<p>The chess knight has a <strong>unique movement</strong>,&nbsp;it may move two squares vertically and one square horizontally, or two squares horizontally and one square vertically (with both forming the shape of an <strong>L</strong>). The possible movements of chess knight are shown in this diagram:</p>

<p>A chess knight can move as indicated in the chess diagram below:</p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/08/18/chess.jpg" style="width: 402px; height: 402px;" />
<p>We have a chess knight and a phone pad as shown below, the knight <strong>can only stand on a numeric cell</strong>&nbsp;(i.e. blue cell).</p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/08/18/phone.jpg" style="width: 242px; height: 322px;" />
<p>Given an integer <code>n</code>, return how many distinct phone numbers of length <code>n</code> we can dial.</p>

<p>You are allowed to place the knight <strong>on any numeric cell</strong> initially and then you should perform <code>n - 1</code> jumps to dial a number of length <code>n</code>. All jumps should be <strong>valid</strong> knight jumps.</p>

<p>As the answer may be very large, <strong>return the answer modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>

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

<pre>
<strong>Input:</strong> n = 1
<strong>Output:</strong> 10
<strong>Explanation:</strong> We need to dial a number of length 1, so placing the knight over any numeric cell of the 10 cells is sufficient.
</pre>

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

<pre>
<strong>Input:</strong> n = 2
<strong>Output:</strong> 20
<strong>Explanation:</strong> All the valid number we can dial are [04, 06, 16, 18, 27, 29, 34, 38, 40, 43, 49, 60, 61, 67, 72, 76, 81, 83, 92, 94]
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> n = 3131
<strong>Output:</strong> 136006598
<strong>Explanation:</strong> Please take care of the mod.
</pre>

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

<ul>
	<li><code>1 &lt;= n &lt;= 5000</code></li>
</ul>

class Solution {
private:
    vector<vector<int>> mat = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}, {-1, 0, -1}};
    int mod = 1e9 + 7;

public:
    int f(int i, int j, int k, vector<vector<vector<long>>> &dp) {
        if (i < 0 || i > 3 || j < 0 || j > 2 || mat[i][j] == -1) return 0;
        
        if (k == 1) return 1;

        if (dp[i][j][k] != -1) return dp[i][j][k];

        long sum = 0;
        sum = (sum + f(i - 2, j + 1, k - 1, dp)) % mod;
        sum = (sum + f(i - 2, j - 1, k - 1, dp)) % mod;
        sum = (sum + f(i + 2, j + 1, k - 1, dp)) % mod;
        sum = (sum + f(i + 2, j - 1, k - 1, dp)) % mod;
        sum = (sum + f(i - 1, j - 2, k - 1, dp)) % mod;
        sum = (sum + f(i + 1, j - 2, k - 1, dp)) % mod;
        sum = (sum + f(i - 1, j + 2, k - 1, dp)) % mod;
        sum = (sum + f(i + 1, j + 2, k - 1, dp)) % mod;

        return dp[i][j][k] = sum;
    }

    int knightDialer(int n) {
        int res = 0;
        vector<vector<vector<long>>> dp(4, vector<vector<long>>(3, vector<long>(n + 1, -1)));
        for (int i = 0; i <= 3; i++) {
            for (int j = 0; j < 3; j++) {
                if (mat[i][j] != -1) {
                    res = (res + f(i, j, n, dp)) % mod;
                }
            }
        }
        return res;
    }
};

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