Enumeration of pyramids of one-dimensional pieces of arbitrary fixed integer length

We consider pyramids made of one-dimensional pieces of fixed integer length a and which may have pairwise overlaps of integer length from 1 to a. We prove that the number of pyramids of size m, i.e. consisting of m pieces, equals (am-1,m-1) for each a >= 2. This generalises a well known result for a = 2. A bijective correspondence between so-called right (or left) pyramids and a-ary trees is pointed out, and it is shown that asymptotically the average width of pyramids is proportional to the square root of the size.