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# (Mono-mcs 6.8.0.105) (judge verdict: Wrong Answer)

using System;
using System.Collections.Generic;
using System.Linq;
class c{
  public static void Main(){
    long n = long.Parse(Console.ReadLine());
    long[] a = Console.ReadLine().Split(' ').Select(s => long.Parse(s)).ToArray();
    long[] sum = new long[n+1];
    long ans=0;
    for(long i=0; i<=n; i++){
      for(long j=i; j<=n; j++){
        sum[j-i] += a[j];
      }
    }
    long[] leaf = new long[n+1];
    leaf[0] = 1;
    for(long i=0; i<n; i++){
      leaf[i+1] = Math.Min(leaf[i]*2, sum[i+1]);
      if(leaf[i+1] < a[i+1]){
        ans = -1;
        break;
      }
      leaf[i+1] -= a[i+1];
    }
    if(ans != -1){
      for(long i=0; i<=n; i++){
        ans += a[i] + leaf[i];
      }
    }
    Console.WriteLine(ans);
  }
}