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: Wrong Answer)

n, x, y = gets.chomp.split(" ").map(&:to_i)
list = {}
ans = []
array = []

n.times do |i|
  list[i+1] = []
  list[i+1].push(i+2) unless i+1 == n
  list[i+1].push(i) unless i+1 == 1
  ans.push([i+1, i+2]) unless i+1 == n
end
list[x].push(y)
list[y].push(x)
# p list

ans.push([x, y])
array += ans

puts ans.length
i = 0

(n-2).times do
  tmp = []
  ans.each do |arr|
    list[arr[1]].each do |v|
      cop = [arr[0], v].sort
      unless array.include?(cop) || arr[0] == v
        tmp.push(cop)
      end
    end
    list[arr[0]].each do |v|
      cop = [arr[1], v].sort
      unless array.include?(cop) || arr[1] == v || tmp.include?(cop)
        tmp.push(cop)
      end
    end
  end
  tmp.pop if i == 0
  ans = tmp
  array += ans
  i+=1
  puts ans.length
end