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: Wrong Answer)

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

n, x, y = gets.chomp.split(/ /).map(&:to_i)

module WarshallFloyd
  class Undirected
    def initialize(n)
      @n = n
      @array = 1.upto(n).map { |i| i.times.map { Float::INFINITY } }

      # puts @array.inspect

      (1..n).each do |i|
        set_distance(i, i, 0)
      end

      # puts @array.inspect

      (1..(n - 1)).each do |from|
        to = from + 1
        set_distance(from, to, 1)
      end

      puts @array.inspect
    end

    attr_reader :array, :n

    def execute
      for k in 1..n
        for from in 1..n
          for to in from..n
            # puts "#{from} -> #{to} : #{distance(from, to).inspect}"
            new_distance = [
              distance(from, to),
              distance(from, k) + distance(k, to)
            ].min
            set_distance(from, to, new_distance)
          end
        end
      end

      self
    end

    private

    def distance(from, to)
      if from <= to
        @array[to - 1][from - 1]
      else
        @array[from - 1][to - 1]
      end
    end

    def set_distance(from, to, distance)
      @array[to - 1][from - 1] = distance
    end
  end
end

warshall_floyd = WarshallFloyd::Undirected.new(n)
warshall_floyd.instance_eval do
  set_distance(x, y, 1)
end

a_flatten = warshall_floyd.execute.array.flatten
h = a_flatten.group_by { |i| i }

1.upto(n - 1) do |i|
  v = h[i]
  if v
    puts v.length
  else
    puts 0
  end
end