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++14 (GCC 5.4.1) (judge verdict: Time Limit Exceeded)

#include <iostream>

#define SIZE 2010
#define INF  1000000

int dist[SIZE][SIZE], ans[SIZE];

int main() {
    int n, x, y;
    std::cin >> n >> x >> y; --x; --y;
    for (int i=0; i<n; ++i) {
        for (int j=0; j<n; ++j) {
            dist[i][j] = INF;
        }
    }

    dist[x][y] = 1; dist[y][x] = 1;
    dist[n-1][n-1] = 0;
    for (int i=0; i<n-1; ++i) {
        dist[i][i] = 0;
        dist[i][i+1] = 1; dist[i+1][i] = 1;
    }
    for (int k=0; k<n; ++k) {
        for (int i=0; i<n; ++i) {
            for (int j=0; j<n; ++j) {
                dist[i][j] = std::min(dist[i][j], dist[i][k] + dist[k][j]);
            }
        }
    }

    for (int i=0; i<n; ++i) {
        for (int j=i+1; j<n; ++j)
            ++ans[dist[i][j]];
    }
    for (int k=1; k<n; ++k) std::cout << ans[k] << '\n';

    return 0;
}