Combinations in Math, PLS help

[triangle127]

There are 19 adjacent parking spots available in a parking lot. A peculiar thing happens each day in this parking lot. None of the drivers are willing to park immediately beside another car, so there are never two adjacent spots that are both occupied. But there are always enough cars in the lot that there are never more than two empty spaces in a row. (So for example, there may be a space, a car, 2 spaces, a car, etc. But there’s never a space, 2 cars, a space, etc. Nor is there ever a car, 3 spaces, a car, etc..) How many different ways may this parking lot be occupied with cars? (Assume cars are identical.)

 

[Deleted member 4993]

There are 19 adjacent parking spots available in a parking lot. A peculiar thing happens each day in this parking lot. None of the drivers are willing to park immediately beside another car, so there are never two adjacent spots that are both occupied. But there are always enough cars in the lot that there are never more than two empty spaces in a row. (So for example, there may be a space, a car, 2 spaces, a car, etc. But there’s never a space, 2 cars, a space, etc. Nor is there ever a car, 3 spaces, a car, etc..) How many different ways may this parking lot be occupied with cars? (Assume cars are identical.)

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Are these spots in one row?

 

[Dr.Peterson]

There are 19 adjacent parking spots available in a parking lot. A peculiar thing happens each day in this parking lot. None of the drivers are willing to park immediately beside another car, so there are never two adjacent spots that are both occupied. But there are always enough cars in the lot that there are never more than two empty spaces in a row. (So for example, there may be a space, a car, 2 spaces, a car, etc. But there’s never a space, 2 cars, a space, etc. Nor is there ever a car, 3 spaces, a car, etc..) How many different ways may this parking lot be occupied with cars? (Assume cars are identical.)

I'd start by experimenting with the concept, to make sure I understand the rules. If I replace cars with 1 and empty spaces with 0, there 19 characters, with no more than one 1 in a row of 2 0's. Valid arrangements would include 1010101010101010101 (with the most 1's) and 00100100100100100100 (with the fewest). You might want to try out other ways to represent these; for example, a valid arrangement might be described by just listing the numbers of consecutive 0's, so that my examples would be 1,1,1,1,1,1,1,1 and 2,2,2,2,2,2,2, with some way to indicate whether it starts with 0 or 1.

These are just initial thoughts without having tried at all to solve the problem. I want to suggest only how you might get started if you have no idea how to solve it.

Now show us some ideas of your own! Also, tell us what you have learned about the subject. Have you just barely learned about combinations and permutations, or do you know other techniques like stars and bars?

Since we are told the spaces are "adjacent", I assume it does mean one row, not for example a circle.