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

Let a₁ = 1 and, for n > 1, aₙ = a₍-₁ + a ₍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 16. 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 n^f (n) 0 and all large enough n.