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 Kotlin (1.0.0) (judge verdict: Accepted)

import java.io.PrintWriter
import java.util.ArrayDeque

fun main(args: Array<String>) {
    PrintWriter(System.out).use { out ->
        solve(out)
        out.flush()
    }
}

/** edge info */
data class Edge<T>(val to: Int, var cost: T)


fun solve(out: PrintWriter) {
    val (n, x, y) = readIntegerList()

    /** incidenceList[fromNodeIndex]: edges to other node */
    val incidenceList = Array(n) { mutableListOf<Edge<Int>>() }
    for (i in 0 until n - 1) {
        incidenceList[i].add(Edge(i + 1, 1))
        incidenceList[i + 1].add(Edge(i, 1))
    }
    incidenceList[x - 1].add(Edge(y - 1, 1))
    incidenceList[y - 1].add(Edge(x - 1, 1))

    val ans = IntArray(n - 1)
    for (i in 0 until n) {
        val costArray = IntArray(n)
        val isChecked = BooleanArray(n)
        val queue = ArrayDeque<Int>()
        queue.addLast(i)
        isChecked[i] = true
        while (queue.isNotEmpty()) {
            val node = queue.removeFirst()

            for (edge in incidenceList[node]) {
                if (isChecked[edge.to] == false) {
                    costArray[edge.to] = costArray[node] + 1
                    isChecked[edge.to] = true
                    queue.addLast(edge.to)
                }
            }
        }

        for (j in (i + 1) until n) {
            ans[costArray[j] - 1]++
        }
    }

    println(ans.joinToString("\n"))
}

fun readInteger() = readLine()!!.toInt()
fun readLong() = readLine()!!.toLong()
fun readStringList() = readLine()!!.split(" ")
fun readIntegerList() = readStringList().map(String::toInt)
fun readLongList() = readStringList().map(String::toLong)
fun readDoubleList() = readStringList().map(String::toDouble)