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 <cmath>
#include <cstdio>
#include <vector>
#include <map>
#include <queue>
#include <list>
#include <iostream>
#include <algorithm>
using namespace std;
#define rep(i, n) for (int i = 0; i < (n); ++i)
using ll = long long;
using P = pair<int, int>;
const int INF = 10000000;

int main() {
  std::cin.tie(0);
  std::ios::sync_with_stdio(false);

  int n,x,y;
  cin >> n >> x >> y;
  vector< vector<int> > d(n, vector<int>(n + 2, INF));
  rep(i, n-1) for (int j = i + 1; j < n; ++j) {
    d[i + 1][j + 1] = j - i;
  }
  // --x;
  // --y;
  d[x][y] = 1;
  for(int i = x; i > 0; --i) for(int j = y; j > 0; --j) {
    if (i >= j) continue;
    if (i == x && j == y) continue;
    int mn = min(d[i+1][j], d[i][j+1]) + 1;
    d[i][j] = min(mn, d[i][j]);
  }
  for(int i = x; i <= n; ++i) for(int j = y; j > 0; --j) {
    if (i >= j) continue;
    if (i == x && j == y) continue;
    int mn = min(d[i-1][j], d[i][j+1]) + 1;
    d[i][j] = min(mn, d[i][j]);
  }
  for(int i = x; i > 0; --i) for(int j = y; j <= n; ++j) {
    if (i >= j) continue;
    if (i == x && j == y) continue;
    int mn = min(d[i+1][j], d[i][j-1]) + 1;
    d[i][j] = min(mn, d[i][j]);
  }
  for(int i = x; i <= n; ++i) for(int j = y; j <= n; ++j) {
    if (i >= j) continue;
    if (i == x && j == y) continue;
    int mn = min(d[i-1][j], d[i][j-1]) + 1;
    d[i][j] = min(mn, d[i][j]);
  }


  vector<int> ret(n-1, 0);
  rep(i, n-1) for (int j = i + 1; j < n; ++j) {
    // cout << i << "," << j << "," << d[i][j] << endl;
    ret[d[i + 1][j + 1] - 1] += 1;
    // if (d[i + 1][j + 1] > 5) cout << i << "," << j << "," << d[i][j] << endl;
  }
  for (auto v : ret) cout << v << endl;

}