| Matrix Representation of Recursive Sequences of Order $3$ and Its Applications |
| Received:September 07, 2017 Revised:January 13, 2018 |
| Key Words:
recursive number sequence of order $3$ matrix representation of recursive number sequences Padovan number sequence Perrin number sequence Tribonacci polynomial sequence
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| Author Name | Affiliation | | Tianxiao HE | Department of Mathematics, Illinois Wesleyan University, Bloomington, Illinois $61702$, USA | | Jeff H.-. LIAO | Department of Mathematics, Taiwan Normal University, Taipei $11677$, Taiwan, China | | Peter J.-S. SHIUE | Department of Mathematical Sciences, University of Nevada, Las Vegas, Nevada, $89154$-$4020$, USA |
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| Abstract: |
| Here presented is a matrix representation of recursive number sequences of order $3$ defined by $a_n=pa_{n-1}+qa_{n-2}+ra_{n-3}$ with arbitrary initial conditions $a_0,$ $a_1=0$, and $a_2$ and their special cases of Padovan number sequence and Perrin number sequence with initial conditions $a_0=a_1=0$ and $a_2=1$ and $a_0=3$, $a_1=0$, and $a_2=2$, respectively. The matrix representation is used to construct many well known and new identities of recursive number sequences as well as Pavodan and Perrin sequences. |
| Citation: |
| DOI:10.3770/j.issn:2095-2651.2018.03.001 |
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