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 PyPy3 (2.4.0) (judge verdict: Runtime Error)




def main():
    import numpy as np
    N, X, Y = map(int, input().split())

    path_len = Y-X
    G = [0]*N
    for i in range(N):
        if i == 0:
            G[0] = [1]
        elif i == N-1:
            G[N-1] = [N-2]
        else:
            G[i] = [i-1, i+1]
        if i+1 == X:
            G[i].append(Y-1)
        if i+1 == Y:
            G[i].append(X-1)
    # print(G)
    D =[0]*N
    for j,g in enumerate(G):
        D[j] = [1 if i in g else np.inf for i in range(N)]
    
    # from pprint import pprint 
    # pprint(D)
    # D =np.array(D)

    # D=[[0]*N]*N
    for k in range(N):
        for i in range(N):
            for j in range(N):
                D[i][j] = min(D[i][j], D[i][k] + D[k][j])
    # pprint(D)
    D=sum(D, [])

    for i in range(1,N):
        tmp = D.count(i)
        if i ==2:
            tmp = tmp - N
        
        print(tmp//2)
    

if __name__ == '__main__':
    main()