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           Control.Monad
import qualified Data.ByteString.Char8       as BS
import           Data.Maybe
import qualified Data.Vector.Unboxed         as V
import qualified Data.Vector.Unboxed.Mutable as VM

readInt = fst . fromJust . BS.readInt
readIntList = map readInt . BS.words
getInt = readInt <$> BS.getLine
getIntList = readIntList <$> BS.getLine

main = do
  [n, x, y] <- getIntList

  h <- VM.new n
  VM.set h (0::Int)

  d <- VM.new (n+1)

  let inf = 10^9 :: Int
      dfs i dist = do
        d_old <- VM.read d i
        if dist > d_old
          then return ()
          else do VM.write d i (min d_old dist)
                  when (i < n) $ dfs (i+1) (dist+1)
                  when (i > 0) $ dfs (i-1) (dist+1)
                  when (i == x) $ dfs y (dist+1)
                  when (i == y) $ dfs x (dist+1)

  forM_ [1..n] $ \i -> do
    VM.set d inf
    dfs i 0
    forM_ [1..n] $ \j -> do
      dist <- VM.read d j
      hh <- VM.read h dist
      VM.write h dist (hh+1)

  forM_ [1..(n-1)] $ \i -> do hh <- VM.read h i
                              print $ hh `div` 2