Of course it can.
Define the 'arithmetic' in anyway you choose that happens to agree with that specific example on those particular numbers. Eg, the Collatz rules except when a_n=1,000,000,000, and anything but division by two there.
If you mean is there another way of defining a Collatz type relation which agrees on ALL values of a_n, then the answer is yes and no. There are equivalent constructions, but functionally they are the same. By that I mean
sin(x) and cos(x-pi/2) are functionally the same, but have different 'labels'