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: Accepted)

#include<bits/stdc++.h>
using namespace std;
using ll=long long;
constexpr ll mod=1e9+7;

int main()
{
  int n,x,y;
  cin>>n>>x>>y;
  x--;y--;
  vector<vector<int>> graph(n);
  graph[0].push_back(1);
  graph[n-1].push_back(n-2);
  for (int i = 1; i < n-1; ++i)
  {
    graph[i].push_back(i-1);
    graph[i].push_back(i+1);
  }
  graph[x].push_back(y);
  graph[y].push_back(x);
  vector<int> ans(n,0);
  for (int start = 0; start < n; ++start)
  {
    queue<int> next;
    vector<int> dist(n,-1);
    dist[start]=0;
    next.push(start);
    while(!next.empty())
    {
      int tmp=next.front();
      next.pop();
      for(const auto& e: graph[tmp]) 
      {
        if(dist[e]!=-1) continue;
        dist[e]=dist[tmp]+1;
        next.push(e);
      }
    }
    for (int i = 0; i < n; ++i)
    {
      ans[dist[i]]++;
    }
  }
  for (int i = 1; i < ans.size(); ++i)
  {
    cout<<ans[i]/2<<endl;
  }
}