i.e. dividing by 2 gives a remainder of $0$ for even and $1$ for odd.
dividing by 4 gives a remainder of $0$ or $1$, for $0$ in secod position, and a remainder of $2$ or $3$, for a $1$ in second position. The first case is an analogue of "even" and the second case is an analgogue of "odd".
But this seems really awkward for a simple idea. Is there a simple way to express it? Or even better, a name for the concept?
BTW This came up in powers of imaginary numbers, which cycle in fours: $1, i, -1, -i$. It seems it would also be helpful for the Taylor Series expansion giving Euler's formula $e^{i\theta} = \cos \theta + i \sin \theta$
Integer division by 2 (i.e. discard remainder), then the even/oddness of the result (i.e. its parity) is what you want.