You have two types of tiles: a 2 x 1 domino shape and a tromino shape. You may rotate these shapes.
Given an integer n, return the number of ways to tile an 2 x n board. Since the answer may be very large, return it modulo 109 + 7.
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.
Example 1
Input: n = 3
Output: 5
Explanation: The five different ways are show above.
Example 2
Input: n = 1
Output: 1
Constraints
- 1 <= n <= 1000
Solution
经典动态规划
class Solution {
public:
int numTilings(int n) {
int mod = 1000000007;
vector<vector<long long>> dp(n, vector<long long>(4));
dp[0][0] = 1;
dp[0][1] = 0;
dp[0][2] = 0;
dp[0][3] = 1;
for (int i = 1; i < n; i++) {
dp[i][0] = dp[i - 1][3] % mod;
dp[i][1] = dp[i - 1][0] % mod + dp[i - 1][2] % mod;
dp[i][2] = dp[i - 1][0] % mod + dp[i - 1][1] % mod;
dp[i][3] = (dp[i - 1][0] % mod + dp[i - 1][1] % mod + dp[i - 1][2] % mod + dp[i - 1][3] % mod) % mod;
}
return dp[n - 1][3];
}
};
