This is what came in y mind.

Consider an as sequence of n tiles.

There can be two case. nth tile is red or nth tile is not red.

Case I nth tile is red.

In that can for an+1 there can be either green tile or black tile in n+1 position. there cant be erd tile (as no consecutive red tile).

So an+1 = an +2 --------- eq1

Now n+1 tile is not red. so n+2 tile can be either of tree color.

so an+2 = an+1 + 3 ---------eq2

adding eq1 and eq2

an+2 = an +5

Case II

nth tile is not red

so n+1 tile can be either of 3.

so an+1 = an +3 ---eq3

And the case where n+1 tile is red,

an+2 = an+1 +2 ------eq4

adding eq3 and eq4

an+2 = an+5

----------------------------------------------------------

So in both the case we are getting recuursive relation as

an+2 = an+5 Answer.

I need to think more over this.

let m know if you feel answer is wrong. We can discuss the problem further.