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

Let a₁ = 1 and, for $n > 1$, aₙ = aₙ₋₁ + an/2. In this paper we will look at congruence properties and the growth rate of this sequence. First we will show that if x ∈ \1, 2, 3, 5, 6, 7 \, then the natural density of n such that aₙ ≡ x 8 exists and equals 1/6. Next we will prove that if m ≤ 15 is not divisible by $4$, then the lower density of n such that aₙ 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 aₙ ≡ x m for any given x and any m not divisible by $32$. Finally, we will show that there exists a function f such that nᶠ⁽ⁿ⁾ < aₙ < nf(n) + ε for all ε > 0 and all large enough n.