Concrete mathematics : a foundation for computer science

330 GENERATING FUNCTIONS
nice property of t2 by retain:ing z2 when we factor the denominator: We need
not take 1 - 4z2 + z4 all the way to a product of four factors (1 - pkz), since
two factors of the form (1 - ()kz’) will be enough to tell us the coefficients. In
other words if we consider the generating function
W(z) =
1
l-42+z2
= w()+w,z+w22+-. ,
we will have V(z) = zW(z’) and U(z) = (1 - z2)W(z2); hence Vzn+l = W,
and U2,, = W,, -W,.- 1. We save time and energy by working with the simpler
function W(z).
The factors of 1 -4z+z1 are (2-2-d) and (z-2+&), and they can
also be written (1 - (2+fi)z) and (1 - (2-fi)z) because this polynomial
is its own reflection. Thus it turns out that we have
VZn+l = wn =
U2n = w, -w,_, =
3-2~6
qq2+J3)“+-(2-ti)“;
3+J3 3-d
-+2+&)?-(2-\/5)n
(2+&l” + (2-m”
= 3 - a 3td3
(7.37)
This is the desired closed form for the number of 3 x n domino tilings.
Incidentally, we can simplify the formula for Uzn by realizing that the
second term always lies between 0 and 1. The number l-lz,, is an integer, so
we have
(7.38)
In fact, the other term (2 -- &)n/(3 + A) is extremely small when n is
large, because 2 - & z 0.268. This needs to be taken into account if we
try to use formula (7.38) in numerical calculations. For example, a fairly
expensive name-brand hand Icalculator comes up with 413403.0005 when asked
to compute (2 + fi)‘O/(3 - a). This is correct to nine significant figures;
but the true value is slightly less than 413403, not slightly greater. Therefore
it would be a mistake to tak.e the ceiling of 413403.0005; the correct answer,
U20 = 413403, is obtained by rounding to the nearest integer. Ceilings can I’ve known slippery
be hazardous.
floors too.
Example 4: A closed form for change.
When we left the problem of making change, we had just calculated the
number of ways to pay 506. Let’s try now to count the number of ways there
are to change a dollar, or a million dollars-still using only pennies, nickels,
dimes, quarters, and halves.