  We consider a very simple Mealy machine (three states over a two-symbol
alphabet), and derive some properties of the semigroup it generates. In
particular, this is an infinite, finitely generated semigroup; we show that the
growth function of its balls behaves asymptotically like n^2.4401..., where
this constant is 1 + log(2)/log((1+sqrt(5))/2); that the semigroup satisfies
the identity g^6=g^4; and that its lattice of two-sided ideals is a chain.
