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: Time Limit Exceeded)

def distanced_nodes(neighbor, start_node, depth_limit):
    visited = [start_node]
    next_nodes = neighbor[start_node]
    depth = 1
    while len(next_nodes) > 0 and depth < depth_limit:
        depth += 1
        queue = next_nodes[:]
        visited += next_nodes
        next_nodes = []
        for q in queue:
            for node in neighbor[q]:
                if node in visited:
                    continue
                else:
                    next_nodes.append(node)
        next_nodes = list(set(next_nodes))
    return next_nodes

N, X, Y = list(map(int, input().split()))
neighbor = { i+1: [] for i in range(N) }
for i in range(1,N):
    neighbor[i].append(i+1)
    neighbor[N-i+1].append(N-i)
neighbor[X].append(Y)
neighbor[Y].append(X)

for i in range(1,N):
    pairs = set()
    for j in range(1,N+1):
        for node in distanced_nodes(neighbor, j, i):
            if node > j:
                pairs.add((j, node))
            else:
                pairs.add((node, j))
    print(len(pairs))