And in order to characterize the ordinate reached by the construction, we can mark the height reached by the bottommost piece whose elevation gain is 1 or 2 at each step of the construction. But as we shall see now, asymptotic estimates can be derived much faster. The coefficient of in counts the number of tilings at height with exactly pieces of any type. Computing the generating functions We now show how to automate the previous computations for any integer . Let be one of them and suppose we want to figure out all the fringes can be derived from.
Similar rules applies if the th digit is a .
No exact expression for this limit is known, although the approximation 1.940215531 is generally agreed on. Suppose for example is a multiple of 6. To tile a chessboard we can put side by side slices of width 6. In this case with the singularity of smallest modulus of the denominator of . If is not a multiple of 6, it suffices to complete with at most 5 vertical stripes of width 1, but this does not change the limit.