\[sec:cycles\].

We finish with a discussion of our computational results. Section \[sec:growthdivfree\] explains why the average growth for some $n$-free sequences is close to the golden ratio and Section \[sec:growthomni\] explains the growth behavior for other values of $n$.

Fibonacci Numbers and $n$-free Fibonacci Sequences {#sec:definitions}
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Let us denote *Fibonacci numbers* by $F_k$. We assume that $F_0=0$ and $F_1=1$. The sequence is defined by the Fibonacci recurrence: $F_{n+1} =F_n + F_{n-1}$ (See A000045). We call an integer sequence $a_n$ *Fibonacci-like* if it satisfies the Fibonacci recurrence: $a_k = a_{k-1} +a_{k-2}$. A Fibonacci-like sequence is similar to the Fibonacci sequence, except it starts with any two integers. The second-famous Fibonacci-like sequence is the sequence of *Lucas numbers* $L_i$ that starts with $L_0=2$ and $L_1=1$: 2, 1, 3, 4, 7, 11, $\ldots$ (See A000032).

An *$n$-free Fibonacci* sequence starts with any two integers: $a_1$ and $a_2$ and is defined by the recurrence $a_k = (a_{k-1} +a_{k-2})/n^i$, where $n^i$ is the largest power of $n$ that is a factor of $a_{k-1} +a_{k-2}$. To continue the tradition we call numbers in the $n$-free Fibonacci sequence that starts with $a_0=0$ and $a_1=1$ *$n$-free Fibonacci numbers*.

In the future we will consider only sequences starting 