Codeforces Round 474 C. Subsequence Counting

Subsequence Counting

time limit per test: 1 second memory limit per test: 256 megabytes input: standard input output: standard output

Pikachu had an array with him. He wrote down all the non-empty subsequences of the array on paper. Note that an array of size n has $2^n-1$ non-empty subsequences in it.

Pikachu being mischievous as he always is, removed all the subsequences in which $\text{M}aximum\_element\_of\_the\_subsequence-Minimum\_element\_of\_subsequence\ge d$

Pikachu was finally left with $\text{X}$ subsequences.

However, he lost the initial array he had, and now is in serious trouble. He still remembers the numbers $\text{X}$ and $d$. He now wants you to construct any such array which will satisfy the above conditions. All the numbers in the final array should be positive integers less than $10^{18}$.

Note the number of elements in the output array should not be more than $10^4$. If no answer is possible, print $-1$.

Input

The only line of input consists of two space separated integers $\text{X}$ and $d$ ( $1\le$ $\text{X}$,$d$ $\le$$10^9$).

Output

Output should consist of two lines.

First line should contain a single integer n ($1\le$ $n$ $\leq$10000)— the number of integers in the final array.

Second line should consist of $n$ space separated integers — $a_1,a_2,a_3,\dots ,a_n$ ($1\le a_i\le 10^{18}$).

If there is no answer, print a single integer $-1$. If there are multiple answers, print any of them.

Example

input

10 5

output

6
5 50 7 15 6 100

Note

In the output of the first example case, the remaining subsequences after removing those with $\text{M}aximum\_element\_of\_the\_subsequence-Minimum\_element\_of\_subsequence\ge 5$ are $[5],[5,7],[5,6],[5,7,6],[50],[7],[7,6],[15],[6],[100]$. There are $10$ of them. Hence, the array $[5,50,7,15,6,100]$ is valid.

Similarly, in the output of the second example case, the remaining sub-sequences after removing those with $\text{M}aximum\_element\_of\_the\_subsequence-Minimum\_element\_of\_subsequence\ge 2$ are $[10],[100],[1000],[10000]$. There are 4 of them. Hence, the array $[10,100,1000,10000]$ is valid.

Tutorial

如果我们有 $n$ 个元素，想要满足条件的子序列最多，那么让这 $n$ 个元素都相等即可，此时可以得到 $2^n-1$ 个子序列，所以我们只需要让题目中的 $x$ 拆分成若干个 $2^{n_i}-1$​ 的次方的和即可

我们可以对 $x$ 的二进制表示着手，从最高位遍历所有的 $2^i-1$，如果此时 $x\le 2^i-1$，那么就从中取一个 $2^i-1$，对于每一个 $2^i-1$，当前元素最多出现两次（因为 $3\times (2^i-1)\ge 2^{i+1}-1$），且下一个元素与当前元素相差 $d$

此解法时间复杂度为 $\mathcal{O}({\log }^{2}x)$

Solution

#include <bits/stdc++.h>
using namespace std;

int main(){
    long long x, d, num = 1;
    cin >> x >> d;
    vector<long long> ans;
    for (int i = 60; i; --i) {
        long long cnt = (1ll << i) - 1;
        while (x >= cnt) {
            for (int j = 0; j < i; ++j) {
                ans.emplace_back(num);
            }
            num += d;
            x -= cnt;
        }
    }
    cout << ans.size() << endl;
    for (long long ai : ans) {
        cout << ai << " ";
    }
    return 0;
}