<p>Given an integer <code>n</code>, find a sequence that satisfies all of the following:</p>

<ul>
	<li>The integer <code>1</code> occurs once in the sequence.</li>
	<li>Each integer between <code>2</code> and <code>n</code> occurs twice in the sequence.</li>
	<li>For every integer <code>i</code> between <code>2</code> and <code>n</code>, the <strong>distance</strong> between the two occurrences of <code>i</code> is exactly <code>i</code>.</li>
</ul>

<p>The <strong>distance</strong> between two numbers on the sequence, <code>a[i]</code> and <code>a[j]</code>, is the absolute difference of their indices, <code>|j - i|</code>.</p>

<p>Return <em>the <strong>lexicographically largest</strong> sequence</em><em>. It is guaranteed that under the given constraints, there is always a solution. </em></p>

<p>A sequence <code>a</code> is lexicographically larger than a sequence <code>b</code> (of the same length) if in the first position where <code>a</code> and <code>b</code> differ, sequence <code>a</code> has a number greater than the corresponding number in <code>b</code>. For example, <code>[0,1,9,0]</code> is lexicographically larger than <code>[0,1,5,6]</code> because the first position they differ is at the third number, and <code>9</code> is greater than <code>5</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> n = 3
<strong>Output:</strong> [3,1,2,3,2]
<strong>Explanation:</strong> [2,3,2,1,3] is also a valid sequence, but [3,1,2,3,2] is the lexicographically largest valid sequence.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> n = 5
<strong>Output:</strong> [5,3,1,4,3,5,2,4,2]
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= n &lt;= 20</code></li>
</ul>

class Solution {
public:
    vector<int> constructDistancedSequence(int n) {
        vector<int> curr (2 * n - 1, -1);
        vector<bool> is_used (n + 1, false);
        return helper(curr, is_used, n, 0);
    }
    vector<int> helper(vector<int> &curr, vector<bool> &is_used, int n, int i)
    {
        // cout << "i = " << i << ", curr:\n";
        // for (int k = 0; k < curr.size(); k++)
        //     cout << curr[k] << '\t';
        // cout << endl << "is_used:\n";
        // for (int k = 0; k < is_used.size(); k++)
        //     cout << is_used[k] << '\t';
        // cout << endl;
        if (i == (2 * n - 1))
        {
            for (int val = 1; val <= n; val++)
            {
                if (is_used[val] == false)
                    return {};
            }
            return curr;
        }

        while ((i < 2 * n - 1) && (curr[i] != -1))
        {
            i++;
        }
        // printf("i = %d\n", i);
        for (int val = n; val > 1; val--)
        {
            if (is_used[val])
                continue;
            if ((i + val < 2 * n - 1) && curr[i + val] == -1)
            {
                // printf("i = %d, use = %d\n", i, val);
                is_used[val] = true;
                curr[i] = val;
                curr[i + val] = val;
                
                vector<int> res = helper(curr, is_used, n, i + 1);
                if (res.size() > 0)
                    return res;

                is_used[val] = false;
                curr[i] = -1;
                curr[i + val] = -1;
            }
        }
        if (!is_used[1])
        {
            curr[i] = 1;
            is_used[1] = true;
            vector<int> res = helper(curr, is_used, n, i + 1);
            if (res.size() > 0)
                return res;
            curr[i] = -1;
            is_used[1] = false;
        }
        for (int val = 1; val <= n; val++)
        {
            if (is_used[val] == false)
                return {};
        }
        return curr;

    }
};
9000