A recipe can
be
decomposed in one or more chunks RpLq,
where Rp
denotes p times direction R and Lq denotes q times
direction L. The values p
and q depend on the input numbers N and D. Define the characteristic
equation N = K*D
+ E,
where K > 0 and 0 < E < D .
3 situations
can
be discerned So, D = 2*E
and N = (2*K+1)*E . Since N and D are relatively prime, the
value of E must be 1, and hence D=2. The
corresponding
recipe, equal to R(N+1)/2 L(N-1)/2
has a primitive
structure.
This
situation is equivalent to E
< D/2. The chunks look like RK+1 LK
, or RK
LK+1, or RK LK
. Their multiplicities can be
derived by comparing multiples of D with the boundaries N and 2*N . The
values
of the multiplicities are respectively (E+1)/2, (E-1)/2, and (D/2-E) .
In
this situation, equivalent to E
> D/2, the chunks look slightly different, namely RK+1
LK+1,
or RK+1 LK , or RK
LK+1 . For
simplifying the formulas we introduce the variable F := D-E . The
multiplicities of the chunks are then respectively (D/2-F), (F+1)/2,
and
(F-1)/2 .
For all 3
situations can be easily verified that the recipe contains N
directions, and
that the number of R directions is 1 more than the number of L
directions.