If I have a 3x3 matrix A and I want to repeatedly multiply it by vector B of length 3, what do I need to make sense of this operation?

I know I'm looking for an expression of the form (a^n + b) / c because I manually performed a few iterations and looked it up on oeis. I sucked shit at linear algebra in college so I really don't have an idea what I'm doing. Apologizes for any abuses of notation.