Concrete mathematics : a foundation for computer science

Anyway those of
us who’ve done
warmup exercise 4
know it.
5.2 BASIC PRACTICE 179
have the form Rk in the resulting expression, somewhat as we did in the
perturbation method of Chapter 2:
m-l-k
k
m-l-k )(-l)k + x (m-;-k)(-)k+’
= R,p, + (-1)‘” - R,p2 - (-l)2(mp’i = R,e, - Rmp2.
(In the next-to-last step we’ve used the formula (-,‘) = (-l)“, which we know
is true when m 3 0.) This derivation is valid for m 3 2.
From this recurrence we can generate values of R, quickly, and we soon
perceive that the sequence is periodic. Indeed,
R, =
1
1
0
-1
-1
0 1
if m mod 6 =
The proof by induction is by inspection. Or, if we must give a more academic
proof, we can unfold the recurrence one step to obtain
0
1
2
3
4
5
R, = (R,p2 - Rmp3) - R,-2 = -Rm-3 ,
whenever m 3 3. Hence R, = Rmp6 whenever m 3 6.
Finally, since Q,, = Rzn, we can determine Q,, by determining 2” mod 6
and using the closed form for R,. When n = 0 we have 2O mod 6 = 1; after
that we keep multiplying by 2 (mod 6), so the pattern 2, 4 repeats. Thus
R1 =l, ifn=O;
Q,, = Rp = R2 = 0, if n is odd;
{ R4=-I, ifn>Oiseven.
This closed form for Qn agrees with the first four values we calculated when
we started on the problem. We conclude that Q,OOOO~~ = R4 = -1.