probability 1, additionally with probability $1/4$ we divide by 4 more, and so on. So the result is 4 to the power $$1+1/4+1/16+\ldots = 4/3.$$

How often on average do we divide? Let us assume that we passed stretches of all even numbers. Each time we divide after that, the previous two numbers are odd. After the division, the remainder is 1, 2, or 3. So the following six cases describe what happens after the division:

If $a \equiv 1$ (mod 4) and $b \equiv 3$ (mod 4), then the following set of remainders might happen until the next division:

-   1;

-   2, 1, 3;

-   3, 2, 1, 3.

If $a \equiv 3$ (mod 4) and $b \equiv 1$ (mod 4), then the following set of remainders might happen until the next division:

-   3;

-   2, 3, 1;

-   1, 2, 3, 1.

\[thm:averagesteps\] The average number of steps between divisions is $8/3$.

We assume that during the division each remainder is generated with probability $1/3$, so the stretches of length 1, 3, and 4 between divisions are equally probable.

We want to build a probabilistic model that reflects some behavior of 4-free Fibonacci sequences. Let us 