co.combinatorics – Why is Schröder numbers equivalent to the number of perfect matchings for triangular grid of n squares and how the graph look like?

In the OEIS entry for the Schröder numbers is A006318. There is a comment which related the sequence to perfect matchings:

The number of perfect matchings in a triangular grid of n squares (n = 1, 4, 9, 16, 25, …). – Roberto E. Martinez II, Nov 05 2001

I am not sure how the triangular grid of n squares look like. Taking my understanding as an example, I draw the graphs as following:
triangle

Let’s ignore the case for n=1, then the perfect matchings for triangular lattice with n=2,3,4 are 2,6,0(which is not matched in A006318: 2, 6, 22, …).

So maybe I misunderstand the comment about the definition of a triangular grid of n squares .

  • Can someone know how the graphs look like?
  • Why is Schröder numbers equivalent to the number of perfect matchings in a triangular grid?

I am very happy to hear from anyone and thank you very much in advance!