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)

n, x, y = map(int, input().split())
x, y = x-1, y-1


def update(dist, queue, d, x):
    if dist[x] >= 0: return dist, queue
    dist[x] = d+1
    queue.append(x)
    return dist, queue

ans = [0 for _ in range(n)]
for i in range(n):
    dist = [-1 for _ in range(n)]
    dist[i] = 0
    queue = [i]
    while True:
        if len(queue) == 0: break
        now = queue.pop(0)
        d = dist[now]
        if now-1 >= 0:
            dist, queue = update(dist, queue, d, now-1)
        if now+1 < n:
            dist, queue = update(dist, queue, d, now+1)
        if now == x:
            dist, queue = update(dist, queue, d, y)
        if now == y:
            dist, queue = update(dist, queue, d, x)
    for d in dist:
        ans[d] += 1

for a in ans[1:]:
    print(a//2)