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.chomp.split(" ").map(&:to_i)
INF = 1000000001
x -= 1
y -= 1
ans = Array.new(n, 0)

def push(v, d)
  return if @arr[v] != INF
  @arr[v] = d
  @que.push(v)
end

(n).times do |i|
  @arr = Array.new(n, INF)
  @que = []
  @que.push(i)
  @arr[i] = 0
  while !@que.empty? do
    v = @que.shift
    d = @arr[v]
    push(v-1, d+1) if v-1 >= 0 
    push(v+1, d+1) if v+1 < n
    push(y, d+1) if v == x 
    push(x, d+1) if v == y 
  end
  (n).times {|j| ans[@arr[j]] += 1 }
end
(n).times {|j| ans[j] /= 2 }
1.upto(n-1) {|i| puts ans[i]}