Problem:
Score : 600 points

Problem Statement

Given is an integer sequence of length N+1: A_0, A_1, A_2, \ldots, A_N. Is there a binary tree of depth N such that, for each d = 0, 1, \ldots, N, there are exactly A_d leaves at depth d? If such a tree exists, print the maximum possible number of vertices in such a tree; otherwise, print -1.

Notes

A binary tree is a rooted tree such that each vertex has at most two direct children.

A leaf in a binary tree is a vertex with zero children.

The depth of a vertex v in a binary tree is the distance from the tree's root to v. (The root has the depth of 0.)

The depth of a binary tree is the maximum depth of a vertex in the tree.

Constraints

0 \leq N \leq 10^5

0 \leq A_i \leq 10^{8} (0 \leq i \leq N)

A_N \geq 1

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_0 A_1 A_2 \cdots A_N

Output

Print the answer as an integer.

Sample Input 1

3
0 1 1 2

Sample Output 1

7

Below is the tree with the maximum possible number of vertices. It has seven vertices, so we should print 7.

Sample Input 2

4
0 0 1 0 2

Sample Output 2

10

Sample Input 3

2
0 3 1

Sample Output 3

-1

Sample Input 4

1
1 1

Sample Output 4

-1

Sample Input 5

10
0 0 1 1 2 3 5 8 13 21 34

Sample Output 5

264

Solution in C# (.NET Core 3.1.201) (judge verdict: Time Limit Exceeded)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace ConsoleApp60
{
    class Program
    {
      static  int n = 0;
        static int[] l;
        static Dictionary<string, res> map = new Dictionary<string, res>();
        static Dictionary<string, int> map2 = new Dictionary<string, int>();
        static void Main(string[] args)
        {
            n = int.Parse( Console.ReadLine());
             l = Console.ReadLine().Split(' ').Select(int.Parse).ToArray();


       
            var idx = 0;
            /*  while( n > idx)
              {
                   ver = calc(ver, l[idx]);
                  if ( ver < 0)
                  {
                      //
                      Console.WriteLine("-1");
                      return;
                  }
                  if ( ver == 0 && idx + 1 != n)
                  {
                      Console.WriteLine("-1");
                      return;
                  }
                  cnt = cnt + ver;

                  idx++;
              }
              Console.WriteLine(cnt);
              */

            var r = calc2( 1, l, idx);
            Console.WriteLine(r);

        }
       struct res
        {
            public int min;
            public int max;
        }
        static res fl = new res { min = -1, max = -1 };

        static res  calc (int a, int b)
        {
            res r;
          
            //頂点より葉っぱが多いとダメ
            var c = a - b;
            if(c < 0)
            {
                r = fl;
              
                return r;
            }

           r =  new res { min = c, max = c* 2 };
            
            return r;
        }
        static  int calc2 ( int now  , int[] list, int idx)
        {
          //  var key = now.ToString() + " " + idx.ToString();
            int r;
           
            var d =   calc(now, list[idx]);
            var key = d.max.ToString() + " " + d.min.ToString() + " " + idx.ToString();
          
            if ( d.min == -1)
            {
                map2[key] = -1;
                return -1;
            }

            if (n > idx + 1)
            {
                var c = d.max - list[idx + 1];
                if (c < 0)
                {
                    map2[key] = -1;
                    return -1;
                }
            }
            if (map2.TryGetValue(key, out r))
            {
                return r;
            }
            if (n == idx)
            {
                if (now == list[idx])
                {
                    map2[key] = now;
                    return now;
                }
                else
                {
                    map2[key] = -1;
                    return -1;
                }
            }

            if (n > idx + 1)
            {
                var m = -1;
                for (var i = d.max; i >= d.min; i--)
                {
                  var  c =  i -  list[idx +1];
                    if( c < 0)
                    {
                        continue;
                    }
                    var a = calc2( i, list, idx + 1);
                    if (m < a)
                    {
                        m = a;
                        break;
                    }

                }
                if (m == -1)
                {
                    map2[key] = -1;
                    return -1;
                }
                map2[key] = now + m;
                return now + m;

            }
            if (n > idx )
            {
                var m = -1;
                for (var i = d.max; i >= d.min; i--)
                {
                    var a = calc2(i, list, idx + 1);
                    if (m < a)
                    {
                        m = a;
                        break;
                    }

                }
                if (m == -1)
                {
                    map2[key] = -1;
                    return -1;
                }
                map2[key] = now + m;
                return now + m;

            }
            map2[key] = -1;
            return -1;
          


        }
    }
}