  Fomin and Zelevinsky show that a certain two-parameter family of rational
recurrence relations, here called the (b,c) family, possesses the Laurentness
property: for all b,c, each term of the (b,c) sequence can be expressed as a
Laurent polynomial in the two initial terms. In the case where the positive
integers b,c satisfy bc<4, the recurrence is related to the root systems of
finite-dimensional rank 2 Lie algebras; when bc>4, the recurrence is related to
Kac-Moody rank 2 Lie algebras of hyperbolic type. Here we investigate the
borderline cases bc=4, corresponding to Kac-Moody Lie algebras of affine type.
In these cases, we show that the Laurent polynomials arising from the recurence
can be viewed as generating functions that enumerate the perfect matchings of
certain graphs. By providing combinatorial interpretations of the individual
coefficients of these Laurent polynomials, we establish their positivity.
