Concrete mathematics : a foundation for computer science

538 ANSWERS TO EXERCISIES
One of these triples applies to every integer n; for example, the six triples in
the first column cover every odd value of n, and the middle column covers all
even n that are not divisible by 6. The remainder of the proof is based on
the fact that A,,,+,, = A,F, 1 + A,+,F,, together with the congruences
for each of the triples (pk, mk, rk). (An improved solution, in which A0 and Al
are numbers of “only” 17 digits each, is also possible [184].)
6.80 The matrix product is
(
K, zIxz,...,x, I) Kn-1(~2,...,~, 1,x,)
Kn I(xI,xL,...,x,-1.1 K~(x~,xz,...,x,~~,x,I ) .
This relates to products of L and R as in (6.137), because we have
The determinant is K, (x1, , x,); the more general tridiagonal determinant
det
x1 1 0 . . . 0
Yr x2 1 0
0 Y3 x3 1 :
. .
1
0 0 . . . yn x,
satisfies tlte recurrence D, = x,D, 1 - ynD,-2.
6.81 Let 0~~~’ = a0 + 1 /(al + l/( a2 + )) be the continued fraction representation
of OL ‘. Then we have
where
aO+
Z
Am(z) =
Ao(z) +
AI (~1 +
1 l-z
= z t .lnaJ )
1
1
A2(z) + /-
Z~4m. I _ z--q”,-l
z-qm--] ’ 4m= L(al,...,a,).
A proof analogous to the text’s proof of (6.146) uses a generalization of Zeckendorf’s
theorem (F’raenkel [104, $41). If z = l/b, where b is an integer 3 2,
TL31