I know that the graph of these two functions is the same:
$$(-1)^{\lfloor x\rfloor} = -2\lfloor x\rfloor + 4\left\lfloor\frac {\lfloor x\rfloor}2\right\rfloor + 1$$
Both of them interchange sign at integer points in the same manner. I'm trying to figure out what identity allows them to be equal, though. I know that I cannot apply a logarithm as that wouldn't do any good. I'm just trying to figure out how I could relate this to floor as an identity rather than just two alternate forms for the exact same function.