Problem:
Score : 700 points

Problem Statement

There is an integer sequence of length 2^N: A_0, A_1, ..., A_{2^N-1}. (Note that the sequence is 0-indexed.)

For every integer K satisfying 1 \leq K \leq 2^N-1, solve the following problem:

Let i and j be integers. Find the maximum value of A_i + A_j where 0 \leq i < j \leq 2^N-1 and (i or j) \leq K.
Here, or denotes the bitwise OR.

Constraints

1 \leq N \leq 18

1 \leq A_i \leq 10^9

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_0 A_1 ... A_{2^N-1}

Output

Print 2^N-1 lines.
In the i-th line, print the answer of the problem above for K=i.

Sample Input 1

2
1 2 3 1

Sample Output 1

3
4
5

For K=1, the only possible pair of i and j is (i,j)=(0,1), so the answer is A_0+A_1=1+2=3.

For K=2, the possible pairs of i and j are (i,j)=(0,1),(0,2).
When (i,j)=(0,2), A_i+A_j=1+3=4. This is the maximum value, so the answer is 4.

For K=3, the possible pairs of i and j are (i,j)=(0,1),(0,2),(0,3),(1,2),(1,3),(2,3) .
When (i,j)=(1,2), A_i+A_j=2+3=5. This is the maximum value, so the answer is 5.

Sample Input 2

3
10 71 84 33 6 47 23 25

Sample Output 2

81
94
155
155
155
155
155

Sample Input 3

4
75 26 45 72 81 47 97 97 2 2 25 82 84 17 56 32

Sample Output 3

101
120
147
156
156
178
194
194
194
194
194
194
194
194
194

Solution in C# (Mono 4.6.2.0) (judge verdict: Time Limit Exceeded)

using System;
using System.Collections.Generic;
using System.Linq;
//using System.Numerics;
class Program
{       
    public void Slove()
    {
        int n = int.Parse(Console.ReadLine());
        var array = Console.ReadLine().Split().Select(int.Parse).ToArray();
        int[] max = new int[1 << n];
        foreach(int i in Enumerable.Range(0, array.Length))
            foreach(int j in Enumerable.Range(0, array.Length))
            {
                if(i == j)
                    continue;
                int index = i | j;
                //if(1 << n >= index)
                {
                    max[index] = Math.Max(max[index], array[i] + array[j]);
                } 
            }
        foreach(int i in Enumerable.Range(0, max.Length).Skip(1))
        {
            max[i] = Math.Max(max[i - 1] , max[i]);
        }
        System.Console.WriteLine(string.Join("\n", max.Skip(1)));
    }
    
    static void Main(string[] args)
    {
        new Program().Slove();
    }
}