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 Python (3.4.3) (judge verdict: Time Limit Exceeded)

#import numpy as np

N = int(input())
A = list(map(int, input().split()))

#dp[S] = A[S], -inf で初期化　Sの部分集合TのうちA[T]が(最大のもの, 次に大きいもの)　を入れていく

dp = [[A[S], -float('inf') ] for S in range(1 << N)]
for i in range(N):
    for T in range( 1 << N):
        if (T >> i) & 1 == 1: # Tがiを含むとき
            merge = dp[T] + dp[T ^ (1 << i)]
            merge.sort(reverse = True)
            dp[T] =merge[:2]
ans = -float('inf')
for S in range(1, 1 << N , 1):
    ans = max(ans, sum(dp[S]))
    print(ans)