{
  "id": "2201.07127",
  "version": "v2",
  "published": "2022-01-14T07:53:47.000Z",
  "updated": "2024-05-14T20:18:24.000Z",
  "title": "Concatenations of Terms of an Arithmetic Progression",
  "authors": [
    "Florian Luca",
    "Bertrand Teguia Tabuguia"
  ],
  "comment": "Substantial revision of the previous version. 15 pages, 27 references",
  "categories": [
    "math.CO",
    "cs.SC"
  ],
  "abstract": "Let $\\left(u(n)\\right)_{n\\in\\mathbb{N}}$ be an arithmetic progression of natural integers in base $b\\in\\mathbb{N}\\setminus \\{0,1\\}$. We consider the following sequences: $s(n)=\\overline{u(0)u(1)\\cdots u(n) }^b$ formed by concatenating the first $n+1$ terms of $\\left(u(n)\\right)_{n\\in\\mathbb{N}}$ in base $b$ from the right; $s_r(n) = \\overline{u(n)u(n-1)\\cdots u(0)}^b$; and $\\left(s_*(n)\\right)_{n\\in\\mathbb{N}}$, given by $s_*(0)=u(0)$, $s_*(n)=\\overline{s_r(n-1)s(n)}^b, n\\geq 1$. We construct explicit formulas for these sequences and use basic concepts of linear difference operators to prove they are not P-recursive (holonomic). We also present an alternative proof that follows directly from their definitions. We implemented $\\left(s(n)\\right)_{n\\in\\mathbb{N}}$ and $\\left(s_r(n)\\right)_{n\\in\\mathbb{N}}$ in the decimal base when $(u(n))_{n\\in\\mathbb{N}}=\\mathbb{N}\\setminus \\{0\\}$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-05-14T20:18:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K31",
      "11Y55",
      "68W30",
      "11-04"
    ],
    "keywords": [
      "arithmetic progression",
      "concatenations",
      "linear difference operators",
      "construct explicit formulas",
      "basic concepts"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}