{
  "id": "2406.09532",
  "version": "v1",
  "published": "2024-06-13T18:43:05.000Z",
  "updated": "2024-06-13T18:43:05.000Z",
  "title": "On the congruence properties and growth rate of a recursively defined sequence",
  "authors": [
    "Wouter van Doorn"
  ],
  "comment": "18 pages",
  "categories": [
    "math.NT"
  ],
  "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 $n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-13T18:43:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "growth rate",
      "recursively defined sequence",
      "congruence properties",
      "natural density",
      "lower density"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}