You are given a fence with n posts and k different colors. Determine the total number of valid ways to paint the fence such that no more than $2$ consecutive posts share the same color.

Constraints:

• $1 \leq$ n $\leq 50$

• $1 \leq$ k $\leq 10^5$

• The test cases are generated such that the answer is within the range $[0, 2^{31} - 1]$ for the given n and k.

Now, let’s take a moment to make sure you’ve correctly understood the problem. The quiz below helps you check if you’re solving the correct problem:

We have a game for you to play. Rearrange the logical building blocks to develop a clearer understanding of how to solve this problem.

Implement your solution in the following coding playground.