AlephNull

PROBLEM STATEMENT

The logician Charles Sanders Pierce proposed a procedure for generating and
ordering all of the positive rational numbers. A rational number is an integer
divided by an integer (n/m where both n and m are integers and m does not equal
zero). 

The procedure proceeds as follows. Start with the two rationals 0/1 and 1/0
(disregarding the fact that 1/0 is not a valid number). Call this generation 1.
To produce the next generation, insert a new rational in between each pair of
rationals in the current generation by summing the numerators (the number being
divided) of the adjacent rationals to produce the new numerator, and by summing
the denominators (the number doing the dividing) of the adjacent rationals to
produce the new denominator. By repeating this procedure indefinitely, all of
the positive rational numbers will be produced in order in their simplest
rational form. 

The first few generations proceed as follows:

G1: 0/1 1/0
G2: 0/1 1/1 1/0
G3: 0/1 1/2 1/1 2/1 1/0
G4: 0/1 1/3 1/2 2/3 1/1 3/2 2/1 3/1 1/0
G5: 0/1 1/4 1/3 2/5 1/2 3/5 2/3 3/4 1/1 4/3 3/2 5/3 2/1 5/2 3/1 4/1 1/0 

Code a method that given a generation number and a zero based index, returns
the numerator and denominator of the rational number at the position indicated
by the index within the generation. If the index is not within the range of
values for the given generation, return the special error value having zero for
both the numerator and denominator.

DEFINITION
Class: AlephNull
Parameters: int, int
Returns: int[]
Method signature: int[] rational(int generation, int item)

(be sure your method is public)

TopCoder will ensure the validity of the inputs.  Inputs are valid if all of
the following criteria are met:

* generation is from 1 to 30 inclusive.
* item is from 0 to 999999999 inclusive. 

HINT
The number of elements in a given generation can be computed as follows:
elements = (2 ^ (generation - 1)) + 1. (The '^' symbol indicates
exponentiation. For example:
Generation 1: 2^0 + 1 = 2 
Generation 2: 2^1 + 1 = 3 
Generation 3: 2^2 + 1 = 5 
Generation 4: 2^3 + 1 = 9
Generation 9: 2^8 + 1 = 257

EXAMPLES
E1:  1,0        ==> [0,1]
E2:  1,1        ==> [1,0]
E3:  1,2        ==> [0,0]      //error value
E4:  4,1        ==> [1,3]
E5:  4,6        ==> [2,1]
E6:  5,11       ==> [5,3]
E7:  8,70       ==> [9,7]
E8:  10,467     ==> [43,12]
E9:  23,4190316 ==> [438,43]
E10: 30,100     ==> [7,157]