Counting LEGO configurations

We discuss the problem of counting certain LEGO structures, primarily those comprising parallel $w \times 1$ tiles. These can be combined, as a single LEGO structure, by interlocking the tiles. %Alternatively, if the interlocking condition is relaxed, so that tiles can also be placed end-to-end, a greater number of possible configurations results. We also study the historically earlier problem of counting the number of ways to combine $2 \times 4$ LEGO tiles, which in this case gives a 3-dimensional structure. In all cases the number of configurations is dominated by an exponential growth term, $\mu^n$ where $n$ is the number of tiles. We present an algorithm for counting these various LEGO configurations, and use the data to estimate the asymptotics. We analyse the data so generated, and conjecture that, for the two-dimensional structures, the number of possible configurations grows like $A(w)\mu(w)^n/n,$ and we give numerical estimates for $A(w)$ and $\mu(w)$ for $w < 11,$ while for the three-dimensional structure the number of possible configurations is conjectured to grow like $A\mu^n/n^{3/2},$ where $\mu = 117.25 \pm 0.05.$ We also study the sequences that arise when we fix the number of tiles $n,$ and vary the tile size $w.$ We prove that the sequences are polynomials of degree $n-1,$ and we give these explicitly for $n=1 \ldots 14.$