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_{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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