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: Accepted)

import qualified Data.Vector.Unboxed as VU
import qualified Data.Vector.Unboxed.Mutable as VUM
import qualified Data.Map as MP
import Data.List
import Data.Tuple
import Control.Monad

type Distance = VUM.IOVector Int
type Table = VUM.IOVector Int

bfs :: Distance -> MP.Map Int [Int] -> [Int] -> IO Distance
bfs distance graph [] = return distance
bfs distance graph (x:xs) = do
  VUM.read distance x >>= \rx -> do
    let Just ys = sort <$> MP.lookup x graph
    ys' <- flip filterM ys $ \y -> do
      VUM.read distance y >>= \ry ->
        if (ry /= 10^9)
           then return False
           else (VUM.write distance y $ rx+1) >> return True
    bfs distance graph (xs ++ ys')

main = do
  [n,a,b] <- map read . words <$> getLine :: IO [Int]
  let vs = let xs = zip [1..n] [2..n] in [(a,b),(b,a)] ++ (map swap xs) ++ xs :: [(Int, Int)]
  let graph = foldl (\m k -> MP.insert k [] m) MP.empty [1..n]
  let graph' = foldl (\m (f,t) -> MP.adjust (t:) f m) graph vs
  
  table <- VUM.replicate (n+1) 0 :: IO Table

  forM_ [1..n] $ \i -> do
    distance <- VUM.replicate (n+1) (10^9) :: IO Distance
    VUM.write distance i 0    
    bfs distance graph' [i]
    distance' <- VU.tail . VU.zip (VU.fromList [0..n]) <$> VU.unsafeFreeze distance

    VU.forM_ distance' $ \(j,d) -> do
      when (j > i) $ do
        VUM.modify table (+1) d

  table' <- VU.tail . VU.init <$> VU.unsafeFreeze table    
  VU.mapM_ print table'