Problem:
Score : 100 points

Problem Statement

Taro and Jiro will play the following game against each other.

Initially, they are given a sequence a = (a_1, a_2, \ldots, a_N).
Until a becomes empty, the two players perform the following operation alternately, starting from Taro:

Remove the element at the beginning or the end of a. The player earns x points, where x is the removed element.

Let X and Y be Taro's and Jiro's total score at the end of the game, respectively.
Taro tries to maximize X - Y, while Jiro tries to minimize X - Y.

Assuming that the two players play optimally, find the resulting value of X - Y.

Constraints

All values in input are integers.

1 \leq N \leq 3000

1 \leq a_i \leq 10^9

Input

Input is given from Standard Input in the following format:

N
a_1 a_2 \ldots a_N

Output

Print the resulting value of X - Y, assuming that the two players play optimally.

Sample Input 1

4
10 80 90 30

Sample Output 1

10

The game proceeds as follows when the two players play optimally (the element being removed is written bold):

Taro: (10, 80, 90, 30) → (10, 80, 90)

Jiro: (10, 80, 90) → (10, 80)

Taro: (10, 80) → (10)

Jiro: (10) → ()

Here, X = 30 + 80 = 110 and Y = 90 + 10 = 100.

Sample Input 2

3
10 100 10

Sample Output 2

-80

The game proceeds, for example, as follows when the two players play optimally:

Taro: (10, 100, 10) → (100, 10)

Jiro: (100, 10) → (10)

Taro: (10) → ()

Here, X = 10 + 10 = 20 and Y = 100.

Sample Input 3

1
10

Sample Output 3

10

Sample Input 4

10
1000000000 1 1000000000 1 1000000000 1 1000000000 1 1000000000 1

Sample Output 4

4999999995

The answer may not fit into a 32-bit integer type.

Sample Input 5

6
4 2 9 7 1 5

Sample Output 5

2

The game proceeds, for example, as follows when the two players play optimally:

Taro: (4, 2, 9, 7, 1, 5) → (4, 2, 9, 7, 1)

Jiro: (4, 2, 9, 7, 1) → (2, 9, 7, 1)

Taro: (2, 9, 7, 1) → (2, 9, 7)

Jiro: (2, 9, 7) → (2, 9)

Taro: (2, 9) → (2)

Jiro: (2) → ()

Here, X = 5 + 1 + 9 = 15 and Y = 4 + 7 + 2 = 13.

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

#include <bits/stdc++.h>

#define ld long double
#define sf scanf
#define pf printf
#define pb push_back
#define mp make_pair
#define PI ( acos(-1.0) )
#define IN freopen("input.txt","r",stdin)
#define OUT freopen("output.txt","w",stdout)
#define FOR(i,a,b) for(int i=a ; i<=b ; i++)
#define FORD(i,a,b) for(int i=a ; i>=b ; i--)
#define INF 1000000000
#define ll long long int
#define eps (1e-8)
#define sq(x) ( (x)*(x) )
#define all(x) x.begin(),x.end()
#define flog2(n) 64 - __builtin_clzll(n) - 1
#define popcnt(n) __builtin_popcountll(n)

using namespace std;

typedef pair < int, int > pii;
typedef pair < ll, ll > pll;

ll d[3005][3005];
ll a[3005];

int main()
{
    ios::sync_with_stdio(0);
    cin.tie(NULL);
    int n;
    cin >> n;
    FOR (i, 1, n) {
    	cin >> a[i];
    }

    FOR (len, 0, n - 1) {
     	FOR (i, 1, n - len) {
     		if (len == 0) {
    			d[i][i] = a[i];
	    		continue;
	    	}

    		int j = i + len;

    		d[i][j] = max(a[j] - d[i][j - 1], a[i] - d[i + 1][j]);
    	}
    }

    cout << d[1][n] << endl;

		

	return 0;
}