enough to find such a sequence that does not need to have positive terms. We can produce such a sequence by building it backwards, similar to what we did in Lemma \[thm:rich\].

There exist arbitrary long division-poor 3-free Fibonacci subsequences.

Let us build an example of a division-poor 3-free sequence, where each division is by exactly 3. Begin with $a_n=1$, $a_{n-1}=1$, and build a sequence that is not necessarily all-positive. That means we will have $a_{n-2k}=3a_{n-2k+2}-a_{n-2k+1}$ and $a_{n-2k-1}=3a_{n-2k+1}-a_{n-2k}$. Here are several terms: $-8$, 7, $-1$, 2, 1, 1. For a positive version we need to add $3^3$ to $-8$, as outlined in Lemma \[thm:adjustement\] as we had two divisions by 3. The adjusted all-positive division-poor sequence is 19, 7, 26, 11, 37, 16.

4-free Fibonacci sequences {#sec:4free}
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Consider the 4-free Fibonacci sequence starting with 0, 1. This sequence is A224382: 0, 1, 1, 2, 3, 5, 2, 7, 9, 1, 10, 11, 21, 2, 23, 25, $\ldots$. It seems that this sequence grows and does not cycle.

In checking many other 4-free Fibonacci sequences, we still did not find any cycles. The behavior of 4-free sequences is completely different from the behavior of 3-free sequences.

For 3-free sequences we 