Concrete mathematics : a foundation for computer science

94 INTEGER FUNCTIONS
Also, as we guessed, we can now compute c, because the third column
has become easy to fathom. It contains d copies of the arithmetic progression
O/m, d/m, 2d/m, . , (m - d)/m, so its sum is
d(;(()+!$).$ = F;
the third column is actually subtracted, not added, so we have
d-m
c = -.
2
End of mystery, end of quest. The desired closed form is
where d = gcd(m, n). As a check, we can make sure this works in the special
cases n = 0 and n = 1 that we knew before: When n = 0 we get d =
gcd(m,O) = m; the last two terms of the formula are zero so the formula
properly gives mLx/ml. And for n = 1 we get d = gcd(m, 1) = 1; the last
two terms cancel nicely, and the sum is just 1x1.
By manipulating the closed form a bit, we can actually make it symmetric
in m and n:
x [T/ =d[???+~n+!$-?!
O$k