Concrete mathematics : a foundation for computer science

90 INTEGER FUNCTIONS
it comes out in closed form so that we can easily check the answer. Our goal
now will be to generalize the sum in (3.26) by finding an expression for Is this a harder sur’n
of floors, or a sum
of harder floors?
integer m > 0, integer n.
Finding a closed form for this sum is tougher than what we’ve done so far
(except perhaps for the discrepancy problem we just looked at). But it’s Be forewarned: This
instructive, so we’ll hack away at it for the rest of this chapter.
As usual, especially with tough problems, we start by looking at small
is the beginning of
a pattern, in that
the last part of the
cases. The special case n = 1 is (3.26), with x replaced by x/m:
chapter consists
= LXJ .
of ihe solution of
some long, difficult
problem, with little
more motivation
to get more data by generalizing
than curiosity.
-Students
And as in Chapter 1, we find it useful
downwards to the case n = 0:
Our problem has two parameters, m and n; let’s look at some small cases
for m. When m = 1 there’s just a single term in the sum and its value is 1x1.
When m = 2 the sum is 1x/2] + [(x + n)/2J. We can remove the interaction
between x and n by removing n from inside the floor function, but to do that
we must consider even and odd n separately. If n is even, n/2 is an integer,
so we can remove it from the floor:
If n is odd, (n - 1)/2 is an integer so we get
The last step follows from (3.26) with m = 2.
These formulas for even and odd n slightly resemble those for n = 0 and 1,
but no clear pattern has emerged yet; so we had better continue exploring
some more small cases. For m = 3 the sum is
and we consider three cases for n: Either it’s a multiple of 3, or it’s 1 more
than a multiple, or it’s 2 more. That is, n mod 3 = 0, 1, or 2. If n mod 3 = 0
Touch& But c’mon,
gang, do you always
need to be to/d
about applications
before you can get
interested in something?
This sum
arises, for example,
in the study of
random number
generation and
testing. But mathematicians
looked
at it long before
computers came
along, because they
found it natural to
ask if there’s a way
to sum arithmetic
progressions that
have been “floored.”
-Your instructor