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 ll long long
using namespace std;
ll a[3000+1];
ll dp[3000+1][3000+1];

ll ans(ll f,ll l,ll n)
{
   if(dp[f][l]!=-1)
   return dp[f][l];
   
   if(f==l)
   {
       if(n%2==0)
       dp[f][l]=-a[f];
       
       else
       dp[f][l]=a[f];
       
       return dp[f][l];
   }
   
   ll x=f-l+1;
   
   if((n%2==0)&&(x%2==0))
   dp[f][l]=max(ans(f+1,l,n)+a[f],ans(f,l-1,n)+a[l]);
   
   else if((n%2==0)&&(x%2!=0))
   dp[f][l]=min(ans(f+1,l,n)-a[f],ans(f,l-1,n)-a[l]);
   
   else if((n%2!=0)&&(x%2==0))
   dp[f][l]=min(ans(f+1,l,n)-a[f],ans(f,l-1,n)-a[l]);
   
   else if((n%2!=0)&&(x%2!=0))
   dp[f][l]=max(ans(f+1,l,n)+a[f],ans(f,l-1,n)+a[l]);
   
   return dp[f][l];
}
int main()
{
    ll n;
    cin>>n;
    
    for(ll i=0;i<=3000;i++)
    {
        for(ll j=0;j<=3000;j++)
        dp[i][j]=-1;
    }

    
    for(ll i=0;i<n;i++)
    cin>>a[i];
    
    cout<<ans(0,n-1,n)<<endl;
    
    
}