Common terms of k-Pell numbers and Padovan or Perrin numbers

Abstract

Let kge 2. A generalization of the well-known Pell sequence is the k-Pell sequence. For this sequence, the first k terms are 0,ldots ,0,1 and each term afterwards is given by the linear recurrence Pn(k)=2Pn-1(k)+Pn-2(k)+⋯+Pn-k(k).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} P_n^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\\cdots +P_{n-k}^{(k)}. \\end{aligned}$$\\end{document}In this paper, we extend the previous work (Rihane and Togbé in Ann Math Inform 54:57–71, 2021) and investigate the Padovan and Perrin numbers in the k-Pell sequence.