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: Accepted)

import itertools

N, X, Y = map(int, input().split())

# dp = [[float('inf')] * (N+1) for _ in range(N+1)]
# # dp[Y][X] = 1
# dp[X][Y] = 1

# for i in range(1,N+1):
#     for j in range(i,N+1):
#         if i == j:
#             dp[i][j] = 0
#         elif i == X and j == Y:
#             dp[X][Y] = 1
#         elif i <= X:
#             dp[i][j] = min(dp[i][j-1] + 1, dp[i][X] + abs(Y - j) + 1)
#         else:
#             dp[i][j] = min(dp[i][j-1]+ 1, abs(i - X)+ abs(Y - j) + 1 )

# dp = list(itertools.chain.from_iterable(dp))
# for i in range(1,N):
#     print(dp.count(i))

result = [0] * N

for i in range(1, N):
    for j in range(i+1, N+1):
        dist = min(abs(j-i), abs(X-i) + 1 + abs(j-Y), abs(Y-i)+1+abs(j-X))
        result[dist] += 1

for i in range(1, N):
    print(result[i])