Concrete mathematics : a foundation for computer science

100 INTEGER FUNCTIONS
43 Find an interesting interpretation of the Knuth numbers, by unfolding
the recurrence (3.16).
44 Show that there are integers aiq’ and diq) such that
ac4)
n
=
D;!, + diq) D(q)
n
+ d(q)
n
q-l = 4 ’
for n > 0,
when DIP’ is the solution to (3.20). Use this fact to obtain another form
of the solution to the generalized Josephus problem:
Jq (n) = 1 + d(‘) + q(n - aCq))
k k ’ for ap’ 6 n < ctp>“,‘, .
45 Extend the trick of exercise 30 to find a closed-form solution to
YO = m,
Y, = 2Yip, - 1 ) for n > 0,
if m is a positive integer.
46 Prove that if n = I( fi’ + fi’-‘)mi , where m and 1 are nonnegative
integers, then Ld-1 = l(&!“’ + fi’)rnl . Use this remarkable
property to find a closed form solution to the recurrence
LO = a, integer a > 0;
Ln = [-\/2LndL-l +l)], for n > 0.
Hint: [&Gi$ZXJ] = [Jz(n + t)J.
47 The function f(x) is said to be replicative if it satisfies
f(mx) = f(x) +f(x+ i) +...+f(x+ v)
for every positive integer m. Find necessary and sufficient conditions on
the real number c for the following functions to be replicative:
a f(x) = x + c.
b f(x) = [x + c is an integer].
c f(x) =max([xJ,c).
d f(x) = x + c 1x1 - i [x is not an integer].
48 Find a necessary and sufficient condition on the real numbers 0 6 a < 1
and B 3 0 such that we can determine cx and J3 from the infinite multiset
of values
{ Inal + 14 ( n > 0 > .