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)

# require 'pry'
INF = 10 ** 8
N, X, Y = gets.split.map(&:to_i)
lists = Array.new(N){Array.new(N, INF)}
g = {}
# binding.pry
(N-1).times do |i|
  # g[i][i+1] = 1
  # g[i+1][i] = 1
  (g[i] ||= []) << i+1
  (g[i+1] ||= []) << i
end
(g[X-1] ||= []) << Y-1
(g[Y-1] ||= []) << X-1
# g.each do |gg|
#   p gg
# end
# p g
# binding.pry
ans = Hash.new(0)
N.times do |i|
  # binding.pry if i == 3
  queue = [] << [i, 1, -1]
  while !queue.empty?
    now = queue.shift
    g[now[0]].each do |to|
      next if to == i || to == now[2]
      if lists[i][to] == INF || lists[i][to] > now[1]
        lists[i][to] = now[1]
        # lists[to][i] = now[1]
        ans[now[1]] += 1
        queue << [to, now[1] + 1]
      end
    end
  end
end
# lists.each do |list|
#   p list
# end
# p ans.sort_by{|k, v|k}.
1.upto(N-1) do |i|
  puts ans[i] / 2
end