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)

#!/usr/bin/env python3
import sys
from pprint import pprint

MOD = 998244353  # type: int

def solve(N: int, a: "List[int]"):
    s = sum(a)
    dp = [[[0] * 2 for _ in range(s + 1)] for _ in range(N + 1)] # dp[idx][sum][]
    cum = [0] * N
    cum[0] = a[0]
    for i in range(N - 1):
        cum[i + 1] = cum[i] + a[i + 1]
    dp[0][0][0] = 1
    dp[0][0][1] = 1
    for i in range(N):
        for j in range(cum[i] + 1 - a[i]):
            dp[i + 1][j + a[i]][0] += dp[i][j][0]
            dp[i + 1][j + a[i]][1] += dp[i][j][1]
            dp[i + 1][j][0] += dp[i][j][0]
            dp[i + 1][j][1] += dp[i][j][1] * 2

    tmp = 0
    border = (s + 1) // 2
    uniq = 0
    #pprint(dp)
    for j in range(border, s + 1):
        tmp += dp[N][j][1]
        uniq += dp[N][j][0]
    #print(tmp, uniq)
    tmp -= uniq * 2
    tmp *= 3

    ret = (3 ** N) % MOD
    col_2 = (2 ** N) % MOD
    ret -= 3 + (col_2 - 2) * 3 + 3
    #print(ret)
    #for i in range(N):
    #    ret *= 3
    #    ret %= MOD
    #    col_2 *= 2
    #    col_2 %= MOD
    ret -= tmp
    print(ret % MOD)
    return


def main():
    def iterate_tokens():
        for line in sys.stdin:
            for word in line.split():
                yield word
    tokens = iterate_tokens()
    N = int(next(tokens))  # type: int
    a = [ int(next(tokens)) for _ in range(N) ]  # type: "List[int]"
    solve(N, a)

if __name__ == '__main__':
    main()