<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 dp[1001][1001];
    const static int mod = 1e9 + 7;
    int solve(int col1, int col2, int cols) {
        if(col1 > cols or col2 > cols) return 0;
        if(col1 == cols and col2 == cols) return 1;
        if(dp[col1][col2] != -1) return dp[col1][col2];
        // can we insert domino
        // inserting vertically
        long sum = 0;
        if(col1 != col2) {
            if(col1 < col2)
            sum += solve(col1 + 2, col2, cols);
            // for col2
            if(col2 < col1)
            sum += solve(col1, col2 + 2, cols);
        } 
        else 
            sum += solve(col1 + 2, col2 + 2, cols);

        // inserting horizontally
        // for col1
       

        // can we insert trimino
        if(col1 == col2) {
            sum += solve(col1 + 2, col1 + 1, cols);
            sum += solve(col1 + 1, col2 + 2, cols);
            sum += solve(col1 + 1, col2 + 1, cols);
        }

        if(col1 == col2 - 1) {
            sum += solve(col1 + 2, col2 + 1, cols);
        }

        if(col2 == col1 - 1) {
            sum += solve(col1 + 1, col2 + 2, cols);
        }

        return dp[col1][col2] = sum % mod;
    }
public:
    int numTilings(int n) {
        memset(dp, -1, sizeof(dp));
        return solve(0, 0, n);
    }
};
11900