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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.
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Linear recurrencesand 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.
Fn = Fn-1+Bn-1
Bn = Fn-1
Fn = Fn-1+Fn-2
n Fn Bn
1 1 0
2 1 1
3 2 1
4 3 2
5 5 3
6 8 5
Check for F2
Theorem:
nn
nF 251
251
51
Proof:
Solution 1
11
1 251
251
51F
Fn = Fn-1+Fn-2
Replace and check
The theorem is true by induction.
Fn = Fn-1+Bn-1
Bn = Fn-1
1 1
1 0
Fn-1
Bn-1
Fn
Bn
=
1 1
1 0
Fn-2
Bn-2
=1 1
1 0
Fn-2
Bn-2
1 1
1 0=
2
F1
B1
1 1
1 0=
n-1 1
0
1 1
1 0=
n-1
Solution 2
1
0
1 1
1 0=
n-1Fn
Bn
1 1
1 0= VDV-1 1 1
1 0= VDn-1V-1
n-1
Solution 2
r 0
0 s
251,
251 srD=
r s
1 1V=
Solution 2
1
0
1 1
1 0=
n-1Fn
Bn
r 0
0 s
r s
1 1
r s
1 1
-1n-1=
1
0
nnn srF
51
Solution 3Fn = Fn-1+Fn-2
(F1,F2,F3,F4,…)={Fi}=F
3211
...0000
...1000
...0100
...0010
L
Rewrite all equations as a vector equation.
L: The left shift operator
L{Fi}={Fi+1}
Fn+2 = Fn+1+Fn
L2F = LF+F(L2-L-I)F = 0(L-rI)(L-sI)F = 0
Solution 3
(L-sI)(L-rI)F = 0Ax = 0
I{Fi}={Fi}
251,
251 sr
(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 = 0an+1 = san
(L-rI)b = 0bn+1 = rbn
an = sn-1a1 bn = rn-1b1
Fn = csn-1+drn-1
Fn = csn-1+drn-1
F1=1 F2=1
Solve for c and d
nn
nF 251
251
51
Fibonacci numbers in nature
2 petals 3 petals
5 petals 34 petals
Fibonacci numbers in nature