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 Haskell (GHC 8.8.3) (judge verdict: Accepted)

import Control.Monad
import qualified Data.ByteString.Char8 as BS
import Data.Maybe
import qualified Data.Vector.Unboxed as VU
import qualified Data.Vector.Unboxed.Mutable as VUM

readInt = fst . fromJust . BS.readInt

getInt = readInt <$> BS.getLine

simulate table a b c = do
  let s = ((a + b) ^ 2 + (b + c) ^ 2 + (c + a) ^ 2) `div` 2
  if s <= 10000
    then do
      p <- VUM.read table s
      if a == b && b == c
        then VUM.write table s (p + 1)
        else
          if a == b || b == c
            then VUM.write table s (p + 3)
            else VUM.write table s (p + 6)
      simulate table a b (c + 1)
    else do
      let s' = ((a + b + 1) ^ 2 + (b + b + 1) ^ 2 + (b + 1 + a) ^ 2) `div` 2
      if s' <= 10000
        then simulate table a (b + 1) (b + 1)
        else do
          let s'' = (a * 2 + 1) ^ 2 `div` 2
          when (s'' <= 10000) $ simulate table (a + 1) (a + 1) (a + 1)

main = do
  n <- getInt
  let n2 = [1 .. 100]
  table <- VUM.replicate 100001 0 :: IO (VUM.IOVector Int)
  simulate table 1 1 1
  vu <- VU.freeze table
  VU.mapM_ print (VU.take n (VU.tail vu))