MS Al-Rawi kindly sent me coordinates of the searchlights created by the Princeton MVPA toolbox, which I made into this image. As before, the center voxel is black, the one-voxel-radius searchlight red, the two-voxel-radius searchlight purple, and the three-voxel-radius searchlight blue.
The number of voxels in each is:
1-voxel radius: 6 surrounding voxels (7 total)
2-voxel radius: 32 surrounding voxels (33 total)
3-voxel radius: 122 surrounding (123 total)
4-voxel radius: 256 surrounding (257 total)
From a quick look at the code generating the coordinates, it appears to "draw" a sphere around the center voxel, then retrieve the voxels falling within the sphere. (anyone disagree with this characterization?)
This "draw a sphere first" strategy explains why I couldn't come up with this number of voxels for a three-voxel-radius searchlight if the shape follows additive rules (e.g. "add voxels that share a face with the next-size-smaller radius"). It's another example of how much it can vary when different people actually implement a description: to me, it was more natural to think of searchlights of different radii as growing iteratively, a rather different solution than the one used in the Princeton MVPA toolbox.