Concrete mathematics : a foundation for computer science

A ANSWERS TO EXERCISES 533
6.55 The stated sum is *(Xzn)(mn’;l)~ by Vandermonde’s convolution.
To get (6.70)~ differentiate and set x = 0.
6.56 First replace k”+’ by ((k - m) + m) n + 1 and expand in powers of
k - m; simplifications occur as in the derivation of (6.72). If m > n or
m < 0, the answer is (-1 )“n! - m”/(“*“‘). Otherwise we need to take the
limit of (5.41) minus the term for k = m, as x + -m; the answer comes to
(-l)nn!+(-l)m+l(~)mn(n+l +mH,, m-mHm).
6.57 First prove by induction that the nth row contains at most three
distinct values A,, 13 B, 3 C,; if n is even they occur in the cyclic order
[C,,,B,,A,,B,,C,], while if n is odd they occur in the cyclic order
[C,,B,,,A,,A,,B,I. Also
A2,+1 = A2n + Bzn;
B2n+1 = B2n +Cz,;
ALn = 2A2n I;
Bin = A2n I +Bzn I;
CZn+l = 2c>,; C2n = B 2n 1+C2n-1.
It follows that Q,, == A, - C, = F,+,. (See exercise 5.75 for wraparound
binomial coefficients of order 3.)
6.58 (a) x n>OF;z'L ~(1 -z)/(l +z)(l -3z+t’) = ;((2-3z)/(l -32+
z2) -2/(1 +zij. (b) 1 naOF;t~n=z(l-2z-z2)/(1-4z-z2)(1+z-z2)=
;(2z/(l -42-z2)+32/(1 +z-2’)). (These formulas are obtained by squaring
or cubing Binet’s formula (6.123) and summing on n, then combining terms
so that @ and $ disappear.) It follows that Fi,, - 4Ff, - FA , = 3(-l)nF,.
(The corresponding recurrence for mth powers has been found by Jarden and
Motakin [163].)
6.59 Let m be fixed. We can prove by induction on n that it is, in fact,
possible to find such an x with the additional condition x $ 2 (mod 4). If x
is such a solution, we can move up to a solution modulo 3”+’ because
F8.3r,m, G 3". F8.3rIm~ , G 3n+1 (mod 3”+‘);
either x or x + 8.3” ’ or x + 16.3” ’ will do the job
6.60 F1 + 1, F2 + 1, F3 + 1, FJ - 1, and F6 - 1 are the only cases. Otherwise
the Lucas numbers of exercise 28 arise in the factorizations
F2m+(-lIm = L,+,F, I; F2m+1+(-lIm = LnF,,,;
F2m-(-lIm = L, IF,+,; F2m+1-(-lim = L+,F,.
(We have F,,, - (-l)nF,p, = L,F, in general.)
6.61 1/F2, = F, ,/F, - FL,,-, /Fz,,, when m is even and positive. The
second sum is 5/4 - F3.pp~/F3.21t, for n 3 1.