Need help with a variation of the domino tiling problem

[Ading]

There are two questions I need some help with. They both involve a ‘honeycomb strip’ (which is just a hexagonal tessellation of two rows), ‘worker bees’ (which take up two hexagons), and larvae (which take up one hexagon).

How can we count the number of ways there are for worker bees and larvae to arrange themselves in an n-cell honeycomb strip? Explain.
Superstitious worker bees will only face up-right. How many ways are there for superstitious worker bees and larvae to arrange themselves in an n-cell honeycomb strip? Why?
There are two questions I need some help with. They both involve a ‘honeycomb strip’ (which is just a hexagonal tessellation of two rows), ‘worker bees’ (which take up two hexagons), and larvae (which take up one hexagon).

1. How can we count the number of ways there are for worker bees and larvae to arrange themselves in an n-cell honeycomb strip? Explain.
2. Superstitious worker bees will only face up-right. How many ways are there for superstitious worker bees and larvae to arrange themselves in an n-cell honeycomb strip? Why?
3. attached is a screenshot of the question for clarity
   
   

This kind of reminded me of that famous domino tiling problem, so it appears that we need to use induction. I tried to build to recurrences: when n = 2k (An) and when n = 2k+1 (Bn). And I got something like An = Bn-1 + An-1 + An-2 + Bn-2 , and Bn = An + Bn-1 + An-1 + Bn-2 , but apparently that’s not right

Thank you to any answers in advanced!

 

[Dr.Peterson]

This kind of reminded me of that famous domino tiling problem, so it appears that we need to use induction. I tried to build to recurrences: when n = 2k (An) and when n = 2k+1 (Bn). And I got something like An = Bn-1 + An-1 + An-2 + Bn-2 , and Bn = An + Bn-1 + An-1 + Bn-2 , but apparently that’s not right

If the domino problem you mention is the one whose answer is the Fibonacci sequence, that was my first thought, too.

When you say your answer is apparently not right, how did you determine that? I'd check by actually counting ways for small n (making a little table along with actual arrangements), and comparing to your formulas; that can also help in correcting a wrong answer (or even in making up a right answer from scratch). I'd much rather decide for myself that something is wrong, than just ask someone else.

Also, can you show your work in more detail? Explain why you wrote each term you did, and maybe in explaining it you'll see something you left out.