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)

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

$ans = [0] * (N+1)
# $ans1 = (N + 10).times.map{[]}

$edges = (N+1).times.map{ [] }
$edges[X] << Y
$edges[Y] << X
(1...N).each{|i|
  $edges[i] << i + 1
  $edges[i + 1] << i
}

def f(start)
  visited = [false] * (N + 1)
  q = [start]
  dist = 0
  visited[start] = true
  while !q.empty?
    dist += 1
    new_q = []
    q.each{|from|
      $edges[from].each{|to|
        next if visited[to]
        visited[to] = true
        $ans[dist] += 1 if start < to
        new_q << to
      }
    }
    q = new_q
  end
end

(1...N).each{|i|
  f(i)
}
(1...N).each{|i|
  p $ans[i]
}