Matrix Representation of Recursive Sequences of Order $3$ and Its Applications 
Received:September 07, 2017 Revised:January 13, 2018 
Key Word:
recursive number sequence of order $3$ matrix representation of recursive number sequences Padovan number sequence Perrin number sequence Tribonacci polynomial sequence

Fund ProjectL: 
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 $a_n=pa_{n1}+qa_{n2}+ra_{n3}$ 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:20952651.2018.03.001 
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