There was this question on one sequence where the expression for the general term contains the floor function. I can clearly see that the floor function is needed for an expression which doesn't burn ones eyes out, but I have no idea how one goes about to construct the explicit formula.
For more examples, there is the sequence A014132 $ n + \left\lfloor 1/2 + \sqrt{2n} \right\rfloor $ which contains every integer but the triangular numbers and A000037 $ n + \left\lfloor1/2 + \sqrt{n-3/4}\right\rfloor $ which misses exactly the square numbers.
So, let's say you are given the sequence $ 4, 6, 7, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25,\ldots $ of non-Fibonacci numbers during an exam (i.e. you cannot use the OEIS to look it up), and you are told to construct a expression like the ones above for the $n$'th term, how does one think to get to the right answer?
Edit: As a side question, is there some quick way to see which one of the floor and ceiling functions yield the nicest expressions?
New edit: I see that the expression for non-Fibonacci numbers is quite complicated, containing base-$\phi$ logarithms. (I didn't research enough, apparently. I expected it to be on par with the two others.) I'll accept a solution for any of the other sequences, or any similar sequence not discussed here.