Fibonacci's rabbits and sunflowers

In a lecture given to high school students, Prof Jeffrey Chasnov derives the Fibonacci numbers by considering rabbits reproducing, and explains the special significance of these numbers and why they may appear unexpectedly in nature.

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[chrisg]

I bought 2 pregnant rabbits but one ate the other, so just 1 was left at the end of the first month. However that remaining rabbit gave birth to 3 rabbits at the end of the 2nd month and promptly died, so 3 were left. I don't know exactly what they got up to during the 3rd month, but by the end of it one rabbit had again been eaten by another, while the third had given birth to 3 rabbits just before dying, so 4 altogether. Of those 4, two gave birth to 3 each next month and died, while yet again 1 rabbit was eaten by another, leaving 7.

2, 1, 3, 4, 7 . . . ? Of course I should have known, they were Lucas rabbits. Their acts of cannibalism each month however follow the Fibonacci sequence: 1, 0, 1, 1, ... I'm sure you can continue.