Concrete mathematics : a foundation for computer science

526 ANSWERS TO EXERCISJES
5.94 This is a consequence of Henrici’s “friendly monster” identity,
f(a,z)f(a,wz)f(a,w"z)
=
F (
;a-+, ia++ 42 3
5a,3a+~i,~a+5,3a-~,fa,~a+~,a I( 9 >) '
where f (a, z) = F(; a; z). This identity can be proved by showing that both
sides satisfy the same differential equation. If we replace 3n by 3n + 1 or
3n + 2, the given sum is zero.
5.95 See [78] for partial reisults. The computer experiments were done by
V. A. Vyssotsky.
5.96 All large n have the property, according to S&k&y [256’]. Paul ErdGs
conjectures that, in fact, ma+, cp ((2,“)) tends to infinity as n + 00.
5.97 The congruence surely holds if 2n + 1 is prime. Steven Skiena has also Ilan Vardi notes
found the example n = 2953, when 2n + 1 = 3.11 .179.
that the condition
holds for
6.1 2314,2431,3241,1342,3124,4132,4213,1423,2143, 3412,4321. 2n+l =p’,
where p is prime,
6.2 { E}n-&, because every such function partitions its domain into k non- if and only if
empty subsets, and there are rnk ways to assign function values for each 2pm’ mod p2 = 1.
partition. (Summing over k gives a combinatorial proof of (6.10).) This yields two
more examples:
6.3 Now dk+’ 6 (center of gravity) --E = 1 -e+(d’ +...+dk)/k. This n= (‘093~-‘)/2;
recurrence is like (6.55) but with 1 - c in place of 1; hence the optimum
solution is dk+’ = (1 - c)Hk. This is unbounded as long as c < 1.
n = (35112-1)/2.
6.4 Hln+’ - :H,,. (Similarly EC”=, (-l)kp’/k = Hz,, - H,.)
6.5 U,(x,y) is equal to
+ ky)n-‘. The
k31 (;)(-l)kp'(~+ky)n-' =
This proves (6.75). Let R,(x,y) =
x~“U,(x,y); then Ro(x,y) =: 0 and R,(x,y) = R,-'(x,y) + l/n+y/x, hence
R,(x,y) = H,+ny/x. (Incidentally, the original sum U, = U,(n, -1) doesn’t
lead to a recurrence such as this; therefore the more general sum, which detaches
x from its dependenice on n, is easier to solve inductively than its
special case. This is another instructive example where a strong induction
hypothesis makes the difference between success and failure.) The Fibonacci recurrence
is additive,
6.6 Each pair of babies bb present at the end of a month becomes a pair but the rabbits are
of adults aa at the end of the next month; and each pair aa becomes an multiplying.