{
  "id": "2407.12868",
  "version": "v1",
  "published": "2024-07-13T16:06:21.000Z",
  "updated": "2024-07-13T16:06:21.000Z",
  "title": "Sum of Consecutive Terms of Pell and Related Sequences",
  "authors": [
    "Navvye Anand",
    "Amit Kumar Basistha",
    "Kenny B. Davenport",
    "Alexander Gong",
    "Steven J. Miller",
    "Alexander Zhu"
  ],
  "comment": "33 Pages. Comments welcome!",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "We study new identities related to the sums of adjacent terms in the Pell sequence, defined by $P_{n} := 2P_{n-1}+P_{n-2}$ for $ n\\geq 2$ and $P_{0}=0, P_{1}=1$, and generalize these identities for many similar sequences. We prove that the sum of $N>1$ consecutive Pell numbers is a fixed integer multiple of another Pell number if and only if $4\\mid N$. We consider the generalized Pell $(k,i)$-numbers defined by $p(n) :=\\ 2p(n-1)+p(n-k-1) $ for $n\\geq k+1$, with $p(0)=p(1)=\\cdots =p(i)=0$ and $p(i+1)=\\cdots = p(k)=1$ for $0\\leq i\\leq k-1$, and prove that the sum of $N=2k+2$ consecutive terms is a fixed integer multiple of another term in the sequence. We also prove that for the generalized Pell $(k,k-1)$-numbers such a relation does not exist when $N$ and $k$ are odd. We give analogous results for the Fibonacci and other related second-order recursive sequences.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-13T16:06:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11Bxx",
      "11B37",
      "11B39",
      "11B50"
    ],
    "keywords": [
      "consecutive terms",
      "related sequences",
      "fixed integer multiple",
      "generalized pell",
      "identities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}