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 OCaml (4.02.3) (judge verdict: Time Limit Exceeded)

let id x = x

let init n f =
  let rec g i ns = if i = 0 then 0 :: ns else g (i - 1) (i :: ns) in
  List.map f (g (n - 1) [])

let (n, x, y) = Scanf.scanf "%d %d %d" @@ fun n x y -> (n, x, y)

let ix = init n (fun i -> i + 1)

let pair = 
  let xs = fst @@ List.fold_left (fun (ns, ys) i -> ((List.map (fun j -> (i, j)) ys)::ns, List.tl ys)) ([], ix) ix in
  List.fold_left (fun xs ys -> List.rev_append xs ys) [] xs

let f (i, j) = 
  let rec min' m = function
    | [] -> m
    | x :: xs -> min' (min m x) xs 
  in
  min' max_int [(abs (i - j)); (abs (i - y)) + 1 + (abs (x - j)); (abs (i - x)) + 1 + (abs (y - j))]

let g xs i =
  match xs with
    | [] -> [(i, 1)]
    | (j, c) :: xs -> if i = j then (j, c + 1) :: xs else (i, 1) :: (j, c) :: xs

let () = 
  let cs = pair 
    |> List.map f
    |> List.fast_sort (fun i j -> i - j)
    |> List.fold_left g [] in
  List.iter (fun k ->
    try Printf.printf "%d\n" (List.assoc k cs)
    with Not_found -> print_endline "0"
  ) (init (n - 1) (fun i -> i + 1))