Problem:
Score : 400 points

Problem Statement

We have an undirected graph G with N vertices numbered 1 to N and N edges as follows:

For each i=1,2,...,N-1, there is an edge between Vertex i and Vertex i+1.

There is an edge between Vertex X and Vertex Y.

For each k=1,2,...,N-1, solve the problem below:

Find the number of pairs of integers (i,j) (1 \leq i < j \leq N) such that the shortest distance between Vertex i and Vertex j in G is k.

Constraints

3 \leq N \leq 2 \times 10^3

1 \leq X,Y \leq N

X+1 < Y

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N X Y

Output

For each k=1, 2, ..., N-1 in this order, print a line containing the answer to the problem.

Sample Input 1

5 2 4

Sample Output 1

5
4
1
0

The graph in this input is as follows:

There are five pairs (i,j) (1 \leq i < j \leq N) such that the shortest distance between Vertex i and Vertex j is 1: (1,2)\,,(2,3)\,,(2,4)\,,(3,4)\,,(4,5).

There are four pairs (i,j) (1 \leq i < j \leq N) such that the shortest distance between Vertex i and Vertex j is 2: (1,3)\,,(1,4)\,,(2,5)\,,(3,5).

There is one pair (i,j) (1 \leq i < j \leq N) such that the shortest distance between Vertex i and Vertex j is 3: (1,5).

There are no pairs (i,j) (1 \leq i < j \leq N) such that the shortest distance between Vertex i and Vertex j is 4.

Sample Input 2

3 1 3

Sample Output 2

3
0

The graph in this input is as follows:

Sample Input 3

7 3 7

Sample Output 3

7
8
4
2
0
0

Sample Input 4

10 4 8

Sample Output 4

10
12
10
8
4
1
0
0
0

Solution in Haskell (GHC 7.10.3) (judge verdict: Time Limit Exceeded)

import Control.Monad
import Data.Bool
import Data.List
import qualified Data.Map as M

ard i j x y = abs (x - i) + abs (y - j) + 1

path i j x y = bool b a (a < b)
  where
    a = j - i
    b = ard i j x y

allpair n = [(x, y) | x <- [1..n], y <- [(x + 1).. n]]

allpath n x y = map (\(a,b) -> (path a b x y, a, b)) $ allpair n

cmp (a1,b1,c1) (a2,b2,c2)
  | a1 == a2 = EQ
  | a1 > a2 = GT
  | otherwise = LT

srt ps = groupBy (\(a1,b1,c1) (a2,b2,c2) -> a1 == a2) $ sortBy cmp ps

srz ps xs = let m = M.fromList $ map (\t@((a,b,c):_) -> (a, length t)) ps
                g (Just x) = x
                g Nothing = 0
             in  map (\t -> g (M.lookup t m)) xs

showResult ps = unlines $ map show ps 

solve n x y = showResult $ srz (srt $ allpath n x y) [1..(n - 1)]

main = do
 [n, x, y] <- (map read) . words <$> getLine
 putStrLn $ solve n x y