UNLV Theses, Dissertations, Professional Papers, and Capstones May 2018 Notes on Linear Divisible Sequences and Their Construction: A Computational Approach Sean Trendell Follow this and additional works at: https://digitalscholarship.unlv.edu/thesesdissertations Part of the Mathematics Commons Repository Citation Trendell, Sean, "Notes on Linear Divisible Sequences and Their Construction: A Computational Approach" (2018). UNLV Theses, Dissertations, Professional Papers, and Capstones. 3336. http://dx.doi.org/10.34917/13568765 This Thesis is protected by copyright and/or related rights. It has been brought to you by Digital Scholarship@UNLV with permission from the rights-holder(s). You are free to use this Thesis in any way that is permitted by the copyright and related rights legislation that applies to your use. 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NOTES ON LINEAR DIVISIBLE SEQUENCES AND THEIR CONSTRUCTION: A COMPUTATIONAL APPROACH by Sean Trendell Bachelor of Science - Computer Mathematics Keene State College 2005 A thesis submitted in partial fulfillment of the requirements for the Master of Science - Mathematical Sciences Department of Mathematical Sciences College of Sciences The Graduate College University of Nevada, Las Vegas May 2018 Copyright © 2018 by Sean Trendell All Rights Reserved Thesis Approval The Graduate College The University of Nevada, Las Vegas April 4, 2018 This thesis prepared by Sean Trendell entitled Notes on Linear Divisible Sequences and Their Construction: A Computational Approach is approved in partial fulfillment of the requirements for the degree of Master of Science – Mathematical Sciences Department of Mathematical Sciences Peter Shiue, Ph.D. Kathryn Hausbeck Korgan, Ph.D. Examination Committee Chair Graduate College Interim Dean Derrick DuBose, Ph.D. Examination Committee Member Arthur Baragar, Ph.D. Examination Committee Member Yi-Tung Chen, Ph.D. Graduate College Faculty Representative ii ABSTRACT NOTES ON LINEAR DIVISIBLE SEQUENCES AND THEIR CONSTRUCTION: A COMPUTATIONAL APPROACH by Sean Trendell Dr. Pete Shiue, Examination Committee Chair Professor of Mathematical Sciences University of Nevada, Las Vegas, USA In this Masters thesis, we examine linear divisible sequences. A linear divisible sequence is any sequence fangn≥0 that can be expressed by a linear homogeneous recursion relation that is also a divisible sequence. A sequence fangn≥0 is called a divisible sequence if it has the property that if njm, then anjam. A sequence of numbers fangn≥0 is called a linear homogeneous recurrence sequence of order m if it can be written in the form an+m = p1an+m−1 + p2an+m−2 + ··· + pm−1an+1 + pman; n ≥ 0; for some constants p1; p2; :::; pm with pm 6= 0 and initial conditions a0; a1; :::; am−1. We focus on taking products, powers, and products of powers of second order linear divisible sequences in order to construct higher order linear divisible sequences. We hope to find a pattern in these constructions so that we can easily form higher order linear divisible sequence. iii ACKNOWLEDGEMENTS First, I would like to thank Dr. Peter Shiue for being an excellent advisor. He helped me come up with a topic for this thesis, which was both interesting and difficult. I would also like to thank the other members of my committee: Dr. Derrick DuBose, Dr. Arthur Baragar, and Dr. Yi-Tung Chen. I would also like to thank Daniel Corral for all the advice he gave me and believing that that I could finish even when I was worried that I could not. I would like to thank Daniel Lautzenheiser for introducing me to Sage, the computer algebra program, which, without its computational efficiency, I would not have been able to complete much of the work in this thesis. I would also like to thank Edward Huynh for helping me with editing this thesis. iv TABLE OF CONTENTS ABSTRACT iii ACKNOWLEDGEMENTS iv LIST OF TABLES vi 1 INTRODUCTION 1 2 SECOND ORDER LINEAR DIVISIBLE SEQUENCES4 3 PRODUCTS OF SECOND ORDER LINEAR DIVISIBLE SEQUENCES 10 3.1 Product of Two Distinct Second Order Linear Divisible Sequences ............... 15 3.2 Product of Three Distinct Second Order Linear Divisible Sequences............... 20 3.3 Product of Four Distinct Second Order Linear Divisible Sequences ............... 28 4 POWERS OF SECOND ORDER LINEAR DIVISIBLE SEQUENCES 48 4.1 Square of a Second Order Linear Divisible Sequences....................... 48 4.2 Cube of a Second Order Linear Divisible Sequences........................ 52 4.3 Fourth Power of a Second Order Linear Divisible Sequences................... 56 4.4 Fifth Power of a Second Order Linear Divisible Sequences.................... 60 4.5 Sixth Power of a Second Order Linear Divisible Sequences.................... 65 5 PRODUCTS OF POWERS 71 5.1 Product of the Square of a Second Order Times a Second Order ................ 71 5.2 Product of the Squares of Two Second Order ........................... 78 6 POLYNOMIAL LINEAR DIVISIBLE SEQUENCES 89 6.1 Products of Polynomial Linear Divisible Sequences........................ 89 6.2 Powers of Polynomial Linear Divisible Sequences......................... 91 6.3 Products of Powers of Polynomial Linear Divisible Sequences .................. 94 7 CONCLUSION 97 APPENDIX: COEFFICIENTS PRODUCT FOUR SEQUENCES 103 BIBLIOGRAPHY 104 CURRICULUM VITAE 105 v LIST OF TABLES 3.1 Terms of the sequence fwn = FnNng ................................ 19 3.2 Terms of the sequence fwn = PnNng ................................ 20 3.3 Terms of the sequence fwn = MnNng ............................... 20 3.4 Terms of the sequence fwn = FnPnMng .............................. 28 3.5 Terms of the sequence fwn = FnPnMnNng ............................ 47 2 4.1 Terms of the sequence wn = Fn ................................. 51 2 4.2 Terms of the sequence wn = Pn ................................. 51 2 4.3 Terms of the sequence wn = Mn ................................. 51 2 4.4 Terms of the sequence wn = Nn ................................. 52 3 4.5 Terms of the sequence wn = Fn ................................. 55 3 4.6 Terms of the sequence wn = Pn ................................. 55 3 4.7 Terms of the sequence wn = Mn ................................. 56 3 4.8 Terms of the sequence wn = Nn ................................. 56 4 4.9 Terms of the sequence wn = Fn ................................. 59 4 4.10 Terms of the sequence wn = Pn ................................. 59 4 4.11 Terms of the sequence wn = Mn ................................. 60 4 4.12 Terms of the sequence wn = Nn ................................. 60 5 4.13 Terms of the sequence wn = Fn ................................. 64 5 4.14 Terms of the sequence wn = Pn ................................. 64 5 4.15 Terms of the sequence wn = Mn ................................. 64 5 4.16 Terms of the sequence wn = Nn ................................. 65 6 4.17 Terms of the sequence wn = Fn ................................. 69 6 4.18 Terms of the sequence wn = Pn ................................. 69 6 4.19 Terms of the sequence wn = Mn ................................. 70 6 4.20 Terms of the sequence wn = Nn ................................. 70 2 5.1 Terms of the sequence wn = Fn Pn ................................ 76 2 5.2 Terms of the sequence wn = Pn Fn ................................ 77 2 5.3 Terms of the sequence wn = Fn Mn ............................... 77 2 5.4 Terms of the sequence wn = MnFn ............................... 77 2 5.5 Terms of the sequence wn = Pn Mn ............................... 78 2 5.6 Terms of the sequence wn = MnPn ............................... 78 2 2 5.7 Terms of the sequence wn = Fn Pn ................................ 86 2 2 5.8 Terms of the sequence wn = Fn Mn ............................... 86 2 2 5.9 Terms of the sequence wn = Fn Nn ................................ 87 2 2 5.10 Terms of the sequence wn = Pn Mn ............................... 87 2 2 5.11 Terms of the sequence wn = Pn Nn ................................ 87 2 2 5.12 Terms of the sequence wn = MnNn ............................... 88 7.1 Products of second order linear divisible sequences to make a specific order .......... 99 vi CHAPTER 1 INTRODUCTION In this thesis we examine the construction of higher order linear divisible sequences. A linear divisible sequence is any sequence of numbers fangn≥0 that can be expressed as a linear homogeneous recurrence relation that is also a divisible sequence. We also look at polynomial linear divisible sequences. A polyno- mial linear divisible sequence is any sequence of polynomials fan(x)gn≥0 that can be expressed as a linear homogeneous recurrence relation that is also a divisible sequence. For the rest of this thesis, we will define fang to mean fangn≥0 and fan(x)g to mean fan(x)gn≥0. A sequence of numbers fang is called a divisibility sequence if it has the property that whenever njm, then anjam. Our definition of divides in the integral domain states that if R is an integral domain and a; b 2 R, then we say ajb if there exists k 2 R such that ak = b. Thus, if fang is a sequence of elements of the ring of integers Z, then anjam means there is a k 2 Z such that ank = am.