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

#result = (n-1).downto(1).to_a
#result[0] += 1
#result[y - x - 1] -= 1

result = Array.new(n).map{ Array.new(n) }
(0 ... n).each{ |i|
	(i + 1 ... n).each{ |j|
		result[i][j] = j - i
	}
}

# ショートカット路を含む長さ
(n - 1).downto(1){ |i|
	# 左端点から先の自由度
	(0 .. i - 1).each{ |j|
		# 左の左
		ll = x - j - 1
		# 左の右
		lr = x + j - 1
		#右の左
		rl = y - (i - j - 1) - 1
		#右の右
		rr = y + (i - j - 1) - 1
		
		if ll >= 0 && rl >= 0 then
			if rl - ll > i then
				result[ll][rl] = i
			end
		end

		if ll >= 0 && rr < n then
			if rr - ll > i then
				result[ll][rr] = i
			end
		end

		if lr < n && rl >= 0 then
			if rl - lr > i then
				result[lr][rl] = i
			end
		end

		if lr < n && rr < n then
			if rr - lr > i then
				result[lr][rr] = i
			end
		end
	}
}

count = Array.new(n, 0)

result.each{ |r|
	r.each{ |rr|
		if rr != nil then
			count[rr] += 1
		end
	}
}

puts count[1..-1].join("\n")