Concrete mathematics : a foundation for computer science

3 EXERCISES 99
39 Prove that the double sum &k~‘og,x &j 1.
40 The spiral function o(n), indicated in the diagram below, maps a nonnegative
integer n onto an ordered pair of integers (x(n), y (n)). For
example, it maps n = 9 onto the ordered pair (1,2).
tY 4
a Prove that if m = [J;;I,
x(n) = (-l)“((n-m(m+l)).[[ZfiJ iseven] + [irnl),
and find a similar formula for y(n). Hint: Classify the spiral into
segments Wk, Sk, Ek, Nk according as [2fij = 4k - 2, 4k - 1, 4k,
4k+ 1.
b Prove that, conversely, we can determine n from o(n) by a formula
of the form
Bonus problems
n = WI2 f (2k+x(n) +y(n)) , k = m=(lx(n)l,lv(n)l).
Give a rule for when the sign is + and when the sign is -.
41 Let f and g be increasing functions such that the sets {f (1)) f (2), . . . } and
{g (1) , g (2)) . . } partition the positive integers. Suppose that f and g are
related by the condition g(n) = f(f(n)) + 1 for all n > 0. Prove that
f(n) = [n@J and g(n) = ln@‘J, where @ = (1 + &)/2.
42 Do there exist real numbers a, (3, and y such that Spec(a), Spec( (3), and
Spec(y) together partition the set of positive integers?