# Bijection between multisets and directed animals?

The number of directed animals (aka polyominoes) of size $n$ (A005773) is enumerated by the generating function $$\frac{1}{2} \left(1+\sqrt\frac{{1+z}}{{1-3 z}}\right).$$ This generating function also enumerates the number of $n$-element multisets of $\{1,\ldots,n\}$ containing no pair of consecutive integers (e.g. $111, 113, 133, 222, 333$ for $n=3$).

The first few numbers in the sequence are $$1, 1, 2, 5, 13, 35, 96, 267, 750, 2123, 6046, 17303, 49721.$$ Is there a good bijection between the two combinatorial classes?