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: Accepted)

require 'set'; require 'prime'
INF=Float::INFINITY; MOD=10**9+7
n,x,y = gets.chomp.split.map(&:to_i)
x -= 1
y -= 1

ans = Array.new(n, 0)

n.times do |start_i|
  q = [start_i]
  d = 0
  ds = Array.new(n, -1)
  ds[start_i] = 0
  until q.empty?
    nq = []
    d+=1
    q.each do |index|
      if index == x && ds[y] == -1
        ds[y] = d
        nq.push y
      elsif index == y && ds[x] == -1
        ds[x] = d
        nq.push x
      end
      if index > 0 && ds[index-1] == -1
        ds[index-1] = d
        nq.push(index - 1)
      end
      if index < n - 1 && ds[index+1] == -1
        ds[index+1] = d
        nq.push(index + 1)
      end
    end
    q = nq
  end
  # p ds
  (start_i + 1).upto(n-1) do |i|
    ans[ds[i]] += 1
  end
end

ans.shift

ans.each do |e|
  puts e
end