Concrete mathematics : a foundation for computer science

510 ANSWERS TO EXERCISES
In both cases it will be helpful to know the number of fractions that are
strictly less than (k - l)/(N + 1) in LPN; this is
1
= i(kN-dtl), d = gcd(k-l,N+l),
by (3.32). Furthermore, the number of fractions equal to (k - l)/(N + 1) in
~PN that should precede it in iPN+l is i (d - 1 - [d even]), by the nature of
organ-pipe order.
IfkNisodd,thendisevenand(k-l)/(N+l)isprecededbyt(kN-l)
elements of ?N; this is just the correct number to make things work. If kN is
even, than d is odd and (k - 1 )/( N + 1) is preceded by i (kN ) elements of ?N.
If d = 1, none of these equ’als (k - l)/(N + 1) and ‘J’N,~N is ‘