On the congruence properties and growth rate of a recursively defined sequence

Let $a_1 = 1$ and, for $n > 1$, $a_n = a_{n-1} + a_{\left \lfloor \frac{n}{2} \right \rfloor}$. In this paper we will look at congruence properties and the growth rate of this sequence. First we will show that if $x \in \{1, 2, 3, 5, 6, 7 \}$, then the natural density of $n$ such that $a_n \equiv x \pmod{8}$ exists and equals $\frac{1}{6}$. Next we will prove that if $m \le 15$ is not divisible by $4$, then the lower density of $n$ such that $a_n$ is divisible by $m$, is strictly positive. To put these results in a broader context, we will then posit a general conjecture about the density of $n$ such that $a_n \equiv x \pmod{m}$ for any given $x$ and any $m$ not divisible by $32$. Finally, we will show that there exists a function $f$ such that $n^{f(n)} < a_n < n^{f(n) + \epsilon}$ for all $\epsilon > 0$ and all large enough $n$.