The problem is to count number of configuration of $9$ bulbs on a $3\times 3$ grid, where no two bulbs that are adjacent are switched on.
I solved this problem in a very ad-hoc kind of manner, the answer is $63$, here's how
(I am representing configuration row wise as : $\mathtt{row_1}$, $\mathtt{row_2}$, $\mathtt{row_3}$)
These are the only unique configurations of the bulbs $101|010|101$ or $010|101|010$ or $100|001|100$ * 4(depending on the $4$ possible position of 2nd row bulb on middle cell of every boundary) or $100|001|010$ * 4 (depending on the position of row 1 bulb on corners)
So a total of $2^5 + (2^4-1) + 4\times 1 + (2^2-1)\times 4$ (subtraction is done to ensure no double counting)
My question is, is there a more systematic way of solving this problem ?