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++ (GCC 9.2.1) (judge verdict: Accepted)

#include <iostream>
#include <vector>
#include <queue>
using namespace std;

void search(const vector<vector<int>> &g, queue<int> &q, vector<bool> &tr, vector<int> &cnt, int k, int i)
{
    int l = q.size();
    for (int j = 0; j < l; j++)
    {
        int s = q.front(); q.pop();
        for (int t:g[s])
        {
            if (tr[t]) 
            {
                q.push(t);
                if (t > i) cnt[k]++;
                tr[t] = false;
            }
        }
    }
    if (q.size() < 1) return;
    else search(g, q, tr, cnt, k+1, i);
}

int main(void)
{
    int n, x, y; cin >> n >> x >> y;
    vector<vector<int>> g(n, vector<int>(0));
    for (int i = 0; i < n - 1; i++)
    {
        g[i].push_back(i + 1);
        g[i + 1].push_back(i);
    }
    g[x - 1].push_back(y - 1);
    g[y - 1].push_back(x - 1);

    vector<int> cnt(n, 0);
    for (int i = 0; i < n; i++)
    {
        queue<int> q; q.push(i);
        vector<bool> tr(n, true); tr[i] = false;
        search(g, q, tr, cnt, 0, i);
    }
    for (int k = 0; k < n - 1; k++)
    {
        cout << cnt[k] << endl;
    }
}