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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Threading.Tasks;
using static System.Console;
class Program
{
	static void Main(string[] args)
	{
		// 整数の入力
		//long a = long.Parse(Console.ReadLine());
		// スペース区切りの整数の入力
		string[] NMK = Console.ReadLine().Split(' ');
		int N = int.Parse(NMK[0]);
		int X = int.Parse(NMK[1]);
		int Y = int.Parse(NMK[2]);

		// i->jに到達済み（1）
		int[,] reach = new int[N+1,N+1];

		// 頂点iから1で到達できる頂点のリスト
		List<List<int>> one = new List<List<int>>();
		for(int i=0; i<=N; i++)
		{
			List<int> a = new List<int>();
			if(i==0)
			{

			}
			else if(i==1)
			{
				a.Add(2);
				reach[i,2] = 1;
			}
			else if(i==N)
			{
				a.Add(i-1);
				reach[i, i-1] = 1;
			}
			else
			{
				a.Add(i+1);
				a.Add(i-1);
				reach[i, i+1] = 1;
				reach[i, i-1] = 1;
			}

			if(i==X)
			{
				a.Add(Y);
				reach[i, Y] = 1;
			}

			else if(i==Y)
			{
				a.Add(X);
				reach[i, X] = 1;
			}

			one.Add(a);
		}

		// 距離kで到達できる頂点のリスト（距離１をコピー）
		List<List<int>> getWithk = new List<List<int>>(one);
		// 距離kで到達できる組み合わせの出力

		Console.WriteLine(N + "");
		for(int k=2;k<N; k++)
		{
			int route = 0; // 組み合わせ数
			for(int p = 1; p<=N; p++)
			{
				List<int> a = getWithk[p];
				List<int> b = new List<int>();
				foreach(int aa in a)
				{
					// 頂点a[q]から距離1で移動できる頂点を加える
					List<int> c = one[aa];
					foreach(int cc in c)
					{
						if(p != cc && reach[p, cc] == 0)
						{
							reach[p, cc] = 1;
							b.Add(cc);

							if(p<cc)
							{
								route++;
							}
						}
					}
				}
				// 頂点pから距離kで到達できる頂点のリストを上書き
				getWithk[p] = b;
			}
			Console.WriteLine(route + "");
		}

		// 文字列の入力
		//string s = Console.ReadLine();
		//出力
		//Console.WriteLine((a+b+c) + " " + s);
	}
}