Concrete mathematics : a foundation for computer science

“Known” like, say,
harmonic numbers.
A. M. Odlyzko and
H. S. Wilf have
shown that
D:’ = [( $)“Cj ,
where
CM 1.622270503.
3.3 FLOOR/CEILING RECURRENCES 81
Fortunately there’s a way to simplify this algorithm if we use the variable
D = 3n + 1 - N in place of N. (This change in notation corresponds to
assigning numbers from 3n down to 1, instead of from 1 up to 3n; it’s sort of
like a countdown.) Then the complicated assignment to N becomes
D := 3n+l- (3n+1-D)-n-1 +(3n+1-D)-n
and we can rewrite the algorithm as follows:
D := 1;
while D < 2n do D := [;Dl ;
Js(n) := 3n+l -D.
Aha! This looks much nicer, because n enters the calculation in a very simple
way. In fact, we can show by the same reasoning that the survivor J4 (n) when
every qth person is eliminated can be calculated as follows:
D := 1;
while D < (q - 1)n do D := [*Dl ;
J,(n) := qn+l -D.
(3.19)
In the case q = 2 that we know so well, this makes D grow to 2m+1 when
n==2”+1; hence Jz(n)=2(2m+1)+1 -2m+1 =21+1. Good.
The recipe in (3.19) computes a sequence of integers that can be defined
by the following recurrence:
D(q) = 1
0 1
D’4’ =
n
L,,(q) for n > 0.
q - 1 n-1 1
(3.20)
These numbers don’t seem to relate to any familiar functions in a simple
way, except when q = 2; hence they probably don’t have a nice closed form.
But if we’re willing to accept the sequence D$’ as “known,” then it’s easy to
describe the solution to the generalized Josephus problem: The survivor Js (n)
is qn+ 1 -Dp’, where k is as small as possible such that D:’ > (q - 1)n.
3.4 ‘MOD’: THE BINARY OPERATION
The quotient of n divided by m is Ln/m] , when m and n are positive
integers. It’s handy to have a simple notation also for the remainder of this