Problem 215

Crack-free Walls

Consider the problem of building a wall out of 2×1 and 3×1 bricks (horizontal×vertical dimensions) such that, for extra strength, the gaps between horizontally-adjacent bricks never line up in consecutive layers, i.e. never form a “running crack”.

For example, the following 9×3 wall is not acceptable due to the running crack shown in red:

There are eight ways of forming a crack-free 9×3 wall, written W(9,3) = 8.

Calculate W(32,10).

没有裂纹的墙

考虑这样一个问题：用大小为2×1和3×1的砖块（均为宽度×高度）来筑墙，而且为了保证强度，两个砖块之间的空隙的上下方均不能是砖块的空隙，也就是说不能有“延伸裂纹”。

例如，如下的9×3的墙就是不可行的，因为其中有用红色标出的延伸裂纹：

筑一面没有裂纹的9×3的墙一共有八种方式，记为W(9,3) = 8。

试计算W(32,10)。