Here's simple 2-D pattern that seems to tile quite efficiently:

>! [![enter image description here][1]][1]

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:

>! [![enter image description here][2]][2]

EDIT: managed to find an even better pattern with $\mathbf{26.\overline6\%}$ utility. Image coming up soon.

  [1]: https://i.sstatic.net/JjBls.png
  [2]: https://i.sstatic.net/Brcca.png