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)

# ABC 160 D
# https://atcoder.jp/contests/abc160/tasks/abc160_d

n, x, y = gets.chomp.split(/ /).map(&:to_i)
a = n.times.map { n.times.map { Float::INFINITY }}

(0..(n - 2)).each do |from|
  to = from + 1
  a[from][to] = 1
  a[to][from] = 1
end

(0..(n - 1)).each do |i|
a[i][i] = Float::INFINITY
end

a[x - 1][y - 1] = 1
a[y - 1][x - 1] = 1

for k in 0..(n - 1)
  for i in 0..(n - 1)
    for j in 0..(n - 1)
      distance = [a[i][j], a[i][k] + a[k][j]].min
      # puts "#{i + 1} -> #{j + 1} : #{distance}"
      a[i][j] = distance
    end
  end
end

# a.each do |ai|
#   puts ai.inspect
# end

a_flatten = a.flatten

1.upto(n - 1) do |i|
  if i == 2
    puts (a_flatten.select { |j| i == j }.length - n) / 2
  else
    puts a_flatten.select { |j| i == j }.length / 2
  end
end