(All measurements refer to Outside Diameter, O.D.)

Here are a few samples of this weave; one done in 5mm rings and one done in 6mm rings.  I used 20g rings but I think 18g would work too, at least for the 6mm chain.  
Try experimenting with other sizes and gauges!

This chain was done in 5mm rings.

This chain was done in 6mm rings with Sterling Silver and Neon Amethyst.

Half Persian 3-1

Step 1. Begin with 2 closed rings. The second ring behind the first.

Step 2. Add a ring, pink, that goes through the intersection of the first 2 rings, as shown and place a closed ring, blue, on it.

Step 3. Now flip the pink ring down and make sure that the blue ring is behind the ring before it.

Step 4. Pass another ring through the intersection of the last 2 rings as shown. Add another ring, like you did before.

Step 5. Flip the pink ring down and make sure the blue ring is behind the ring before it. Repeat steps 4 and 5 until you have the desired length. Once you get the hang of this weave it goes very quickly.