after $2n$ coin flips is $(2\cdot 3 \cdot 7 \cdot 13)^{1/4}/5$, which is below 0.97. So the overall trend for this sequence is to decrease.

Based on our computational experiments and probabilistic discussions above we conjecture:

Any 3-free Fibonacci sequence ends in a cycle.

So 2-free Fibonacci sequences provably end in a cycle, 3-free sequences conjecturally end in a cycle. Will 4-free Fibonacci sequences end in cycles too? Before discussing 4-free Fibonacci sequences, we want to make a detour and prove the promised result that an arbitrarily long division-poor sequence exists (See Lemma \[thm:poor\]) as a corollary to a much stronger and a more general theorem.

Customized-division subsequences {#sec:customized}
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We promised to give a proof that there exist arbitrary long division-poor sequences that are 3-free Fibonacci sequences. Now we want to prove a stronger statement. We want to allow any $n$ and show that we can build a customized $n$-free Fibonacci sequence that will have a division by a prescribed power of $n$ with the prescribed remainder.

Let us correspond to any $n$-free Fibonacci sequence the list of numbers by which we divide at every step. We call this list *a signature*. For example, a 3-free sequence 5, 4, 