How many ways are there to cover a $2\times 16$ rectangle with $2\times 2,$ $2\times 3$ and $2\times 4$ rectangles?
I already dealt with a similar problem, which is how many ways are there to cut a $1\times 8$ rectangle into $1\times 1$ and $1\times 2$ rectangles. The answer to this problem can be calculated using the number of ways to divide $1\times k$ strip for $3\le k\le8$ sequentially, to arrive at $34$. However, this problem is completely different. How can I solve this problem?