expected that all of them cycle. Here it might be possible that none of them cycles.

Before making any claims, let us see how these sequences behave modulo 4.

A 4-free Fibonacci sequence contains an odd number.

Suppose there exists a 4-free Fibonacci sequence containing only even numbers. Then ignoring the initial terms, all the elements of the sequence equal 2 modulo 2. Therefore, we divide by a power of 4 every time. This cannot last forever.

After the first occurrence of an odd number, a 4-free Fibonacci sequence cannot have two even numbers in a row.

Start with the first odd number. The steps that do not include division generate a parity pattern: odd, odd, even, odd, odd, even and so on. So there are no two even numbers in a row. That means we can get a multiple of 4 only after summing two odd numbers. We might get an even number after the division, but the next number must be odd again.

Let us analyze the 4-free Fibonacci sequences probabilistically, similar to what we did for $n=3$.

\[thm:average4\] An average division is by a factor of $4^{4/3} \approx 6.35$.

When we divide, we divide by 4 with 