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

TL;DR

This paper investigates the congruence properties and growth rate of a recursively defined sequence, establishing density results, conjecturing general behaviors, and bounding its growth with a specific function.

Contribution

It provides new results on the density of sequence terms modulo integers, proves positivity of divisibility densities, and conjectures broader density patterns, along with growth bounds.

Findings

Density of certain residues modulo 8 is 1/6.

Lower density of divisibility by certain m is positive.

Sequence growth is bounded between n^{f(n)} and n^{f(n)+ε}.

Abstract

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 $n$.