Concrete mathematics : a foundation for computer science

296 SPECIAL NUMBERS
6 An explorer has left a pair of baby rabbits on an island. If baby rabbits
become adults after one month, and if each pair of adult rabbits produces
one pair of baby rabbits every month, how many pairs of rabbits are
present after n months’? (After two months there are two pairs, one of
which is newborn.) Find a connection between this problem and the “bee
tree” in the text.
7 Show that Cassini’s identity (6.103) is a special case of (6.108), and a
special case of (6.134).
8 Use the Fibonacci number system to convert 65 mi/hr into an approximate
number of km/hr.
9 About how many square kilometers are in 8 square miles?
10 What is the continued fraction representation of $?
Basics
11 What is I:,(-l)“[t], the row sum of Stirling’s cycle-number triangle
with alternating signs, when n is a nonnegative integer?
12 Prove that Stirling numbers have an inversion law analogous to (5.48):
g(n) = G {t}(--1 lkf(k) W f(n) = $ [L] (-l)kg(k).
13 The differential operators D = & and 4 = zD are mentioned in Chapters
2 and 5. We have
a2 = z2D2+zD,
because a2f(z) = &f’(z) = z&zf’(z) = z2f”(z) + zf’(z), which is
(z2D2+zD)f(z). Similarly it can be shown that a3 = z3D3+3z2D2+zD.
Prove the general formulas
for all n 3 0. (These can be used to convert between differential expressions
of the forms tk cxkzkfik’(z) and xk fikakf(z), as in (5.1og).)
14 Prove the power identity (6.37) for Eulerian numbers.
15 Prove the Eulerian identity (6.39) by taking the mth difference of (6.37).