Folia

AtCoder

IDnomura2020_c

Time2000ms

Memory256MB

Difficulty—

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 $-1$. ## 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: $N$ $A_0$ $A_1$ $A_2$ $\cdots$ $A_N$ [samples] ## 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.

Samples

Input #1

3
0 1 1 2

Output #1

7

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

![image](https://img.atcoder.jp/nomura2020/0d8d99d13df036f23b0c9fcec52b842b.png)

Input #2

4
0 0 1 0 2

Output #2

Input #3

2
0 3 1

Output #3

\-1

Input #4

1
1 1

Output #4

\-1

Input #5

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

Output #5

API Response (JSON)

{
  "problem": {
    "name": "Folia",
    "description": {
      "content": "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 $d$? ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "nomura2020_c"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "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 $d$? ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments