<p>You have two types of tiles: a <code>2 x 1</code> domino shape and a tromino shape. You may rotate these shapes.</p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/07/15/lc-domino.jpg" style="width: 362px; height: 195px;" />
<p>Given an integer n, return <em>the number of ways to tile an</em> <code>2 x n</code> <em>board</em>. Since the answer may be very large, return it <strong>modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>

<p>In a tiling, every square must be covered by a tile. Two tilings are different if and only if there are two 4-directionally adjacent cells on the board such that exactly one of the tilings has both squares occupied by a tile.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/07/15/lc-domino1.jpg" style="width: 500px; height: 226px;" />
<pre>
<strong>Input:</strong> n = 3
<strong>Output:</strong> 5
<strong>Explanation:</strong> The five different ways are show above.
</pre>

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

<pre>
<strong>Input:</strong> n = 1
<strong>Output:</strong> 1
</pre>

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

<ul>
	<li><code>1 &lt;= n &lt;= 1000</code></li>
</ul>

class Solution {
private:
    map<int, int> myWidthToPatterns;
public:
    Solution()
    {
        myWidthToPatterns.insert(make_pair(1, 1));
        myWidthToPatterns.insert(make_pair(2, 2));
        myWidthToPatterns.insert(make_pair(3, 5));
    }

    unsigned int GetPatterns(const int aWidth)
    {
        if (myWidthToPatterns.contains(aWidth))
            return myWidthToPatterns[aWidth];

        const int nPatterns = (2*GetPatterns(aWidth-1) + GetPatterns(aWidth-3))%(1000000007);
        myWidthToPatterns.insert(make_pair(aWidth, nPatterns));
        return nPatterns;
    }

    int numTilings(int n) {
        return GetPatterns(n);        
    }
};
10800