E17. A 1-2-3 counting problem

The following problem seems at first to be quite difficult, but if you look at it the right way it isn't.

How many n-digit integers are there that contain no digits other than 1, 2, or 3, subject to the condition that any two consecutive digits differ by exactly 1.

This problem (for the n = 10 case) appeared in the ATMIM newsletter, Winter 2002, where it is credited to http://www.mathkangaroo.org, an interesting math enrichment and contest Web site.

I think this problem is too easy for me to post an answer, but if anyone asks for one, I will.