Problem:
Score : 600 points

Problem Statement

You are given N integers. The i-th integer is a_i.
Find the number, modulo 998244353, of ways to paint each of the integers red, green or blue so that the following condition is satisfied:

Let R, G and B be the sums of the integers painted red, green and blue, respectively. There exists a triangle with positive area whose sides have lengths R, G and B.

Constraints

3 \leq N \leq 300

1 \leq a_i \leq 300(1\leq i\leq N)

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
a_1
:
a_N

Output

Print the number, modulo 998244353, of ways to paint each of the integers red, green or blue so that the condition is satisfied.

Sample Input 1

4
1
1
1
2

Sample Output 1

18

We can only paint the integers so that the lengths of the sides of the triangle will be 1, 2 and 2, and there are 18 such ways.

Sample Input 2

6
1
3
2
3
5
2

Sample Output 2

150

Sample Input 3

20
3
1
4
1
5
9
2
6
5
3
5
8
9
7
9
3
2
3
8
4

Sample Output 3

563038556

Solution in Python (3.4.3) (judge verdict: Time Limit Exceeded)

import numpy as np 

MOD = 998244353

def check(a,b,c):
  return 0 < b+c-a and 0 < a - np.abs(b-c) 

N=int(input())
a = np.empty(N, dtype=np.int)
for i in range(N):
  a[i]=int(input())

# N = 4
# a = np.array([1,1,1,2], dtype=np.int)


# N = 300
# a = np.array([1]*N, dtype=np.int)


dp = [{} for i in range(N+1)]
dp[0][(0,0,0)] = 1

for i in range(0,N):
    for k in dp[i]:
        for sw in [(1,0,0),(0,1,0),(0,0,1)]:
            new_k = (k[0]+sw[0]*a[i], k[1]+sw[1]*a[i], k[2]+sw[2]*a[i])
            if not new_k in dp[i+1]:
                dp[i+1][new_k] = 0
            dp[i+1][new_k] += dp[i][k]
    for k in dp[i+1]:
        dp[i+1][k] %= MOD
    

# print(a)

# for i in range(N+1):
#     print(dp[i])

sumv = 0
for k in dp[N]:
    if check(*k):
        sumv += dp[N][k]
sumv %= MOD
print(sumv)