An analogue of the Thue-Morse sequence

We consider the finite binary words Z(n), n is an element of N, defined by the following self-similar process: Z(0) := 0, Z(1) := 01, and Z(n + 1) := Z(n) center dot <(Z(n -1))over bar>, where the dot center dot denotes word concatenation, and (w) over bar the word obtained from w by exchanging the zeros and the ones. Denote by Z(infinity) = 01110100... the limiting word of this process, and by z(n) the n'th bit of this word. This sequence z is an analogue of the Thue-Morse sequence. We show that a theorem of Bacher and Chapman relating the latter to a "Sierpinski matrix" has a natural analogue involving z. The semi-infinite self-similar matrix which plays the role of the Sierpinski matrix here is the zeta matrix of the poset of finite subsets of N without two consecutive elements, ordered by inclusion. We observe that this zeta matrix is nothing but the exponential of the incidence matrix of the Hasse diagram of this poset. We prove that the corresponding Mobius matrix has a simple expression in terms of the zeta matrix and the sequence z.