{
  "id": "1102.5111",
  "version": "v1",
  "published": "2011-02-24T21:52:58.000Z",
  "updated": "2011-02-24T21:52:58.000Z",
  "title": "Arithmetic properties of the sequence of degrees of Stern polynomials and related results",
  "authors": [
    "Maciej Ulas"
  ],
  "comment": "16 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Let $B_{n}(t)$ be a $n$-th Stern polynomial and let $e(n)=\\op{deg}B_{n}(t)$ be its degree. In this note we continue our study started in \\cite{Ul} of the arithmetic properties of the sequence of Stern polynomials and the sequence $\\{e(n)\\}_{n=1}^{\\infty}$. We also study the sequence $d(n)=\\op{ord}_{t=0}B_{n}(t)$. Among other things we prove that $d(n)=\\nu(n)$, where $\\nu(n)$ is the maximal power of 2 which dividies the number $n$. We also count the number of the solutions of the equations $e(m)=i$ and $e(m)-d(m)=i$ in the interval $[1,2^{n}]$. We also obtain an interesting closed expression for a certain sum involving Stern polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-02-24T21:52:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "arithmetic properties",
      "related results",
      "maximal power",
      "th stern polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1102.5111U"
    }
  }
}