 
Summary: FIBONACCI AND BINET
AN INTRODUCTION TO GENERATING FUNCTIONS (FOR YOUR EDIFICATION)
1. The Fibonacci Numbers
As in the text, we define the Fibonacci numbers by a recurrence relation. For n
N {0}, we let an denote the "nth Fibonacci number", we give the first two:
a0 = 0 a1 = 1
and declare
an+2 = an+1 + an
which determines all other values.
The multiplication of rabbits (counting the number of "breeding pairs") is the example
usually presented as motivation for this recurrence relation. But it appears in other settings,
and the Fibonacci numbers themselves appear in flower petals, pineapples, and a whole
myriad of living things. The first few Fibonacci numbers are:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144
1.1. A particular instance. In how many ways can you walk up a staircase of n stairs,
taking the stairs either 1 or 2 at a time? If we let bn denote the number of ways to do
this, then clearly b1 = 1, and b2 = 2 Now let's consider bn for n 3 and we'll see that the
recurrence above is satisfied.
We are at the bottom of a staircase having n stairs. We have a choice; to take the first
two stairs in one step, or to step onto the first stair. If we step onto the second stair, we
