Should Heightmap Resolution be one greater than Detail Resolution?

In the Terrain examples I've found, the Heightmap/Detail Resolutions are usually 513/512 or 1025/1024.

Is there some reason they're not both regular multiples of eight, and is there any relationship between these two values that I should think about for performance or memory concerns?

Terrain heightmaps need to be power of two plus one, because they construct a mesh, which is made out of quads (two triangles), and the heights are the values at the vertices. For example, a 3x3 heightmap:

+----+----+
|    |    |
|    |    |
+----+----+
|    |    |
|    |    |
+----+----+

No resolution has any relationship to any other resolution, other than if you use a really high heightmap res and a really low detail res, it will probably look bad. The larger the values, the more memory is used.