I'm beginning to deduce some of the magic required here to make the gears mesh.

I stumbled upon a compatible gear set by accident originally, but now I think I understand what is actually going on.

The rules:

• The sun gear tooth count must be divisible by both 2 and <num planets>.
• The ring gear tooth count must be divisible by both 2 and <num planets>.
• Stationary ring gear tooth count - sun tooth count must be an even number.
• Output ring gear tooth count = Ring tooth count +- <num planets>, with adjusted gear pitch to equal the stationary ring gear diameter.  The adjustment to diametrical pitch = <ring_tooth_pitch>*(ring_tooth_count> +-<num planets> )/<ring_tooth_count>

So basically... for 3 planets I will need multiples of 6 for both the sun and ring gear tooth count.

For the test I've settled on 18 sun teeth, 60 ring teeth, and 57 output teeth.  

Planet tooth count is a always a function of the sun/ring tooth count and ends up being = (<ring_tooth_count> - <sun_tooth_count>)/2.  So for this test, they will have 21 teeth.

I increased my pitch a little so it still fits on a Nema 23, from 0.965 to 1.05 and I'm using a pressure angle of 24.

This is what that looks like.

Sliced.

Printing.