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 (Python):
n = int(input())

f = 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66,0,12,36,78,36,42,0,36,0,78,24,48,24,18,30,90,15,96,18,27,21,54,0,24,21,48,27,84,18,48,42,36,6,66,12,108,12,42,0,48,9,27,60,21,36,84,6,36,18,42,18,54,18,60,0,60,15,72,9,99,42,12,12,60,0,57,42,18,18,78,0,42,6,36,36,63,9,42,15,30,18,78,6,24,48,42,0,102,12,75,45,42,30,84,24,66,9,24,30,90,24,42,0,24,3,84,21,84,33,66,24,36,6,36,48,51,18,72,0,66,18,54,30,24,18,48,18,30,0,102,12,45,36,24,0,102,3,42,15,42,54,54,21,66,24,66,30,102,3,66,0,48,12,24,30,45,36,54,6,78,9,48,24,39,36,96,0,66,6,27,27,60,30,18,30,48,12,90,24,60,36,6,0,42,12,78,36,36,24,114,6,60,0,18,6,69,18,84,0,36,6,48,6,39,24,42,24,54,6,72,48,36,42,60,0,48,0,36,18,138,24,30,30,42,18,75,18,48,18,24,0,78,0,78,57,60,9,48,9,36,30,30,42,42,39,84,0,24,12,87,27,30,42,24,24,90,0,36,24,60,36,84,0,42,0,30,21,87,12,72,24,42,12,60,30,72,21,60,0,96,12,30,42,39,18,66,12,84,12,63,42,42,24,42,0,18,6,78,15,75,45,54,12,54,6,84,30,30,18,54,0,42,24,39,27,96,18,42,21,48,0,54,18,33,36,66,0,84,0,75,48,30,42,54,12,54,15,42,30,96,18,42,0,18,6,123,15,48,24,48,39,42,6,21,54,45,12,60,0,90,12,30,36,45,18,72,21,78,12,30,12,60,30,18,0,78,6,60,36,33,18,60,12,30,0,60,18,90,30,54,0,36,1,78,27,84,30,18,15,102,9,27,30,24,57,90,0,42,9,48,42,42,12,72,27,48,9,108,18,48,48,18,0,84,6,66,24,48,21,114,6,78,24,30,30,42,18,60,0,18,12,63,9,51,66,48,15,54,0,66,60,36,18,48,0,66,6,36,30,96,9,36,30,30,24,93,24,72,24,66,0,60,24,48,30,24,30,72,6,60,12,30,12,93,18,72,0,30,0,81,30,36,48,36,30,90,6,54,24,57,24,36,0,54,18,36,24,78,18,72,33,12,0,51,30,48,24,42,0,90,3,78,30,30,27,138,6,30,18,36,24,75,24,12,0,54,12,42,18,84,36,30,18,66,6,45,36,36,48,114,0,24,6,30,21,108,15,48,18,36,18,57,24,63,42,54,0,78,6,81,36,18,24,42,18,120,0,24,24,66,12,54,0,48,12,66,18,42,36,48,18,72,6,48,48,33,6,102,0,54,24,39,54,60,12,90,18,36,3,111,24,51,48,42,0,90,0,45,36,54,48,54,24,18,18,51,18,72,0,54,0,18,12,54,36,57,18,42,18,120,0,63,42,27,36,48,0,54,6,39,30,81,18,72,33,54,9,72,12,54,54,24,0,54,15,75,24,60,12,78,6,30,12,39,54,72,6,54,0,72,6,78,12,54,39,30,15,48,6,57,48,18,24,90,0,90,18,42,24,39,42,24,30,42,0,84,6,66,30,30,0,30,0,27,42,39,33,90,24,84,18,51,36,69,24,54,0,42,18,87,33,51,18,42,9,126,9,84,36,66,18,54,0,36,0,66,12,48,15,54,30,18,18,60,18,96,54,42,0,102,0,27,36,30,36,108,15,54,0,24,36,60,24,66,0,30,0,54,15,75,48,36,18,48,24,36,36,54,15,108,0,84,18]

for i in range(n):
    print(f[i])

Solution (Python):
import sys, math, itertools, collections, bisect 
input = lambda: sys.stdin.buffer.readline().rstrip().decode('utf-8') 
inf = float('inf') ;mod = 10**9+7 
mans = inf ;ans = 0 ;count = 0 ;pro = 1

n = int(input())
for i in range(1, n+1):
    cnt = 0
    for x in range(1, int(math.sqrt(i+1))):
        for y in range(1, int(math.sqrt(i+1))):
            d = (x+y)**2 -4*(x*x + y*y + x*y - i)
            if d < 0:
                pass
            else:
                z = (-(x+y)+math.sqrt(d))/2
                if z == int(z) and z >= 1:
                    cnt += 1
    print(cnt)

Solution (Python):
"""
(x+y+z)^2 = (x^2+y^2+z^2) + 2(xy+yz+zx)
"""
def func(x, y, z): return (x + y + z) ** 2 - x * y - y * z - z * x


def main():
    num = int(input())
    counter = [0] * 20000
    
    
    for x in range(1, 100):
        for y in range(1, 100):
            for z in range(1, 123 - x - y + 1):
                tmp = func(x, y, z)
                counter[tmp] += 1
    for key, count in enumerate(counter):
        if key == 0:
            continue
        if key > num:
            break
        print(count)


if __name__ == '__main__':
    main()

Solution (Python):


N = int(input())

a = [0 for i in range(100**2 * 6 + 2)]

for x in range(1, 100):
    for y in range(1, 100):
        for z in range(1, 100):
            a[ x**2 + y**2 + z**2 + x*y + y*z + z*x ] += 1

for i in range(1,N+1):
    print(a[i])

Solution (Python):


from math import sqrt
def count(k, t):
    if k*k - 4*t < 0: return 0
    if t == 0: return 0

    r = (-k + sqrt(k*k - 4*t)) / 2
    r = int(r)

    if r*r + k*r + t == 0 and 1 <= r: return 1
    return 0

def f(N):
    ret = 0
    for x in range(1, N+1):
        for y in range(1, N+1):
            k = x+y
            t = x*x + y*y + x*y - N
            
            ret += count(k, t)
    return ret


ans = dict()

for i in range(1, 10**4+1):
    ans[i] = 0

for x in range(1, 10**2+1):
    for y in range(1, 10**2+1):
        if (x**2 + y**2 > 10**4): continue
        for z in range(1, 10**2+1):
            n = x*x + y*y + z*z + x*y + x*z + y*z
            if n > 10**4: continue
            ans[n] += 1

N = int(input())
for i in range(1, N+1):
    print(ans[i])

Solution (Python):
def fx(x, y, z):
    return x**2 + y**2 + z**2 + x*y + y*z + z*x


n = int(input())

ind = {_: 0 for _ in range(1, 10**4 + 1)}
cnt = 0
for i in range(1, 10**4 + 1):
    if fx(i, 1, 1) > 10 ** 4:
        break
    for j in range(1, 10 ** 4 + 1):
        if fx(1, j, 1) > 10 ** 4:
            break
        for k in range(1, 10 ** 4 + 1):
            if fx(i, j, k) <= 10 ** 4:
                ind[fx(i, j, k)] += 1
            else:
                break

for num in range(1, n + 1):
    print(ind[num])

Solution (Python):
n = int(input())

a = [0] * 10010


for x in range(1, 101):
  for y in range(1, 101):
    for z in range(1, 101):
      if x*x + y*y + z*z + x*y + y*z + z*x > n:
        continue
      a[x*x + y*y + z*z + x*y + y*z + z*x] += 1


for i in range(1, n+1):
  print(a[i])

Solution (Python):
n = int(input())

func = [0]*n

for x in range(1,100):
    for y in range(x, 100):
        for z in range(y,100):
            temp = x**2 + y**2 + z**2 + x*y + y*z + z*x
            if temp <= n:
                if x == y and y == z :
                    func[temp-1] += 1
                elif x == y or y == z:
                    func[temp-1] += 3
                else:
                    func[temp-1] += 6

for i in range(n):
    print(func[i])