A295229 Number of tilings of the n X n grid, using diagonal lines to connect the grid points.

Each of the n^2 cells can be tiled with either / or \. 
These behave differently than a binary coloring under reflection or rotation. 
In particular, the symbols are interchanged by a horizontal or vertical reflection or
by rotation through a quarter turn, but remain the same under a diagonal or antidiagonal reflection or
by rotation through a half turn.

A formula can be derived using Burnsides lemma considering the number of solutions that are invariant under each of the 8 symmetries.

All solutions without regard to symmetry:
  2^(n^2)

Invariant under diagonal or antidiagonal reflection:
  2^(n*(n+1)/2)

For other symmetries it is necessary to consider whether n is even or odd.
Case even n:
  Invariant under horizontal/vertical reflection:
     2^(n^2/2)
  Invariant under quarter turn:
     2^(n^2/4)
  Invariant under half turn:
     2^(n^2/2)

Case odd n:
  Invariant under horizonatl/vertical reflection:
     0
  Invariant under quarter turn:
     0
  Invariant under half turn:
     2^((n^2+1)/2)

Applying Burnside's lemma gives combined formula:
  Even n:   a(n) = (2^(n^2) + 2*2^(n*(n+1)/2) + 3*2^(n^2/2) + 2*2^(n^2/4)) / 8.
  Odd n:    a(n) = (2^(n^2) + 2*2^(n*(n+1)/2) + 2^((n^2+1)/2)) / 8.