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 Python (3.8.2) (judge verdict: Accepted)

import math

n = int(input())
folia = list(map(int, input().split()))
#print(n, folia)

seq = [[folia[-1], folia[-1]]]

for i in range(n): # 深さ'n'分
    tmp = folia[-(i+2)]
    sml = math.ceil(seq[i][0] / 2) + tmp 
    big = seq[i][1] + tmp
    seq += [[sml, big]]
#print(seq)

if not (seq[-1][0] <= 1 <= seq[-1][1]):
    print(-1)
    exit()

ans =[1]
for j in range(n): # 深さ'n'分
    tmp = min((ans[j] - folia[j]) * 2, seq[-(j+2)][1])
    ans.append(tmp)
print(sum(ans))