Problem of the Week

I drove to a park this weekend for a hike. As I drove eastward, I noticed a small creek flowing from the north. I passed over the creek several times on a number of bridges as it meandered back and forth along side the road for a few miles, after which the creek continued flowing on to the south.

Now, there is only one way for a creek flowing from the north to the south to cross an east-west road only once, while there are two distinct ways for a creek to cross a road three times in this way. See the diagrams below.

There are eight ways for this to occur with five crossings.

How many ways can this occur with seven crossings?

For the more daring:  Find necessary and sufficient conditions that characterize these paths.  In particular, find an upper bound for the number of distinct ways for a creek to cross a road a total of n times in the way described above.

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