the Fibonacci Sequence

  • was not important to Fibonacci and came from a problem in Liber Abbaci:[1]
    A certain man put a pair of rabbits in a place surrounded by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

  • assumptions:
    1. each pair of rabbits matures and mates after 1 month
    2. after mating, it takes 1 more month to produce another pair of rabbits
    3. rabbits never die
    4. rabbits are always born in pairs, with one female and one male


so what happens?

  • The problem defines a sequence Fn where:
    • F1 = 1,
    • F2 = 1 and
    • Fn = Fn-1 + Fn-2 for all natural n > 2

  • F1: 1 pair (1 baby, 0 mature pairs)
  • F2: 1 pair (0 baby, 1 mature pairs)
  • F3: 2 pair (1 baby, 1 mature pairs)
  • F4: 3 pair (1 baby, 2 mature pairs)
  • F5: 5 pair (2 baby, 3 mature pairs)
  • F6: 8 pair (3 baby, 5 mature pairs)
  • F7: 13 pair (5 baby, 8 mature pairs)
  • F8: 21 pair (8 baby, 13 mature pairs)
  • F9: 34 pair (13 baby, 21 mature pairs)
  • F10: 55 pair (21 baby, 34 mature pairs)
  • F11: 89 pair (34 baby, 55 mature pairs)
  • F12: 144 pair (55 baby, 89 mature pairs)
  • You're still here?
    Notice that at any Fn in the sequence, there are Fn-1 mature rabbits and Fn-2 baby rabbits (assuming n>2).

     

     

     

Fibonacci Numbers

  • Are the numbers of pairs of rabbits that exist after n months along the sequence Fn corresponding to Fibonacci's rabbit exercise.


  • Can be computed via the recursive function:
    F(n) =  1
     F(n-1) + F(n-2)     		 if n <= 2
     if n > 2

  • Are easy to generate if you're a computer.
    Don't believe me? Try typing a number in this box and hitting Calculate. (this uses a faster, non-recursive algorithm. the one above is horrible.)

so what?

  • the Fibonacci Sequence is closely related to the Golden Ratio

  • Some people claim that Fibonacci Numbers and the Golden Ratio exhibit themselves all over the place in the natural world.[2]

  • We will look into this later.

  • First, we must examine the golden ratio.

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