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 C++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <bits/stdc++.h> 

using namespace std;
using ll = long long;

int main() {
	const ll MOD = 998244353;
	int N;
	cin >> N;
	vector<ll> a(N);
	int sum = 0;
	for (int i = 0; i < N; ++i) {
		cin >> a[i];
		sum += a[i];
	}

	vector<vector<ll>> dp(N + 1, vector<ll>(sum + 1, 0));
	dp[0][0] = 1;
	auto dp2 = dp;
	for (int i = 0; i < N; ++i) {
		for (int j = 0; j + a[i] <= sum; ++j) {
			(dp[i + 1][j + a[i]] += dp[i][j]) %= MOD;
			(dp[i + 1][j] += dp[i][j] * 2) %= MOD;

			(dp2[i + 1][j + a[i]] += dp2[i][j]) %= MOD;
			(dp2[i + 1][j] += dp2[i][j]) %= MOD;
		}
	}
				
	ll res = 1; 
	for (int i = 1; i <= N; ++i) { (res *= 3) %= MOD; }

	ll rm = 0;
	for (int i = sum / 2 + 1; i <= sum; ++i) {
		(rm += (dp[N][i] * 3)) %= MOD;
	}
	if (sum % 2 == 0) {
		(rm += (dp[N][sum / 2] - dp2[N][sum / 2] * 2) * 3) %= MOD;
		(rm += (dp2[N][sum / 2] * 3)) %= MOD;
	}

	res = (res - rm + MOD) % MOD;
	cout << res << endl;

	return 0;
}