<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 {

    int MOD = 1000000007;

    long long numTilingLDP(
        unordered_map<int, long long> &cache, 
        unordered_map<int, long long> &cacheL, 
        int n
    ){
        if(n <= 0)return 0;
        if(n == 1)return 1;
        if(cacheL.find(n) == cacheL.end()){
            cacheL[n] = (
                (numTilingLDP(cache, cacheL, n-1)%MOD) + // Place one Horizontal 
                (numTilingDP(cache, cacheL, n-1)%MOD)   // PLace one L shaped 
            )%MOD;
        }
        return cacheL[n];
    }



    long long numTilingDP(
        unordered_map<int, long long> &cache, 
        unordered_map<int, long long> &cacheL, 
        int n
    ){
        if(n < 0)return 0;
        if(n == 0)return 1;
        if(cache.find(n) == cache.end()){
            cache[n] = (
                (numTilingLDP(cache, cacheL, n-2)%MOD) + // Place one L shaped - L down
                (numTilingLDP(cache, cacheL, n-2)%MOD) + //place one L shaped - L up
                (numTilingDP(cache, cacheL, n-2)%MOD) + // Place two horizontal
                (numTilingDP(cache, cacheL, n-1)%MOD)  // Place one vertical
            )%MOD;
        }
        return cache[n];
    }

public:
    int numTilings(int n) {
        unordered_map<int, long long> cache;
        unordered_map<int, long long> cacheL;
        return numTilingDP(cache, cacheL, n);
    }
};
11300