This is homework, please only provide hints.
I've been given a problem: consider a 1-by-n chessboard. Coloring each square with one of two colors, red or blue. Let $a_n$ be the number of colorings in which no two squares that are red are adjacent.
So, I started off with two cases: 1) possible colorings where the last square is blue. 2) colorings where the last square is red.
length(n = 1) (last square blue): $\left \{ B\right \}$ (last square red): $\left \{ R\right \}$ which leaves us with 2 possible choices
length(n = 2) blue: $\left \{ RB\right \}$ $\left \{ BB \right \}$ red: $\left \{ BR \right \}$ which leaves us with 3 possible choices
etc..
At this point, I'm basically stuck. I noticed these numbers begin to look like the Fibonacci sequence. So obviously my solution is (I confirmed in the book):
$$a_n = a_{n-1} + a_{n-2}$$
My issue is, I only know that because I recognized the sequence. If this was a random sequence, how should I set it up to determine the relation?