PROOFS YOU CAN COUNT on by Helen K. Jenne a Thesis Submitted
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2017 Kmumc (Pdf)
Sixth Annual Kennesaw Mountain Undergraduate Mathematics Conference Program and Abstracts February 17{18, 2017 Welcome Welcome to the sixth annual Kennesaw Mountain Undergraduate Mathematics Conference! We are thrilled that this year KMUMC attracted over 135 participants from 21 universities in 6 states! We hope you will enjoy the talks, activities, food, great weather, and the beautiful Kennesaw State University campus and come back next year! We would also appreciate any feedback and any suggestions you have. Please fill out the feedback form included in your registration materials or send comments to Dr. Yuliya Babenko ([email protected]). Sincerely, KMUMC Organizing Committee Yuliya Babenko Ken Keating Sandra R. Chandler David Glassmeyer Erik E. Westlund Ludmila Orlova-Shokry Dhruba Adhikari Accessing KSU WiFi Network 1. Select \KSUGuest" from the list of available wireless networks. 2. Enter \kennesaw" as security key. 3. Open a web browser. 4. Login with your email address. 5. You are now connected to the WiFi network. Note: Guests have limited bandwidth, will only be able to access the Network between 6am and midnight, and are restricted to Internet connectivity through a web browser. Location of Talks The conference will take place in the Clendenin, Science, and Science Lab Buildings, abbreviated CL, SC, and SL, respectively. See the campus map in your registration packet for directions. Registration and all breaks will be in the Clendenin and Science Lab Building Atriums (CL 1000 and SL 1001). KSU Campus Bookstore The KSU Campus Bookstore is located about 200 yards from the main entrance to the Science building (see parking map on the next page). -
Note on Ward-Horadam H (X)-Binomials' Recurrences And
Note on Ward-Horadam H(x)- binomials' recurrences and related interpretations, II Andrzej Krzysztof Kwa´sniewski Member of the Institute of Combinatorics and its Applications, Winnipeg, Manitoba, Canada PL-15-674 Bia lystok, Konwaliowa 11/11, Poland e-mail: [email protected] Summary This note is a continuation of [48, 2010]. Firstly, we propose H(x)-binomials' recurrence formula appointed by Ward-Horadam H(x) = hHn(x)in≥0 functions' sequence i.e. any functions' sequence solution of the second order recurrence with functions' coefficients. As a method this comprises H ≡ H(x = 1) num- ber sequences V -binomials' recurrence formula determined by the primordial Lucas sequence of the second kind V = hVnin≥0 as well as its well elaborated companion fundamental Lucas sequence of the first kind U = hUnin≥0 which gives rise in its turn to the known U-binomials' recurrence as in [1, 1878] , [6, 1949], [8, 1964], [12, 1969], [17, 1989] or in [18, 1989] etc. Then we deliver a new type H(x)-binomials' "mixed" recurrence, straightforward n n formula for special case of Ward-Horadam sequences H(x) = hp (x) + q (x)in≥0. Here general H(x)-binomials' array is appointed by Ward-Horadam sequence of functions which in predominantly considered cases where chosen to be polyno- mials. For the sake of combinatorial interpretations and in number theory H(x = 1) = hHn(x = 1)in≥0 is usually considered to be natural or integer numbers valued sequence. Number sequences H = H(x = 1) = hHnin≥0 were also recently called by several authors: Horadam sequences. -
Combinatorially Thinking SIMUW 2008: July 14–25
Combinatorially Thinking SIMUW 2008: July 14–25 Jennifer J. Quinn [email protected] Philosophy. We want to construct our mathematical understanding. To this end, our goal is to situate our problems in concrete counting contexts. Most mathematicians appreciate clever combinatorial proofs. But faced with an iden- tity, how can you create one? This course will provide you with some useful combinatorial interpretations, lots of examples, and the challenge of finding your own combinatorial proofs. Throughout the next two weeks, your mantra should be to keep it simple. Contents 1 Basic Tools 3 1.1 SomeCombinatorialInterpretations . ........... 3 1.2 Counting Technique 1: Ask a Question and Answer Two Ways . ........... 6 1.3 Counting Technique 2: Description-Involution-Exception(DIE) ............ 7 2 Binomial Identities 8 2.1 WorkingTogether ................................. .... 8 2.2 OnYourOwn....................................... 9 n 2.3 What’s the Parity of k ? ................................. 10 3 Fibonacci Identities 13 3.1 TilingwithSquaresandDominoes . ........ 13 3.2 WorkingTogether ................................. .... 14 3.3 OnYourOwn....................................... 15 3.4 CombinatorialProofofBinet’sFormula . ........... 16 4 Generalizations—Lucas, Gibonacci, and Linear Recurrences 18 4.1 MotivatingGeneralization . ......... 18 4.2 WeightingtheTilings ... ... .. ... ... ... ... ... .. ... ...... 19 4.3 WorkingTogether ................................. .... 20 4.4 OnYourOwn...................................... -
The Tilings of a (2 × N)-Board and Some New Combinatorial Identities
1 2 Journal of Integer Sequences, Vol. 20 (2017), 3 Article 17.5.4 47 6 23 11 The Tilings of a (2 × n)-Board and Some New Combinatorial Identities Reza Kahkeshani Department of Pure Mathematics Faculty of Mathematical Sciences University of Kashan Kashan Iran [email protected] Abstract We know that the Fibonacci numbers count the tilings of a (1×n)-board by squares and dominoes, or equivalently, the number of tilings of a (2×n)-board by dominoes. We use the tilings of a (2 × n)-board by colored unit squares and dominoes to obtain some new combinatorial identities. They are generalization of some known combinatorial identities and in the special case give us the Fibonacci identities. 1 Introduction The Fibonacci numbers (A000045), (Fn)n≥0, are the sequence defined by F0 = 0, F1 = 1, and Fn = Fn−1 + Fn−2 for n ≥ 2 [3, 4]. It is easy to see that the tilings of a (1 × n)-board with (1 × 1) squares and (1 × 2) dominoes can be counted by the Fibonacci numbers. In fact, if fn counts such tilings then fn = Fn+1. From now on, we use the word “square(s)” to mean “(1 × 1) square(s)”. The Pell numbers (A000129) are defined by P0 = 0, P1 = 1, and for n ≥ 2, Pn = 2Pn−1 + Pn−2. The Pell number Pn+1 counts the tilings of a (1 × n)-board with dominoes and two colors of squares. Benjamin, Quinn, Plott, and Sellers [1, 2] proved a number of combinatorial identities using these interpretations. McQuistan and Lichtman [6] also studied the tilings of a (2 × n)-board by squares and dominoes for placing dimers on a lattice and proved that kn =3kn−1 + kn−2 − kn−3 1 for n ≥ 3, where kn is the number of such tilings (see Cipra’s comment on (A030186) in OEIS [7]). -
Arxiv:1510.03159V3 [Math.CO] 6 Aug 2016
ANALOGUES OF A FIBONACCI-LUCAS IDENTITY GAURAV BHATNAGAR Abstract. Sury’s 2014 proof of an identity for Fibonacci and Lucas numbers (Identity 236 of Benjamin and Quinn’s 2003 book: Proofs that count: The art of combinatorial proof) has excited a lot of comment. We give an alternate, telescoping, proof of this—and associated— identities and generalize them. We also give analogous identities for other sequences that satisfy a three-term recurrence relation. 1. Lucas’ 1876 identity and its associates Let Fk and Lk be the kth Fibonacci and kth Lucas numbers, respectively, for k = 0, 1, 2,... Both sequences of numbers satisfy a common recurrence relation, for n = 0, 1, 2,... xn+2 = xn+1 + xn; with differing initial values: F0 = 0, F1 = 1 but L0 = 2,L1 = 1. Fibonacci numbers satisfy many identities. The classic 1876 identity is due to Lucas himself: F0 + F1 + F2 + · · · + Fn−1 = Fn+1 − 1, which (in view of the relation Fk+1 + Fk−1 = Lk, for k ≥ 1) can be written as n (Lk − Fk+1)= Fn+1, (1.1) k X=0 where we used 1 = L0 − F1, F0 = L1 − F2, F1 = L2 − F3, and so on. Next consider Identity 236 of Benjamin and Quinn [4, p. 131]: n k n+1 2 Lk = 2 Fn+1. (1.2) k X=0 A recent proof of this identity by Sury [15] excited a lot of comment. Kwong [10] gave an arXiv:1510.03159v3 [math.CO] 6 Aug 2016 alternate proof, and Marques [11] (see also Martinjak [12]) found an analogous identity that replaces 2 by 3. -
View This Volume's Front and Back Matter
AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL 27 CountProofs the Really The Art of Combinatorial Proof Arthur T. Benjamin Jennifer J. Quinn i i \master" – 2012/1/30 – 15:35 – page i – #1 i i 10.1090/dol/027 Proofs That Really Count The Art of Combinatorial Proof i i i i i i \master" – 2012/1/30 – 15:35 – page ii – #2 i i c 2003 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 2003108524 Print edition ISBN: 978-0-88385-333-7 Electronic edition ISBN: 978-1-61444-208-0 Printed in the United States of America Current Printing (last digit): 10 9 8 7 6 5 4 i i i i i i \master" – 2013/4/23 – 18:00 – page iii – #3 i i The Dolciani Mathematical Expositions NUMBER TWENTY-SEVEN Proofs That Really Count The Art of Combinatorial Proof Arthur T. Benjamin Harvey Mudd College and Jennifer J. Quinn University of Washington Tacoma Published and Distributed by The Mathematical Association of America i i i i i i \master" – 2012/1/30 – 15:35 – page iv – #4 i i DOLCIANI MATHEMATICAL EXPOSITIONS Committee on Publications GERALD ALEXANDERSON, Chair Dolciani Mathematical Expositions Editorial Board DANIEL J. VELLEMAN, Editor EDWARD J. BARBEAU DONNA L. BEERS ROBERT BURCKEL ROBERT DEVANEY JERROLD GROSSMAN LESTER H. LANGE SANFORD SEGAL WILLIAM S. ZWICKER i i i i i i \master" – 2012/1/30 – 15:35 – page v – #5 i i The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathematical Association of America was established through a generous gift to the Association from Mary P.