FIBONACCI AND BINET AN INTRODUCTION TO GENERATING FUNCTIONS (FOR YOUR EDIFICATION) 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 Collections: Mathematics