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 Python (3.8.2) (judge verdict: Accepted)

#!/usr/bin/env python3
def main():
    from collections import deque
        
    N, X, Y = map(int, input().split())

    G = [[1]] + [[x - 1, x + 1] for x in range(1, N - 1)] + [[N - 2]]
    G[X - 1].append(Y - 1)
    G[Y - 1].append(X - 1)

    dist = [[-1] * N for _ in [0] * N]
    for s in range(N):
        q = deque([s])
        dist[s][s] = 0
        while q:
            v = q.popleft()
            for nv in G[v]:
                if dist[s][nv] == -1:
                    dist[s][nv] = dist[s][v] + 1
                    q.append(nv)
    res = [0] * N
    for i in range(N):
        for j in range(i + 1, N):
            res[dist[i][j]] += 1
    for d in range(1, N):
        print(res[d])


if __name__ == '__main__':
    main()