Three-dimensional tilings of blocks and bracelets and related {2,1,2} sequences

Abstract

This paper discusses the combinatorial interpretation of the H_n numbers, where each Hn denotes the number of ways to tile a 2 2 n block with 2 2 1 plates and 6-block L shapes. It then investigates a closely related tiling sequence, which is tiling a 22n bracelet with the same two building blocks, and discusses its relation with Hn. The recursive equation for both integer sequences are found using one to one correspondence, induction and Newtons Sum. Additionally, in the case of H_n numbers, its related Lucas Style sequence P_n is found. The relationship between the P_n numbers, the Bn numbers and the Hn numbers is similar to that of the Lucas numbers and the Fibonacci numbers. Finally, identities concerning H_ns generating function and its relations with other existing {2,1,2} sequences are discussed, and a theorem that generalizes the generating function of all tiling sequences is proposed and proved.