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)

N, X, Y = gets.split(" ").map(&:to_i)

dp = (N + 2).times.map { [0] * (N + 2) }
dp[X][Y] = 1
(1..(N - 1)).each do |x|
  dp[x][x + 1] = 1
end

(1..(N - 1)).each do |i|
  (1..(N - 1)).each do |x|
    row = dp[x]
    ((x + 1)..N).each do |y|
      next if dp[x][y] > 0

      case
      when dp[x - 1][y] == i,
           dp[x][y - 1] == i,
           dp[x + 1][y] == i,
           dp[x][y + 1] == i
        dp[x][y] = i + 1
      end
    end
  end
end

numbers = [0] * N
dp.each do |row|
  row.each do |n|
    next if n.nil?

    numbers[n] += 1
  end
end

numbers[1..-1].each {|n| puts n }