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

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

srz ps n = let m = group $ sort ps
           in modlist m 1 n

modlist [] i n 
     | i < n = replicate (n - i) 0
     | otherwise = []
modlist t@(x:xs) i n = let d = head x
                           ln = length x
                         in if i == n + 1 
                              then []
                              else if d > i 
                                  then (replicate (d - i) 0) ++ modlist t d n
                                  else ln : modlist xs (i + 1) n

solve n x y = unlines $ map show $ srz (allpath n x y) n

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