Concrete mathematics : a foundation for computer science

3 EXERCISES 97
20 Find the sum of all multiples of x in the closed interval [(x.. fi], when
x > 0.
21 How many of the numbers 2", for 0 6 m < M, have leading digit 1 in
decimal notation?
22 Evaluate the sums S, = &, [n/2k + ij and T, = tk3, 2k [n/2k + i] 2.
23 Show that the nth element of the sequence
1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,...
is [fi + 51. (The sequence contains exactly m occurrences of m.)
24 Exercise 13 establishes an interesting relation between the two multisets
Spec(oL) and Spec(oc/(ol- l)), when OL is any irrational number > 1,
because 1 /OL + ( OL - 1 )/OL = 1. Find (and provej an interesting relation
between the two multisets Spec(a) and Spec(oL/(a+ l)), when OL is any
positive real number.
25 Prove or disprove that the Knuth numbers, defined by (3.16), satisfy
K, 3 n for all nonnegative n.
26 Show that the auxiliary Josephus numbers (3.20) satisfy
for n 3 0.
27 Prove that infinitely many of the numbers DF’ defined by (3.20) are
even, and that infinitely many are odd.
28 Solve the recurrence
a0 = 1;
an= an-l + lJan-l.l, for n > 0.
29 Show that, in addition to (3.31), we have
D(oL,n) 3 D(oI’, 1an.J) - 0~~’ -2.
30 Show that the recurrence
X0 = m,
x, = x:-,-2, for n > 0,
has the solution X, = [01~“1, if m is an integer greater than
a + 0~~’ = m and OL > 1. For example, if m = 3 the solution is
x, = [@2n+’ 1 )
l+Js
4=-y-, a = a2.
2, where