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 ll = int_fast64_t;
using P = std::pair<ll,ll>;

int main(){
        ll n, x, y;
        scanf("%ld %ld %ld", &n, &x, &y);

        std::vector<std::vector<ll>> links(n, std::vector<ll>(0));
        for(ll i=1; i<n; i++){
                links[i-1].push_back(i);
                links[i].push_back(i-1);
        }
        links[x-1].push_back(y-1);
        links[y-1].push_back(x-1);

        std::vector<ll> ans(n, 0);
        for(ll i=0; i<n; i++){
                std::vector<int> dist(n, 1e5), used(n, 0);
                std::queue<P> que;
                que.push(P(i, 0));
                while(!que.empty()){
                        int v, dis;
                        std::tie(v, dis) = que.front();
                        que.pop();
                        used[v] = 1;
                        for(ll to: links[v]){
                                if(!used[to]){
                                        dist[to] = std::min(dist[to], dis);
                                        que.push(P(to, dis+1));
                                }
                        }
                }
                for(ll j=0; j<n; j++){
                        if(dist[j]!=1e5) ans[dist[j]]++;
                }
        }
        for(ll i=0; i<n-1; i++) printf("%ld\n", ans[i]/2);
        return 0;
}