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 Pascal (FPC 2.6.2) (judge verdict: Wrong Answer)

uses math;
const fi='DEQUE.INP';
      fo='DEQUE.OUT';
      maxn=3000;

var n:longint;
    a:array[1..maxn] of longint;
    s:array[0..maxn] of int64;
    f:array[1..maxn,1..maxn] of int64;
//============================
procedure doc;
var i:longint;
begin
  readln(n);

  for i:=1 to n do read(a[i]);

end;
//============================
procedure tinhs;
var i:longint;
begin
  s[0]:=0;
  s[1]:=a[1];
  for i:=2 to n do s[i]:=s[i-1]+a[i];
end;
//============================
function sum(i,j:longint):int64;
begin
  sum:=s[j]-s[i-1];
end;
//============================
procedure xuli;
var l,r,i:longint;
begin
  tinhs;

  for i:=1 to n do f[i,i]:=a[i];

  for l:=1 to n-1 do
    for r:=l+1 to n do
      f[l,r]:=max(sum(l,r)-f[l+1,r],sum(l,r)-f[l,r-1]);

  write(f[1,n]-(sum(1,n)-f[1,n]));
end;
//============================
BEGIN
  //assign(input,fi); reset(input);
  //assign(output,fo); rewrite(output);
  doc;
  xuli;
  //close(input); close(output);
END.