There are $10$ lightbulbs in a row, on or off. How many combinations of on and off lightbulbs can we have if no two turned on bulbs can be next to each other?
It seems like it forms a Fibonacci sequence if we start from a base case of $1$ bulb and work up, but I don't understand intuitively why this is the case.
$$F(1) = 2\\ F(2) = 3\\ F(3) = 5$$
Where $F(X)$ is the number of combinations we can have with $X$ light bulbs in a row.
Thanks for any help.