We consider a variation on Euler Problem 215.

Walls are still built out of 2x1 or 3x1 bricks, but this time is it OK to have "running cracks" of length at most 2, as in the following 9x3 wall:

However, longer running cracks are disallowed.  The following 9x3 wall is not acceptable because it has one running crack of length 3.

There are possible 9x3 walls in which the maximal length of running cracks is 2 or 1.

There are possible 10x4 walls in which the maximal length of running cracks is 2 or 1.

There are possible 11x5 walls in which the maximal length of running cracks is 2 or 1.

Compute the value of .

Hint: consider a graph whose vertices represent pairs of layers, such that an edge represents in acceptable superposition of the three layers , , and .