Combinatorics Through Guided Discovery, 2004a

86 4. Generating Functions
◦ Problem 212. Suppose we deposit $5000 in a savings certificate that pays
ten percent interest and also participate in a program to add $1000 to the
certificate at the end of each year (from the end of the first year on) that
follows (also subject to interest.) Assuming we make the $5000 deposit at
the end of year 0, and letting a i be the amount of money in the account at
the end of year i, write a recurrence for the amount of money the certificate
is worth at the end of year n. Solve this recurrence. How much money do
we have in the account (after our year-end deposit) at the end of ten years?
At the end of 20 years?
4.3.2 Fibonacci numbers
The sequence of problems that follows (culminating in Problem 222) describes a
number of hypotheses we might make about a fictional population of rabbits. We
use the example of a rabbit population for historic reasons; our goal is a classical
sequence of numbers called Fibonacci numbers. When Fibonacci5 introduced them,
he did so with a fictional population of rabbits.
4.3.3 Second order linear recurrence relations
• Problem 213. Suppose we start (at the end of month 0) with 10 pairs of
baby rabbits, and that after baby rabbits mature for one month they begin to
reproduce, with each pair producing two new pairs at the end of each month
afterwards. Suppose further that over the time we observe the rabbits, none
die. Let a n be the number of rabbits we have at the end of month n. Show
that a n = a n−1 +2a n−2 . This is an example of a second order linear recurrence
with constant coefficients. Using a method similar to that of Problem 211,
show that
∞∑
a i x i 10
=
1 − x − 2x 2 .
i=0
This gives us the generating function for the sequence a i giving the population
in month i; shortly we shall see a method for converting this to a
solution to the recurrence.
• Problem 214. In Fibonacci’s original problem, each pair of mature rabbits
produces one new pair at the end of each month, but otherwise the situation
is the same as in Problem 213. Assuming that we start with one pair of baby
rabbits (at the end of month 0), find the generating function for the number
of pairs of rabbits we have at the end on n months. (h)
5Apparently Leanardo de Pisa was given the name Fibonacci posthumously. It is a shortening of
“son of Bonacci” in Italian.