Here's simple 2-D pattern that seems to tile quite efficiently:
The area of the each tile (blue square) is $21\times21 = 441$ tiles, and it contains $4\times14=56$ generators tiles, for a ratio of $\frac{56}{441} \approx 12.7\%$
The trick here is that
it's easy to double the density to $\frac{112}{441} \approx \mathbf{25.4\%}$ by copying the pattern and moving the copy 10 tiles diagonally.
The final pattern looks like this:
