Combinatorics Through Guided Discovery, 2004a

4.3. Generating Functions and Recurrence Relations 89
(c) Explain why there is a real number b such that, for large values of n,
the value of the nth Fibonacci number is almost exactly (but not quite)
some constant times b n . (Find b and the constant.)
(d) Find an algebraic explanation (not using the recurrence equation) of
what happens to make the square roots of five go away. (h)
(e) As a challenge (which the author has not yet done), see if you can find
a way to show algebraically (not using the recurrence relation, but
rather the formula you get by removing the square roots of five) that
the formula for the Fibonacci numbers yields integers.
Problem 223. Solve the recurrence a n =4a n−1 − 4a n−2 .
4.3.5 Catalan Numbers
⇒
Problem 224.
(a) Using either lattice paths or diagonal lattice paths, explain why the
Catalan Number c n satisfies the recurrence
∑n−1
C n = C i−1 C n−i .
i=1
(h)
(b) Show that if we use y to stand for the power series ∑ ∞
n=0 c n x n , then
we can find y by solving a quadratic equation. Find y. (h)
(c) Taylor’s theorem from calculus tells us that the extended binomial
theorem
∞∑ ( ) r
(1 + x) r = x i
i
holds for any number real number r, where ( r )
i
is defined to be
r i
i!
i=0
r(r − 1) ···(r − i +1)
= .
i!
Use this and your solution for y (note that of the two possible values
for y that you get from the quadratic formula, only one gives an actual
power series) to get a formula for the Catalan numbers. (h)