Problem:
Score : 300 points

Problem Statement

Let f(n) be the number of triples of integers (x,y,z) that satisfy both of the following conditions:

1 \leq x,y,z

x^2 + y^2 + z^2 + xy + yz + zx = n

Given an integer N, find each of f(1),f(2),f(3),\ldots,f(N).

Constraints

All values in input are integers.

1 \leq N \leq 10^4

Input

Input is given from Standard Input in the following format:

N

Output

Print N lines. The i-th line should contain the value f(i).

Sample Input 1

20

Sample Output 1

0
0
0
0
0
1
0
0
0
0
3
0
0
0
0
0
3
3
0
0

For n=6, only (1,1,1) satisfies both of the conditions. Thus, f(6) = 1.

For n=11, three triples, (1,1,2), (1,2,1), and (2,1,1), satisfy both of the conditions. Thus, f(6) = 3.

For n=17, three triples, (1,2,2), (2,1,2), and (2,2,1), satisfy both of the conditions. Thus, f(17) = 3.

For n=18, three triples, (1,1,3), (1,3,1), and (3,1,1), satisfy both of the conditions. Thus, f(18) = 3.

Solution in OCaml (4.10.0) (judge verdict: Accepted)

let create_int_array n = Array.init n (fun _ -> 0)

let calc x y z = x * x + y * y + z * z + x * y + y * z + z * x

let case x y z =
  if x = y then (if y = z then 1 else 3)
  else if y = z then 3
  else 6

let () =
  let n = read_int () in
  let a = create_int_array n in

  let rec f x =
    let k = 6 * x * x in
    if k <= n then
      let rec g y =
        let k = x * x + 3 * y * y + 2 * x * y in
        if n < k then f (x + 1)
        else
          let rec h z =
            let k = calc x y z in
            if n < k then g (y + 1)
            else
              let w = case x y z in
              a.(k - 1) <- a.(k - 1) + w;
              h (z + 1)
          in h y
      in g x
  in

  f 1;
  ignore (Array.map (fun i -> Printf.printf "%d\n" i) a);
  ()