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-csc 3.5.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.IO;
using System.Text;
using System.Numerics;
using System.Threading;
using static System.Math;
using static System.Array;
using static AtCoder.IO_ShortCut;
using static AtCoder.Tool;
using static AtCoder.ModInt;
namespace AtCoder
{
    class AC
    {
        const int MOD = 1000000007;
        //const int MOD = 998244353;

        const int INF = int.MaxValue / 2;
        const long SINF = long.MaxValue / 2;
        const double EPS = 1e-8;
        static readonly int[] dI = { 0, 1, 0, -1, 1, -1, -1, 1 };
        static readonly int[] dJ = { 1, 0, -1, 0, 1, 1, -1, -1 };
        //static List<List<int>> G = new List<List<int>>();
        //static List<List<Edge>> G = new List<List<Edge>>();
        //static List<Edge> E = new List<Edge>();
        static void Main(string[] args)
        {
            var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false }; Console.SetOut(sw);

            var th = new Thread(Run, 1 << 26);
            th.Start();
            th.Join();

            //Run();
            Console.Out.Flush();
        }

        static void Run()
        {
            int Testcase = 1;
            for (var _ = 0; _ < Testcase; _++) Solve();
        }
        static void Solve()
        {
            var N = Int;
            var A = _ReadSplitLong;
            if (N == 0) { OutL(A[0] == 1 ? 1 : -1);return; }
            if (A[0] != 0) { OutL(-1);return; }

            long ans = 1;
            long v = 1;
            long res = A.Sum() - 1;
            for(var i = 1; i <= N; i++)
            {
                v -= A[i - 1];
                if (v <= 0) { OutL(-1);return; }
                long ad = Min(res, v);
                v += ad;
                if (v > 1e18) { Environment.Exit(-1); }
                res -= ad;

                ans += v;
            }
            OutL(res == 0 ? ans : -1);
        }
        public struct Edge
        {
            public int from;

            public int to;
            public long dist;

            public Edge(int t, long c)
            {
                from = -1;
                to = t;
                dist = c;
            }
            public Edge(int f, int t, long c)
            {
                from = f;
                to = t;
                dist = c;
            }
        }
    }

    struct ModInt
    {
        public long value;
        private const int MOD = 1000000007;
        //private const int MOD = 998244353;
        public ModInt(long value) { this.value = value; }
        public static implicit operator ModInt(long a)
        {
            var ret = a % MOD;
            return new ModInt(ret < 0 ? (ret + MOD) : ret);
        }
        public static ModInt operator +(ModInt a, ModInt b) => (a.value + b.value);
        public static ModInt operator -(ModInt a, ModInt b) => (a.value - b.value);
        public static ModInt operator *(ModInt a, ModInt b) => (a.value * b.value);
        public static ModInt operator /(ModInt a, ModInt b) => a * Modpow(b, MOD - 2);
        public static ModInt operator <<(ModInt a, int n) => (a.value << n);
        public static ModInt operator >>(ModInt a, int n) => (a.value >> n);
        public static ModInt operator ++(ModInt a) => a.value + 1;
        public static ModInt operator --(ModInt a) => a.value - 1;
        public static ModInt Modpow(ModInt a, long n)
        {
            var k = a;
            ModInt ret = 1;
            while (n > 0)
            {
                if ((n & 1) != 0) ret *= k;
                k *= k;
                n >>= 1;
            }
            return ret;
        }
        private static readonly List<long> Factorials = new List<long>() { 1 };
        public static ModInt Fac(long n)
        {
            for (var i = Factorials.Count(); i <= n; i++)
            {
                Factorials.Add((Factorials[i - 1] * i) % MOD);
            }
            return Factorials[(int)n];
        }
        public static ModInt nCr(long n, long r)
        {
            return n < r ? 0 : Fac(n) / (Fac(r) * Fac(n - r));
        }
        public static explicit operator int(ModInt a) => (int)a.value;
    }
    static class IO_ShortCut
    {
        public static string[] _ReadSplit => Console.ReadLine().Split();
        public static int[] _ReadSplitInt => ConvertAll(Console.ReadLine().Split(), int.Parse);
        public static long[] _ReadSplitLong => ConvertAll(Console.ReadLine().Split(), long.Parse);
        public static double[] _ReadSplit_Double => ConvertAll(Console.ReadLine().Split(), double.Parse);
        public static string Str => Console.ReadLine();
        public static int Int => int.Parse(Console.ReadLine());
        public static long Long => long.Parse(Console.ReadLine());
        public static double Double => double.Parse(Console.ReadLine());
        public static T Conv<T>(string input)
        {
            if (typeof(T).Equals(typeof(ModInt)))
            {
                return (T)(dynamic)(long.Parse(input));
            }
            return (T)Convert.ChangeType(input, typeof(T));
        }
        public static void Input<T>(out T a) => a = Conv<T>(Console.ReadLine());
        public static void Input<T, U>(out T a, out U b)
        { var q = _ReadSplit; a = Conv<T>(q[0]); b = Conv<U>(q[1]); }
        public static void Input<T, U, V>(out T a, out U b, out V c)
        { var q = _ReadSplit; a = Conv<T>(q[0]); b = Conv<U>(q[1]); c = Conv<V>(q[2]); }
        public static void Input<T, U, V, W>(out T a, out U b, out V c, out W d)
        { var q = _ReadSplit; a = Conv<T>(q[0]); b = Conv<U>(q[1]); c = Conv<V>(q[2]); d = Conv<W>(q[3]); }
        public static void Input<T, U, V, W, X>(out T a, out U b, out V c, out W d, out X e)
        { var q = _ReadSplit; a = Conv<T>(q[0]); b = Conv<U>(q[1]); c = Conv<V>(q[2]); d = Conv<W>(q[3]); e = Conv<X>(q[4]); }

        public static void OutL(object s) => Console.WriteLine(s);
        public static void Out_Sep<T>(IEnumerable<T> s) => Console.WriteLine(string.Join(" ", s));
        public static void Out_Sep<T>(IEnumerable<T> s, string sep) => Console.WriteLine(string.Join($"{sep}", s));
        public static void Out_Sep(params object[] s) => Console.WriteLine(string.Join(" ", s));
        public static void Out_One(object s) => Console.Write($"{s} ");
        public static void Out_One(object s, string sep) => Console.Write($"{s}{sep}");
        public static void Endl() => Console.WriteLine();
    }
    public static class Tool
    {
        static public void Initialize<T>(ref T[] array, T initialvalue)
        {
            array = ConvertAll(array, x => initialvalue);
        }
        static public void Swap<T>(ref T a, ref T b)
        {
            T keep = a;
            a = b;
            b = keep;
        }
        static public void Display<T>(T[,] array2d, int n, int m)
        {
            for (var i = 0; i < n; i++)
            {
                for (var j = 0; j < m; j++)
                {
                    Console.Write($"{array2d[i, j]} ");
                }
                Console.WriteLine();
            }
        }
        static public long Gcd(long a, long b)
        {
            if (a == 0 || b == 0) return Max(a, b);
            return a % b == 0 ? b : Gcd(b, a % b);
        }
        static public long LPow(int a, int b) => (long)Pow(a, b);
        static public bool Bit(long x, int dig) => ((1L << dig) & x) != 0;
        static public int Sig(long a) => a == 0 ? 0 : (int)(a / Abs(a));
    }
}