[Discrete Math] Recursive Relations Word Problem

I've PM'ed the author of this problem (TrevTutor), but I'm eager to find the solution.

Take a look at the video:

https://youtu.be/K4RXwgHdLV0?t=37m35s

Over here, he says that if the chip in the nth position is blue, then the previous chip in the n-1 position is limited to only 3 options. Then, under that condition, the chip in the n-2 position can be any of the 4 colors.

What I don't understand is why he multiplies 3 by a_n-2.

If the n-2 position can be 4 options, and n-1 position can be 3 options, how did he come up with 3 options * a_n-2?

Doesn't that mean that you're still only limiting the chip in the n-2 position to 3 options?

I know there's something wrong with my thought process but I can't pin point what I'm doing wrong.

My progress in trying to write it out myself

EDIT: Also, since he multiplies 3(a_n-2) if n is blue because you have 3 options for the n-1 position, why doesn't he multiply the a(n-1) position by 4 if you have 4 options if n =/= blue?