Concrete mathematics : a foundation for computer science

7.3 SOLVING RECURRENCES 327
When Q(z) has repeated roots, the calculations become more difficult,
but we can beef up the proof of the theorem and prove the following more
general result:
General Expansion Theorem for Rational Generating Functions.
If R(z) = P(t)/Q(z), where Q(z) = qo(1 - ~12)~' . ..(l - p~z)~[ and the
numbers (PI,. . , pi) are distinct, and if P(z) is a polynomial of degree less
than dl + . . . + dl, then
[z"] R(z) = f,ln)p; + ... + ft(n)p; for all n 3 0, (7.30)
where each fk(n) is a polynomial of degree dk - 1 with leading coefficient
This can be proved by induction on max(dl , . . . , dl), using the fact that
al(dl -l)! al(dl - l)!
R(z) - (1py - . . . -
(1 - WldL
(7.31)
is a rational function whose denominator polynomial is not divisible by
(1 - pkz)dk for any k.
Example 2: A more-or-less random recurrence.
Now that we’ve seen some general methods, we’re ready to tackle new
problems. Let’s try to find a closed form for the recurrence
go = g1 = 1 ;
Sn = gn-l+2g,~~+(-l)~, for n 3 2. (7.32)
It’s always a good idea to make a table of small cases first, and the recurrence
lets us do that easily:
No closed form is evident, and this sequence isn’t even listed in Sloane’s
Handbook [270]; so we need to go through the four-step process if we want
to discover the solution.