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: Wrong Answer)


from math import ceil
N = int(input())
A = list(map(int, input().split()))
if(A[0] != 0):
    print(-1)
    exit(0)
# curは作り出せる最大ノード数
cur = 0
S = [0] * (N + 1)
for i in range(N, 0, -1):
    cur += A[i]
    S[i - 1] = cur
# print(S)

ans, cur = 0, 2
node = [0] * (N + 1)
a = [0] * (N + 1)
node[0] = 2
for i in range(1, N):
    if(A[i] > cur):
        print(-1)
        exit(0)
    cur = 2 * (cur - A[i])
    node[i] = cur

    if(node[i] >= S[i]): node[i] = S[i]
    a[i] = node[i - 1] + cur
    #print("i={}, cur={}".format(i, cur))
# print(node)
# print(a)
# 最初の頂点
print(sum(node) + 1)
'''
cur = 0
for i in range(N, 0, -1):
    cur += A[i]
    if(cur >= 2**(i - 1)): cur = 2**(i - 1)
    #if(cur >= 2 * A[i - 1]): cur = 2 * A[i - 1]
    ans += cur
    print(cur, ans)
print(ans)

'''