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: Wrong Answer)

import sys
input = sys.stdin.readline

N, X, Y = map(int, input().split())
dp = [[N] * N for _ in range(N)]
for i in range(N):
  dp[i][i] = 0
  for j in range(i, N - 1):
    dp[i][j + 1] = min(dp[i][j + 1], dp[i][j] + 1)
    if j == X - 1:
      dp[i][Y - 1] = min(dp[i][Y - 1], dp[i][j] + 1)

  for j in range(N - 1, 0, -1):
    dp[i][j - 1] = min(dp[i][j - 1], dp[i][j] + 1)
    if j == Y - 1:
      dp[i][X - 1] = min(dp[i][X - 1], dp[i][j] + 1)

  for j in range(i, N - 1):
    dp[i][j + 1] = min(dp[i][j + 1], dp[i][j] + 1)
    if j == X - 1:
      dp[i][Y - 1] = min(dp[i][Y - 1], dp[i][j] + 1)

  for j in range(N - 1, 0, -1):
    dp[i][j - 1] = min(dp[i][j - 1], dp[i][j] + 1)
    if j == Y - 1:
      dp[i][X - 1] = min(dp[i][X - 1], dp[i][j] + 1)
table = [0] * N
for i in range(N):
  for j in range(i, N):
    if dp[i][j] < N: table[dp[i][j]] += 1

for k in range(1, N):
  print(table[k])