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 adding a copy of the pattern, staggered so that the required empty spaces (marked in pink in the image above) overlap. This happens nicely as long as the copy is moved 9 to 12 tiles both horizontally and vertically.
The final pattern looks like this:
EDIT: managed to find an even better pattern with $\mathbf{26.\overline6\%}$ utility. Image coming up soon.