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 (3.4.3) (judge verdict: Time Limit Exceeded)

import numpy as np
from scipy.sparse.csgraph import shortest_path,floyd_warshall,dijkstra,bellman_ford,johnson
from scipy.sparse import csr_matrix
N,X,Y = map(int,input().split())

#隣接リストの整備
"""
route_list = np.zeros((N,N))
route_list[0][1] = 1
for i in range(1,N-1):
    route_list[i][i-1] = 1
    route_list[i][i+1] = 1
print(route_list)
route_list[-1][-2] = 1
route_list[X-1][Y-1] = 1
route_list[Y-1][X-1] = 1
"""
a = np.array([1]*(N-1))
route_list = np.diag(a,k=1)
#b = np.diag(a,k=1)
#c = np.diag(a,k=-1)
#route_list = b | c
route_list[X-1][Y-1] = 1
route_list[Y-1][X-1] = 1

shortest = shortest_path(route_list,directed=False)

for i in range(1,N):
    print(np.count_nonzero(shortest == i)//2)