
4
3
2
5
5
3
6
8
5
Fn = Fn-1+Bn-1
Bn = Fn-1
Fn = Fn-1+Fn-2
Solution 1
1 5  1 5  
1
Theorem:







F n 5  2   2  
 
 

n
n
1
1

Proof:
1   1  5   1  5  






F1 5  2   2  
 
 

Check for F2
Fn = Fn-1+Fn-2
Replace and
check
The theorem is true by induction.
Solution 2
Fn
=
Bn
1 1 Fn-1
1 0 Bn-1
=
Fn = Fn-1+Bn-1
Bn = Fn-1
1 1 1 1 Fn-2
1 0 1 0 Bn-2
=
1 1
1 0
n-1
F1
B1
=
=
1 1
2
Bn-2
1 0
1 1
1 0
n-1
Fn-2
1
0
Solution 2
Fn
=
Bn
1 1
1 0
1 1
=
1 0
D=
n-1
r 0
0 s
VDV-1
1
0
1 1
n-1
= VDn-1V-1
1 0
r s r   1  5 , s   1 
V=
 2 
 2



1 1
5



Solution 2
Fn
Bn
=
=
1 1
n-1
1 0
r s r 0
1
0
n-1
1 1 0 s
r s
1 1

1 n n
Fn  5 r  s

-1
1
0
Solution 3
Rewrite all
equations as a
(F1,F2,F3,F4,…)={Fi}=F
vector equation.
Fn = Fn-1+Fn-2
0

0
0

0


1 0 0 ...  1 
 
0 1 0 ...  1 
0 0 1 ...  2 
 
0 0 0 ...  3 



     
L
L: The left shift
operator
L{Fi}={Fi+1}
Solution 3
Fn+2 = Fn+1+Fn
L2F = LF+F
I{Fi}={Fi}
(L2-L-I)F = 0
Ax = 0
(L-rI)(L-sI)F = 0
(L-sI)(L-rI)F = 0
1 5 
1 5 
, s  

r  

 2 
2




(L-rI)(L-sI)F = 0
(L-sI)(L-rI)F = 0
Everything in the null space of (L-sI)
and everything in the null space of
(L-rI) is a solution.
(L-sI)a = 0
(L-rI)b = 0
an+1 = san
an = sn-1a1
bn+1 = rbn
bn = rn-1b1
Fn = csn-1+drn-1
Fn = csn-1+drn-1
F1=1
F2=1
Solve for c and d
n
n

1   1  5   1  5  
F n  5   2    2  
 
 

Fibonacci numbers in nature
2 petals
3 petals
5 petals
34 petals
Fibonacci numbers in nature