276. Paint-Fence

difficulty: Easy

There is a fence with n posts, each post can be painted with one of the k colors.

You have to paint all the posts such that no more than two adjacent fence posts have the same color.

Return the total number of ways you can paint the fence.

Note: n and k are non-negative integers.

Example:

Input: n = 3, k = 2
Output: 6
Explanation: Take c1 as color 1, c2 as color 2. All possible ways are:
            post1  post2  post3      
 -----      -----  -----  -----       
   1         c1     c1     c2 
   2         c1     c2     c1 
   3         c1     c2     c2 
   4         c2     c1     c1  
   5         c2     c1     c2
   6         c2     c2     c1

Method One

class Solution {
    public int numWays(int n, int k) {
        if(n == 0){
            return 0;
        }
​
        if( n == 1){
            return k;
        }
        int prevPrev = k;
        int prev = k * k;
        for(int i = 3; i <= n; i++ ){
            int cur = (prevPrev + prev ) * (k - 1);
            prevPrev = prev;
            prev = cur;
        }
        return prev;
    }
}
​
/**
State Transition Equation: 状态转移方程
​
total[i] = same[i] + diff[i];
diff[i] =  total[i - 1] * (k - 1);
same[i] =  diff[i - 1] * 1;  diff[i - 1] 就是 i - 1 和 i - 2 不同的所有方法
=> total[i] = total[i - 1] * (k - 1) + diff[i - 1];
=> total[i] = ( total[i - 1]  + total[i - 2] ) *(k - 1) ;
​
Boundary Conditions:
total[0] : 0; 这个无法给出正确的total[2], 所以这个题必须给出 1，2 的情况作为边界条件
total[1] : k;
total[2] : k*k;
*/
​