{
  "id": "0801.1097",
  "version": "v2",
  "published": "2008-01-07T19:49:15.000Z",
  "updated": "2008-05-29T10:17:04.000Z",
  "title": "A Combinatorial Interpretation for Certain Relatives of the Conolly Sequence",
  "authors": [
    "B. Balamohan",
    "Zhiqiang Li",
    "Stephen Tanny"
  ],
  "comment": "13 pages, 6 figures, 1 table",
  "journal": "Journal of Integer Sequences, Vol. 11 (2008), Article 08.2.1",
  "categories": [
    "math.CO"
  ],
  "abstract": "For any integer s >= 0, we derive a combinatorial interpretation for the family of sequences generated by the recursion (parameterized by s) h_s(n) = h_s(n - s - h_s(n - 1)) + h_s(n - 2 - s - h_s(n - 3)), n > s + 3, with the initial conditions h_s(1) = h_s(2) = ... = h_s(s+2) = 1 and h_s(s+3) = 2. We show how these sequences count the number of leaves of a certain infinite tree structure. Using this interpretation we prove that h_s sequences are \"slowly growing\", that is, h_s sequences are monotone nondecreasing, with successive terms increasing by 0 or 1, so each sequence hits every positive integer. Further, for fixed s the sequence h_s(n) hits every positive integer twice except for powers of 2, all of which are hit s+2 times. Our combinatorial interpretation provides a simple approach for deriving the ordinary generating functions for these sequences.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-05-29T10:17:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "11B37",
      "11B39"
    ],
    "keywords": [
      "combinatorial interpretation",
      "conolly sequence",
      "infinite tree structure",
      "initial conditions",
      "sequences count"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Journal of Integer Sequences",
      "year": 2008,
      "month": "May",
      "volume": 11,
      "number": 2,
      "pages": 21
    },
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008JIntS..11...21B"
    }
  }
}