Concrete mathematics : a foundation for computer science

304 SPECIAL NUMBERS
80 Show that continuant polynomials appear in the matrix product
(i A)(; J2)-.(Y iI)
1:
and in the determinant
I -1 Xl x2 1 0 1 0 0
0 0 -1x31 ,.. 0 -1
det
. . .
-1 . . .
0 0
1 :
81 Generalizing (6.146), find a continued fraction related to the generating
function En21 z LnaJ, when 01 is any positive irrational number.
82 Let m and n be odd, positive integers. Find closed forms for
%I = & F2,,*+:+F ;
m "J = x Fzmk+:-Fm'
k>O
Hint: The sums in exercise 62 are S:,3 - ST,,,,, and S1,s - ST,~,+~.
83 Let o( be an irrational number in (0,l) and let al, a2, as, . . . be the
partial quotients in its continued fraction representation. Show that
ID (01, n) 1 < 2 when n = K( al, . . . , a,), where D is the discrepancy
defined in Chapter 3.
84 Let Q,, be the largest denominator on level n of the Stern-Brocot tree.
(Thus (Qo, QI, Q2, Q3,Qh,. . .) = (1,2,3,5,8,. . .) according to the diagram
in Chapter 4.) Prove that Q,, = F,+2.
85 Characterize all N such that the Fibonacci residues
x,
{FomodN, FI modN, FzmodN, . ..}
form the complete set {0, 1,. . . , N - l}. (See exercise 59.)
Research problems
86 What is the best way to extend the definition of {t} to arbitrary real
values of n and k?
87 Let H, be written in lowest terms as an/b,, as in exercise 52.
a Are there infinitely many n with 11 \a,?
b Are there infinitely many n with b, = lcm(l,2,. . . ,n)? (Two such
values are n = 250 and n = 1000.)
88 Prove that y and eY are irrational.