Concrete mathematics : a foundation for computer science

178 BINOMIAL COEFFICIENTS
some data for checking our results. Here are the nonzero terms and their sums
for the first four values of rt.
n Qll
0 (2 =1 =1
’ (3 - (3 =1-l =o
2 (i) - (;) + (i) = 1 - 3 + 1 = -1
3 @-((:)+($)-(;;)+(;)=l-7+15-lO+l= 0
We’d better not try the next case, n = 4; the chances of making an arithmetic
error are too high. (Computing terms like (‘4’) and (‘:) by hand, let alone
combining them with the others, is worthwhile only if we’re desperate.)
So the pattern starts out 1, 0, -1, 0. Even if we knew the next term or
two, the closed form wouldn’t be obvious. But if we could find and prove a
recurrence for Q,, we’d probably be able to guess and prove its closed form.
To find a recurrence, we need to relate Qn to Q,--1 (or to Qsmaiier vaiues); but
to do this we need to relate a term like (12:J13), which arises when n = 7 and
k = 13, to terms like (“,;“). This doesn’t look promising; we don’t know
any neat relations between entries in Pascal’s triangle that are 64 rows apart.
The addition formula, our main tool for induction proofs, only relates entries
that are one row apart.
But this leads us to a key observation: There’s no need to deal with
entries that are 2”-’ rows apart. The variable n never appears by itself, it’s
always in the context 2”. So the 2n is a red herring! If we replace 2” by m, Oh, the sneakiness
all we need to do is find a closed form for the more general (but easier) sum of the instructor
who set that exam.
integer m 3 0;
then we’ll also have a closed form for Q,, = Rz~. And there’s a good chance
that the addition formula will give us a recurrence for the sequence R,.
Values of R, for small m can be read from Table 155, if we alternately
add and subtract values that appear in a southwest-to-northeast diagonal.
The results are:
There seems to be a lot of cancellation going on.
Let’s look now at the formula for R, and see if it defines a recurrence.
Our strategy is to apply the addition formula (5.8) and to find sums that