1. Linear recurrences
and Fibonacci numbers

2. 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.

3. n Fn Bn
Fn = Fn-1+Bn-1
Bn = Fn-1
Fn = Fn-1+Fn-2

4. Solution 1 Check for F2 Replace and check Fn = Fn-1+Fn-2
Theorem:
Proof:
Check for F2
Replace and check
Fn = Fn-1+Fn-2
The theorem is true by induction.

5. 2 n-1 n-1 Solution 2 Fn = Fn-1+Bn-1 Fn Bn 1 1 1 0 Fn-1 Bn-1 =
1 1
1 0
Fn-1
Bn-1 =
Bn = Fn-1 2
Fn-2
Bn-2
1 1
1 0
1 1
1 0
Fn-2
Bn-2
1 1
1 0 = =
n-1
n-1 F1 B1 1
1 1
1 0
1 1
1 0 = =

6. n-1 n-1 Solution 2 Fn Bn 1 1 1 0 1 = 1 1 1 0 1 1 1 0 = VDV-1
1 1
1 0 1 =
n-1
1 1
1 0
1 1
1 0
= VDV-1
= VDn-1V-1
r 0
0 s
r s
1 1 D= V=

7. Solution 2
n-1 Fn Bn
1 1
1 0 1 =
n-1 -1
r s
1 1
r 0
0 s
r s
1 1 1 =

8. Solution 3 Rewrite all equations as a vector equation. Fn = Fn-1+Fn-2
(F1,F2,F3,F4,…)={Fi}=F
L: The left shift operator
L{Fi}={Fi+1} L

9. Solution 3 Fn+2 = Fn+1+Fn I{Fi}={Fi} L2F = LF+F Ax = 0 (L2-L-I)F = 0
(L-rI)(L-sI)F = 0
(L-sI)(L-rI)F = 0

10. (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
bn+1 = rbn
an = sn-1a1
bn = rn-1b1
Fn = csn-1+drn-1

11. Fn = csn-1+drn-1
F1= F2=1
Solve for c and d

12. Fibonacci numbers in nature
2 petals
3 petals
5 petals
34 petals

13. Fibonacci numbers in nature