<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 {
public:
    map<pair<int, int>, long long> m;
    const long long MOD = 1000000007LL;
    long long solve(int n, int type) {

        if (m.contains({n, type})){
            return m[{n,type}];
        }

        long long tmp = 0;
        if (type==0) {
            tmp += solve(n-1, 0);
            tmp %= MOD;
            if (n>=2)
                tmp += solve(n-2, 0);
            tmp %= MOD;
            tmp += solve(n-1, 1);
            tmp %= MOD;
            tmp += solve(n-1, 2);
            tmp %= MOD;

        } else if (type==1) {
            tmp += solve(n-2, 0);
            tmp += solve(n-1, 2);
            tmp %= MOD;
        } else {
            tmp += solve(n-2, 0);
            tmp += solve(n-1, 1);
            tmp %= MOD;
        }
        m[{n, type}] = tmp;
        return tmp;

    }
    int numTilings(int n) {
        m[{0, 0}] = 0;
        m[{0, 1}] = 0;
        m[{0, 2}] = 0;
        m[{1, 0}] = 1;
        m[{1, 1}] = 0;
        m[{1, 2}] = 0;
        m[{2, 0}] = 2;
        m[{2, 1}] = 1;
        m[{2, 2}] = 1;

        long long ans =  solve(n, 0);

        return ans;
    }
};
13400