Concrete mathematics : a foundation for computer science

“inventive genius
requires pleasurable
mental activity as
a condition for its
vigorous exercise.
‘Necessity is the
mother of invention’
is a silly proverb.
‘Necessity is the
mother of futile
dodges’is much
nearer to the truth.
The basis of the
growth of modern
invention is science,
and science is almost
wholly the
outgrowth of pleasurable
intellectual
curiosity.”
-A. N. Whitehead
[303]
then n/3 and 2n/3 are integers, so the sum is
3.5 FLOOR/CEILING SUMS 91
If n mod 3 = 1 then (n - 1)/3 and (2n - 2)/3 are integers, so we have
Again this last step follows from (3.26), this time with m = 3. And finally, if
n mod 3 = 2 then
The left hemispheres of our brains have finished the case m = 3, but the
right hemispheres still can’t recognize the pattern, so we proceed to m = 4:
At least we know enough by now to consider cases based on n mod m. If
n mod 4 = 0 then
Andifnmod4=1,
The case n mod 4 = 3 turns out to give the same answer. Finally, in the case
n mod 4 = 2 we get something a bit different, and this turns out to be an
important clue to the behavior in general:
This last step simplifies something of the form [y/2] + [(y + 1)/2J, which
again is a special case of (3.26).