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 Ruby (2.3.3) (judge verdict: Time Limit Exceeded)

io = STDIN
n,x,y=io.gets.split.map(&:to_i)
ar=Array.new(n){Array.new(n){Float::INFINITY}}
ans=[0]*(n-1)

(0...n-1).each do |i|
  ar[i][i+1]=1
  ar[i+1][i]=1
  ar[i][i]=0
  ans[ar[i][i+1]-1]+=1
end
ar[n-1][n-1]=0
ar[x-1][y-1]=1
ar[y-1][x-1]=1
ans[ar[x-1][y-1]-1]+=1
n.times do |k|
  n.times do |i|
    n.times do |j|
      before=ar[i][j]
      if ar[i][j] > ar[i][k] + ar[k][j]
        ar[i][j] = ar[i][k] + ar[k][j]
        if before==Float::INFINITY
          ans[ar[i][j]-1]+=1
        else
          ans[before-1]-=1
          ans[ar[i][j]-1]+=1
        end
      end
    end
  end
end
ans[0]=ans[0]*2
puts ans.map{|i|i/2}