  We study the cluster variables and "imaginary" elements of the semicanonical
basis for the coefficient-free cluster algebra of affine type $A_1^{(1)}$. A
closed formula for the Laurent expansions of these elements was obtained by
P.Caldero and the author in math.RT/0604054. As a by-product, there was given a
combinatorial interpretation of the Laurent polynomials in question, equivalent
to the one obtained by G.Musiker and J.Propp in math.CO/0602408. The arguments
in math.RT/0604054 used a geometric interpretation of the Laurent polynomials
due to P.Caldero and F.Chapoton (math.RT/0410184). This note provides a quick,
self-contained and completely elementary alternative proof of the same results.
