Contentspga/conference.pdf · History of our Association and Quarterly The Fibonacci Association is...

Contents

1 History of our Association and Quarterly 1

2 Conference Program 5

3 Abstracts 13

Peter G. Anderson, More Properties of the Zeckendorf Array . . 13

Suman Balasubramanian, A simple Binet Form for GeneralizedFibonacci and Fibonacci Type Numbers and Applications toGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Christian Ballot, Lucas Sequences and Some Classical Congru-ences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Arthur Benjamin, Mathemagics: Secrets of Mental Math, & theArt of Mental Calculation . . . . . . . . . . . . . . . . . . . . 15

Arthur Benjamin, Fibonomial Identities in Search of Combina-torial Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Gerald Bergum and Marjorie Johnson, The Fibonacci Asso-ciation and its Quarterly . . . . . . . . . . . . . . . . . . . . 16

Charles K. Cook and Michael R. Bacon, Some PolygonalNumber Summation Formulas . . . . . . . . . . . . . . . . . . 16

3

4 CONTENTS

Charles K. Cook, Michael R. Bacon, and Sharon M. Mos-grove, Some Pyramidal and Other Higher Dimensional Fig-urate Number Summation Formulas . . . . . . . . . . . . . . . 16

Curtis Cooper, Mersenne Primes and GIMPS . . . . . . . . . . 17

Curtis Cooper, Algebraic Statements Similar to Those in Ra-manujan’s “Lost Notebook” . . . . . . . . . . . . . . . . . . . . 18

Daryl DeFord, Enumerating Distinct Chessboard Tilings . . . . 19

Karl Dilcher, Zeros and Irreducibility of Chebyshev-Like Polyno-mials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Karl Dilcher, Pairs of Reciprocal Quadratic Congruences Involv-ing Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Patrick Dynes, Xixi Edelsbrunner, and Kimsy Tor, Benford-ness of Generalized Zeckendorf Decompositions . . . . . . . . 20

Larry Ericksen, Patterns in Art . . . . . . . . . . . . . . . . . . 21

Larry Ericksen, Reducibility and Irreducibility of (0, 1) Stern Poly-nomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Mirac Cetin Firengiz and Naim Tuglu, On q-Seidel Matrix . 22

Rigoberto Florez, Using Fibonacci Numbers to Count Dyck Paths 22

Josep Freixas and Sascha Kurz, Fibonacci Sequences and theGolden Number in Voting Systems . . . . . . . . . . . . . . . 23

Heather M. Gaddy and Kenneth S. Berenhaut, Propagationof Values in Binary Sequences . . . . . . . . . . . . . . . . . 24

George Grossman and Yifan Zhang, Diophantine Triples andthe Extendibility of {1, 2, 5}, {1, 5, 10} . . . . . . . . . . . . . 24

Jaime Gutierrez, On the Linear Complexity and Lattice Test ofNonlinear Pseudorandom Number Generators . . . . . . . . . 24

CONTENTS 5

Nathan Hamlin and William Webb, Exotic Compositions Whichare Counted by Recurrence Sequences . . . . . . . . . . . . . . 25

Heiko Harborth, Generalized Pascal Steinhaus Triangles . . . . 25

Russell Jay Hendel, Jump Sum Formulae and Recursions . . . . 26

Mohand Ouamar Hernane and Jean-Louis Nicolas, Zeros ofCyclotomic Polynomials of Fibonacci Type . . . . . . . . . . . 27

Ben Kaufman, Brian McDonald and Madeleine Weinstein,Cookie Monster Meets the Fibonacci Numbers Mmmmmm —Theorems! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Clark Kimberling and Peter Moses, The Infinite FibonacciTree and Other Trees Generated by Rules . . . . . . . . . . . . 28

Ron Knott, Fibonacci and Phi: Puzzles, Pictures and Proof . . . 29

Takao Komatsu, Higher-Order Identities for Fibonacci Numbers 29

Takao Komatsu, Computing Arbitrary Integer Powers for CertainType of Band Matrices with Fibonacci Numbers . . . . . . . . 29

Jeffrey C. Lagarias, Facts and Conjectures about Factorizationsof Fibonacci and Lucas Numbers . . . . . . . . . . . . . . . . . 30

Kalman Liptai, Generalizations of Balancing Numbers . . . . . . 30

Karyn McLellan, A New Computation of Viswanath’s Constant 31

Steven J. Miller, Mind the Gap: Distribution of Gaps in Gener-alized Zeckendorf Decompositions . . . . . . . . . . . . . . . . 31

Sam Northshield, Two Analogues of Stern’s Diatomic Sequence . 32

Tomasz Nowicki, Fibonacci Unimodal Maps . . . . . . . . . . . . 33

Gopal Krishna Panda, Balancing-like Sequences Associated withIntegral Standard Deviations of Consecutive Natural Numbers 34

6 CONTENTS

C. N. Phadte and S. P. Pethe, Trigonometric Pseudo FibonacciSequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Marc Renault, Symmetries of Fibonacci Points Modulo m . . . 36

Anthony G. Shannon, Charles K. Cook, and Rebecca A.Hillman, An Extension of Some Results Due to Jarden . . . 36

Anthony G. Shannon and Krassimir T. Atanassov, SecondType of Pulsated Fibonacci Sequence . . . . . . . . . . . . . . 36

Lawrence Somer, On Primes in Lucas Sequences . . . . . . . . . 37

Pante Stanica, Counting Affine Equivalence Classes in PrimePower Dimension . . . . . . . . . . . . . . . . . . . . . . . . . 38

Marton Szikszai, A Generalization of the Jacobsthal Function . . 39

William Webb and Nathan Hamlin, When is the Number ofCompositions Given by a Recurrence Sequence? . . . . . . . . 39

William Webb, A Wiki for Fibonacci Identities . . . . . . . . . . 40

Michael D. Weiner, Daniel Birmajer, and Juan B. Gil, OnConvolutions of Linear Recurrence Sequences . . . . . . . . . 40

Paul Young, Symmetries of Stirling Number Series . . . . . . . . 40

4 List of Participants 41

Chapter 1

History of our Association andQuarterly

The Fibonacci Association is a non-profit 501(c)(3) corporation, incorporatedin 1962 by Verner E. Hoggatt, Jr. and I. Dale Ruggles, San Jose State College;and Brother Alfred Brousseau, St. Mary’s College. The Fibonacci Quarterlywas first published in 1963 with Editor V. E. Hoggatt, Jr. and ManagingEditor Brother Alfred. V. E. Hoggatt was editor for 18 years, 1963–1980;Gerald E. Bergum, editor for 18 years, 1980–1998; and current editor, CurtisCooper, 1998 to date.

Brother Alfred was treasurer 1963–1975; Leonard Klosinski, 1976–1979; Mar-jorie Bicknell-Johnson, 1979–1998; and current treasurer, Peter G. Anderson,1998 to date. Marjorie Bicknell-Johnson was secretary 1963–2010; currentsecretary, Art Benjamin, 2010 to date.

The International Conferences on Fibonacci Numbers and Their Applicationshave been held for 20 years.

1984 Patras, Greece

1986 San Jose State, San Jose, California

1988 Pisa, Italy

1990 Wake Forest, North Carolina

1

2 CHAPTER 1. HISTORY OF OUR ASSOCIATION AND QUARTERLY

1992 St. Andrews University, Scotland

1994 Washington State, Pullman, Washington

1996 Technische Universitat, Graz, Austria

1998 Rochester Institute of Technology, Rochester, New York

2000 Luxembourg City, Luxembourg

2002 North Arizona University, Flagstaff, Arizona

2004 Technische Universitat, Braunschweig, Germany

2006 San Francisco State, San Francisco, California

2008 University of Patras, Greece

2010 Universidad Nacional Autonoma de Mexico, Morelia, Mexico

2012 Eszterhazy Karoly College, Eger, Hungary

2014 Back to Rochester Institute of Technology

by Marjorie Bicknell-Johnson and Gerald E. Bergum

4 CHAPTER 1. HISTORY OF OUR ASSOCIATION AND QUARTERLY

Chapter 2

Conference Program

Sunday, July 20

7 PM–9 PM Welcome! Wine & Cheese Reception

University Gallery

Vignelli Center for Design Studies

5

6 CHAPTER 2. CONFERENCE PROGRAM

Monday, July 21

8:30 AM–Noon Morning Session moderated by Peter Anderson

8:30 AM Welcome from RIT’s President Bill Destler

9:00 AM Marton Szikszai: A Generalization of the Jacobsthal Func-tion

9:30 AM William Webb and Nathan Hamlin: When is the Num-ber of Compositions Given by a Recurrence Sequence?

10:00 AM Nathan Hamlin and William Webb: Exotic Composi-tions Which are Counted by Recurrence Sequences

10:30 AM Break

11:00 AM Paul Young: Symmetries of Stirling Number Series

11:30 AM Tomasz Nowicki: Fibonacci Unimodal Maps

12:00 PM Lunch

1:30 PM–5:30 PM Afternoon Session moderated by Bill Webb

1:30 PM Karl Dilcher: Pairs of Reciprocal Quadratic CongruencesInvolving Primes

2:00 PM Pante Stanica: Counting Affine Equivalence Classes inPrime Power Dimension

2:30 PM Lawrence Somer: On Primes in Lucas Sequences

3:00 PM Break

3:30 PM Anthony G. Shannon, Charles K. Cook, and RebeccaA. Hillman: An Extension of Some Results Due to Jarden

4:00 PM Peter G. Anderson: More Properties of the ZeckendorfArray

4:30 PM Gopal Krishna Panda: Balancing-like Sequences Associ-ated with Integral Standard Deviations of Consecutive Nat-ural Numbers

5:00 PM Suman Balasubramanian: A simple Binet Form for Gen-eralized Fibonacci and Fibonacci Type Numbers and Appli-cations to Graphs

7

Tuesday, July 22

8:30 AM–Noon Morning Session moderated by Clark Kimberling

8:30 AM Sam Northshield: Two Analogues of Stern’s DiatomicSequence

9:00 AM Christian Ballot: Lucas Sequences and Some ClassicalCongruences

9:30 AM Mohand Ouamar Hernane and Jean-Louis Nico-las: Zeros of Cyclotomic Polynomials of Fibonacci Type

10:00 AM Group PhotoMeet in atrium of the Golisano building

10:30 AM Break

11:00 AM Ron Knott: Fibonacci and Phi: Puzzles, Pictures andProof

12:00 PM LunchBoard Meeting Global Village conference room

1:30 PM–4:30 PM Afternoon Session moderated by Pante Stanica

1:30 PM Karl Dilcher: Zeros and Irreducibility of Chebyshev-Like Polynomials

2:00 PM Clark Kimberling and Peter Moses: The InfiniteFibonacci Tree and Other Trees Generated by Rules

2:30 PM Curtis Cooper: Algebraic Statements Similar to Thosein Ramanujan’s “Lost Notebook”

3:00 PM Break

3:30 PM Anthony G. Shannon and Krassimir T.Atanassov: Second Type of Pulsated FibonacciSequence

3:45 PM Gerald Bergum and Marjorie Johnson: The Fi-bonacci Association and its Quarterly

4:45 PM–10:00 PM Dinner Cruise on the Erie Canal (optional)

4:45 PM Board bus


Wednesday, July 23

9:00 AM–Noon Morning Session moderated by Karl DilcherThis session is open to the public.Talks be held in the Main Auditorium,Golisano Building, Ground Floor

9:00 AM The Edouard Lucas Memorial LectureJeffrey C. Lagarias: Facts and Conjectures about Fac-torizations of Fibonacci and Lucas Numbers

10:00 AM Larry Ericksen: Patterns in Art

10:30 AM Break

11:00 AM Arthur Benjamin: Mathemagics: Secrets of MentalMath, & the Art of Mental Calculation

12:00 Noon Genesee Country MuseumMeet at entrance of Golisano building(There are lunch restaurants at the Museum.)

6:00 PM Picnic at Peter and Jane Anderson’s home

9

Thursday, July 24

8:30 AM–Noon Morning Session moderated by Art Benjamin

8:30 AM Kalman Liptai: Generalizations of Balancing Numbers

9:00 AM Karyn McLellan: A New Computation of Viswanath’sConstant

9:30 AM Curtis Cooper: Mersenne Primes and GIMPS

10:00 AM Russell Jay Hendel: Jump Sum Formulae and Recur-sions

10:30 AM Break

11:00 AM Problem Session by Clark Kimberling

12:00 PM Lunch

1:00 PM–5:00 PM Afternoon Session moderated by Christian Ballot

1:00 PM Josep Freixas and Sascha Kurz: Fibonacci Sequencesand the Golden Number in Voting Systems

1:30 PM Arthur Benjamin: Fibonomial Identities in Search ofCombinatorial Proofs

2:00 PM Heiko Harborth: Generalized Pascal Steinhaus Trian-gles

2:30 PM Takao Komatsu: Higher-Order Identities for FibonacciNumbers

3:00 PM Break

3:30 PM Marc Renault: Symmetries of Fibonacci PointsModulo m

4:00 PM Michael D. Weiner, Daniel Birmajer, and Juan B.Gil: On Convolutions of Linear Recurrence Sequences

4:30 PM C. N. Phadte and S. P. Pethe: Trigonometric PseudoFibonacci Sequence

5:45 PM–9:45 PM Conference Banquet at Artisan Works

5:45 PM Board bus in Lot “S” by Global Village


Friday, July 25

8:30 AM–Noon Morning Session moderated by Curtis Cooper

8:30 AM Steven J. Miller: Mind the Gap: Distribution of Gaps inGeneralized Zeckendorf Decompositions

9:00 AM Jaime Gutierrez: On the Linear Complexity and LatticeTest of Nonlinear Pseudorandom Number Generators

9:30 AM Daryl DeFord: Enumerating Distinct Chessboard Tilings

10:00 AM William Webb: A Wiki for Fibonacci Identities

10:30 AM Break

11:00 AM Mirac Cetin Firengiz and Naim Tuglu: On q-SeidelMatrix

11:30 AM Rigoberto Florez: Using Fibonacci Numbers to CountDyck Paths

12:00 PM Lunch

1:00 PM–5:30 PM Afternoon Session moderated by Ron Knott

1:00 PM Ben Kaufman, Brian McDonald and Madeleine We-instein: Cookie Monster Meets the Fibonacci Numbers Mm-mmmm — Theorems!

1:30 PM Charles K. Cook and Michael R. Bacon: Some Polyg-onal Number Summation Formulas

2:00 PM Charles K. Cook, Michael R. Bacon, and Sharon M.Mosgrove: Some Pyramidal and Other Higher DimensionalFigurate Number Summation Formulas

2:30 PM Takao Komatsu: Computing Arbitrary Integer Powers forCertain Type of Band Matrices with Fibonacci Numbers

3:00 PM Break

3:30 PM Heather M. Gaddy and Kenneth S. Berenhaut: Prop-agation of Values in Binary Sequences

4:00 PM Patrick Dynes, Xixi Edelsbrunner, and Kimsy Tor:Benfordness of Generalized Zeckendorf Decompositions

4:30 PM Larry Ericksen: Reducibility and Irreducibility of (0, 1)Stern Polynomials

5:00 PM George Grossman and Yifan Zhang: DiophantineTriples and the Extendibility of {1, 2, 5}, {1, 5, 10}

11

Saturday, July 26

Trip to Niagara Falls (optional)

9:15 AM Bus pickup at Global Village

9:30 AM Bus pickup at RIT Inn

6:00 PM Return to RIT

Chapter 3

Abstracts

More Properties of the Zeckendorf Array

Peter G. Anderson

The Zeckendorf array contains every positive integer exactly once (Zeckendorf’s theo-rem). Extend the Zeckendorf array infinitely to the left by precursion, and the entriescorresponding to sums of negatively subscripted Fibonacci numbers give every non-zero integer exactly once (Bunder’s theorem). Finally, every pair of positive numbersoccurs exactly once in the entire two-way infinite array (Morrison’s theorem). We re-fine the third statement and show how to locate the given pairs in the array. Similarresults hold for related sequences: Tribonacci numbers, etc.

13

14 CHAPTER 3. ABSTRACTS

A simple Binet Form for Generalized Fibonacci andFibonacci Type Numbers and Applications to Graphs

Suman Balasubramanian

Fibonacci Numbers and generalized k− Fibonacci numbers are numbers of the formFn = Fn−1 +Fn−2 for n ≥ 0, F0 = 0, F1 = 1 and F k

n = F kn−1 + . . .+F k

n−k for k ≥ 2.Generating a Binet Formula for such numbers have been well studied. In this talk,we study the generation of a simplified Binet Formula for generalization of the aboveFibonacci numbers. We also study their relationships to Fibonacci type labelings ofgraphs.

Lucas Sequences and Some Classical Congruences

Christian Ballot

In an 1862 paper, Wolstenholme proved that for all primes p ≥ 5∑0<t<p

1

t≡ 0 (mod p2) and

(2p− 1

p− 1

)≡ 1 (mod p3).

These congruences have had many generalizations. But a new direction of general-ization for both congruences which uses Lucas sequences was discovered by Kimballand Webb in the 1990’s. We will review the work done in this direction within thelast six years before presenting further recent results.

15

Mathemagics: Secrets of Mental Math,& the Art of Mental Calculation

Arthur Benjamin

Dr. Arthur Benjamin is a mathematician and a magician. In his entertaining andfast-paced performance, he will demonstrate and explain how to mentally add andmultiply numbers faster than a calculator, how to figure out the day of the week ofany date in history, and other amazing feats of mind. He has presented his mixtureof math and magic to audiences all over the world.Dr. Benjamin has appeared on many television and radio programs, including TheToday Show, CNN, National Public Radio, and The Colbert Report. He has beenprofiled in The New York Times, Los Angeles Times, USA Today, Scientific American,Discover, Omni, Esquire, People Magazine, and Reader’s Digest. He has presented 3TED talks which have been viewed over 10 million times online.In their “Best of America” issue, Reader’s Digest recently called him “America’s BestMath Whiz.”

Fibonomial Identities in Search ofCombinatorial Proofs

Arthur Benjamin

In 2011, Bruce Sagan and Carla Savage derived two very nice combinatorial inter-pretations of Fibonomial coefficients in terms of tilings created by lattice paths. Webelieve that this interpretation should lead to combinatorial proofs of Fibonomialidentities. We provide a list of simple looking identities that are still in need ofcombinatorial proof.


The Fibonacci Association and its Quarterly

Gerald Bergum and Marjorie Johnson

The Fibonacci Quarterly is now in its 52nd year of publication. Jerry Bergum wasthe second Editor, serving for 18 years.Marjorie Johnson was Secretary of the Association 1963–2010 and Treasurer for 19of those years.They will bring the history of our group to life.

Some Polygonal Number Summation Formulas

Charles K. Cook and Michael R. Bacon

Various relationships involving the polygonal (or figurate) numbers are investigated.Several summation formulas for the general case as well as examples of specific typesof polygonal numbers are obtained.

Some Pyramidal and Other Higher DimensionalFigurate Number Summation Formulas

Charles K. Cook, Michael R. Bacon, and Sharon M. Mosgrove

Analogous to a paper by Cook and Bacon concerning various sums of the polygonal(figurate) numbers, similar sums will be considered for some pyramidal and otherhigher dimensional figurate–like numbers.

17

Mersenne Primes and GIMPS

Curtis Cooper

We will discuss Mersenne primes and the Great Internet Mersenne Prime Search(GIMPS). In particular, we will discuss the discovery of the Mersenne prime257885161 − 1.


Algebraic Statements Similar to Thosein Ramanujan’s “Lost Notebook”

Curtis Cooper

Ramanujan’s “lost notebook” contains algebraic statements

if g4 = 5, then5√

3 + 2g − 5√

4− 4g5√

3 + 2g + 5√

4− 4g= 2 + g + g2 + g3,

and

if g5 = 2, then√

1 + g2 =g4 + g3 + g − 1√

5.

In this paper we will discover algebraic statements similar to those in Ramanujan’s“lost notebook”. For example, we will prove algebraic statements like

if g5 = 2, then3√

5g2 + 1 + 3√

35g2 + g − 433√

5g2 + 1− 3√

35g2 + g − 43=

2 + g + 2g2 + 2g3 + 2g4

g,

and

if g5 = 8, then√

2g2 − 3 =g4 + 2g3 − 2g2 − 2

2√

5.

References:

• M. D. Hirschhorn and V. Sinescu, Elementary algebra in Ramanujan’s lost note-book, The Fibonacci Quarterly, 51.2 (2013), 123–129.

• S. Ramanujan, The Lost Notebook and Other Unpublished Papers, New Delhi,Narosa, 1988, p. 344.

19

Enumerating Distinct Chessboard Tilings

Daryl DeFord

Counting the number of distinct colorings of various discrete objects, via Burnside’sLemma and Polya Counting, is a traditional problem in combinatorics. We address arelated question for more general tiling situations: Given an m× n chessboard and afixed set of (possibly colored) tiles, how many distinct tilings exist, up to symmetry?More specifically, we are interested in the recurrent sequences formed by counting thenumber of distinct tilings of boards of size (m × 1), (m × 2), (m × 3), . . ., for a fixedset of tiles and some natural number m. The terms of these sequences can be usedto construct upper bounds on the orders of recurrences satisfied by other classes oftiling problems not reduced by symmetry.We present explicit results and closed forms for several well–known classes of tilingproblems, including domino tilings and tilings with squares of arbitrary sizes. Severalof these cases have convenient representations in terms of the combinatorial Fibonaccinumbers. Finally, we give a characterization of all 1 × n tiling problems in terms ofthe generalized Fibonacci numbers and colored Fibonacci tilings.

Zeros and Irreducibilityof Chebyshev-Like Polynomials

Karl Dilcher

By way of a nonlinear recurrence relations we define a sequence of polynomials re-sembling the Chebyshev polynomials of the first kind. Among other properties weobtain results on their irreducibility and zero distribution. We then study the 2 × 2Hankel determinants of these polynomials, which have interesting zero distributions.Furthermore, if these polynomials are split into two halves, then the zeros of one halflie in the interval (−1, 1), while those of the other half lie on the unit circle. Somefurther extensions and generalizations of these results are indicated.(Joint work with Kenneth B. Stolarsky.)


Pairs of Reciprocal Quadratic CongruencesInvolving Primes

Karl Dilcher

Using Pell equations and known solutions that involve Lucas sequences, we find allsolutions of the reciprocal pair of quadratic congruences p2 ≡ ±1 (mod q), q2 ≡ ±1(mod p) for odd primes p, q. In particular, we show that there is exactly one solution(p, q) = (3, 5) when the right-hand sides are −1 and 1. When the right-hand sides areboth −1, there are four known solutions, all of them pairs of Fibonacci primes, andwhen the right-hand sides are both 1, there are no solutions. With similar methodsone can completely characterize the solutions of p2 ≡ ±N (mod q), q2 ≡ ±N (mod p)for N = 2 and 4, and give partial results for N = 3 and 5. (Joint work with John B.Cosgrave).

Benfordness of Generalized ZeckendorfDecompositions

Patrick Dynes, Xixi Edelsbrunner, and Kimsy Tor

Zeckendorf proved that every positive integer can be expressed uniquely as a sumof non-consecutive Fibonacci numbers; this result has been extended to decompo-sitions arising from many other recurrence relations. The building blocks of thesedecompositions satisfy Benford’s law of digit bias, which means that the probabilityof a first digit of d in our sequence is log10(1 + 1/d). We discuss the Benfordness ofcertain subsets of Fibonacci numbers, such as those used in summands in Zeckendorfdecompositions, and will mention generalizations to other decompositions as timepermits.

21

Patterns in Art

Larry Ericksen

Mathematical patterns have been used over the centuries in artwork, from theParthenon and classical sculpture to folkloric textiles and famous paintings. Thegolden ratio has been used in art from classical times. The Fibonacci sequence pro-vided a compositional tool used in the golden rectangle and the Fibonacci spiral.Artists from Da Vinci and Michelangelo to Dali and Mondrian have employed thesemathematical structures.Several art movements based on these mathematical objects developed in Europeand North America. Within the fine arts and crafts, colored block patterns arearranged in paintings and quilts to evoke visual responses, emotional or intellectual.Colored squares in lattice patterns can be placed randomly or distributed accordingto mathematical rules by congruences and periodicities over a number array grid. Wealso examine the geometric constructs of polygon tessellations, Escher designs, andPenrose tilings.

Reducibility and Irreducibilityof (0, 1) Stern Polynomials

Larry Ericksen

As polynomial extensions of the classical Stern (diatomic) sequence, we define poly-nomials by at(2n;x) = at(n;xt) and at(2n + 1;x) = x at(n;xt) + at(n + 1;xt) at anyinteger t ≥ 2. These polynomials generalize the Stern polynomials a(n;x) at t = 2given by Dilcher and Stolarsky.Polynomials at(n;x) are Newman polynomials with only 0 and 1 as coefficients. Re-ducibility and irreducibility properties for these generalized Stern polynomials at(n;x)will be proven at t = 2 and t > 2. Cyclotomic polynomials will be identified as factorsof the reducible Stern polynomials. This is joint work with Karl Dilcher.


On q-Seidel Matrix

Mirac Cetin Firengiz and Naim Tuglu

In this paper, we define q-Seidel matrix by akn(x, q) = xqn+2k−3ak−1n (x, q) + ak−1n+1(x, q)with k ≥ 1, n ≥ 0 for an initial sequence a0n(x, q) = an(x, q), n ≥ 0. By using q-Seidelmatrix, we obtain several properties of q-analogs of the generalized Fibonacci andLucas polynomials.

Using Fibonacci Numbers to Count Dyck Paths

Rigoberto Florez

A Dyck word is a word in the letters X and Y with as many X’s as Y ’s and in whichno initial segment has more Y ’s than X’s. Each Dyck word gives rise to a path (Dyck)in the xy-plane. Dyck paths start at the origin, end on the x-axis, and do not crossthe x-axis. We say that a Dyck path P is non-decreasing if the y-coordinates of thelocal minima (valleys) of the path P form a non-decreasing sequence.Using power series in several variables we count the number of local maxima (peaks),the pyramid weights, and give some other statistics on non-decreasing Dyck paths interms of Fibonacci numbers.

23

Fibonacci Sequences and the Golden Numberin Voting Systems

Josep Freixas and Sascha Kurz

Binary voting systems, or simple games, are structures that, up to isomorphism,are countable as a function of the number of voters and / or different additionalparameters. In the late nineteenth century, Dedekind investigated the problem ofcounting certain Boolean functions (simple games) and got some seminal results, inthe mid-twentieth century May enumerated the class of symmetric games, in whichall voters play an equivalent role and therefore belong to the same equivalent class.For certain voting systems some scholars have been determined upper bounds fortheir number; however it is very difficult in general to determine the exact numberof them. Slightly surprising, Fibonacci sequences appear regularly for games withfew types of equivalent players and many of the counts differ asymptotically by amultiplicative factor which turns out to be the golden number or a power of it.It is nice to observe that these voting systems are very common in practice andare frequently used to govern many democratic institutions, as councils, counties,parliaments, but also in the boards of many private companies.The paper summarizes the known counts for significant classes of binary voting sys-tems that follow Fibonacci sequences.Basic References:

1. J. Freixas, X. Molinero, S. Roura. Complete voting systems with two classesof voters: weightedness and counting. Annals of Operations Research, 193:273-289, 2012.

2. J. Freixas, S. Kurz. The golden number and Fibonacci sequences in the designof voting systems. European Journal of Operational Research, 226: 246-257,2013.


Propagation of Values in Binary Sequences

Heather M. Gaddy and Kenneth S. Berenhaut

In this talk we consider the propagation of values in recursive binary sequences. Suchsequences have been heavily studied in the context of feedback shift registers. Herewe prove some results on convergence of sequences in terms of greatest commondivisors of elements in underlying delay sets. An application to difference equationsis included.

Diophantine Triples and the Extendibilityof {1, 2, 5}, {1, 5, 10}

George Grossman and Yifan Zhang

In this paper we consider Diophantine triples, (denoted D(n)-3-tuples,){1, 2, 5}, {1, 5, 10} for the case n = −1. We show by elementary methods, using prop-erties of Lucas and Fibonacci numbers that neither of 3-tuples {1, 2, 5}, {1, 5, 10} canbe extended to a D(−1)-4-tuple.

On the Linear Complexity and Lattice Test ofNonlinear Pseudorandom Number Generators

Jaime Gutierrez

In this paper we study several measures for asserting the quality of pseudorandomsequences, involving generalizations of linear complexity and lattice tests and relationsbetween them.

25

Exotic Compositions Which areCounted by Recurrence Sequences

Nathan Hamlin and William Webb

In the presentation “When is the number of compositions counted by a recurrencesequence?” a necessary and sufficient condition for this to occur was given, when theconstraint on the type of composition was the restriction that all parts must be ina given set S. Here we examine a number of examples of more exotic compositionssubject to other types of constraints where the number of compositions is still givenby a recurrence sequence. One way this can be done is by finding a one-to-onecorrespondence between compositions subject to the more exotic constraint, and onesknown to be counted by a recurrence.

Generalized Pascal Steinhaus Triangles

Heiko Harborth

Steinhaus triangles consist of a row with n plus and minus signs and in the followingrows a plus is written under two equal signs and a minus is written under two differentsigns. Hugo Steinhaus in 1963 has asked whether a first row can be chosen such thatthe numbers of plus and of minus signs are the same.With digits 0 and 1 instead of + and - the operation is as in a Pascal triangle modulo2. In a generalized Pascal triangle modulo 2 the operation is generalized to the sumof s consecutive digits of the preceding row. Here it will be discussed whether thereexist corresponding triangles with equal numbers of digits 0 and 1.


Jump Sum Formulae and Recursions

Russell Jay Hendel

For positive integers n, j and non-negative integer c define the jump function,

S(n, c, j) =∑

k≡c (mod j)

(n

k

),

so for example, for all n, S(n, 0, 1) = 2n, and for c = 0, 1, S(n, c, 2) = 2n−1.The 1974 Putnam exam asked for discovery that S(100, 1, 3) = 2100−1

3. Larson included

this problem in his book Problem-solving through Problems. At the 10th Fibonacciconference, Hillman and Cook, gave explicit formulae for S(n, c, j) for j = 3, 4, 5,where the formulae depend on the congruence classes of n and c modulo j. Hillmanand Cook also studied functions defined similarly to S(n, c, j) for other triangles ofcoefficients.This paper extends the Hillman-Cook results: (i) we introduce improved, more com-pact proof methods, (ii) we present (and prove) formulae for S(n, c, j) for j ≤ 20, forall n and c where formulae depend on congruence classes modulo j, (iii) we presentarrays of coefficients derived from these formulae and indicate several conjectures.To illustrate new formulae let j = 6, c = 2, and n ≡ i (mod 6) : Then

6S(n, 2, 6))− 2n = (−1)bn6cci27b

n6c + bi,

with 〈ci : 0 ≤ i ≤ 5〉 = 〈1, 3, 6, 9, 9, 0〉, and 〈bi : 0 ≤ i ≤ 5〉 = 〈−1, 1, 2, 1,−1− 2〉.Notice also that the sequence Gn = 6S(6n + k, 2, 6) − 26n+k, n ≥ 1, satisfies therecursion Gn = −27Gn−1 + dk28 with 〈dk : 0 ≤ k ≤ 4〉 = 〈−1,−1, 2, 1,−1〉. Similarrecursions are presented for the sequences Gn = jS(jn+k, c, j)−2jn+k and propertiesof the resulting triangle of coefficients are studied.

27

Zeros of Cyclotomic Polynomialsof Fibonacci Type

Mohand Ouamar Hernane and Jean-Louis Nicolas

Let us define Gn(z) = (ϕ − 1)zn+1 − ϕzn + 1 and Hn(z) = ϕzn+1 − (ϕ − 1)zn − 1where ϕ = (1 +

√5)/2, is the golden ratio.

We give explicit estimations of some roots of these two families of trinomials by twodifferent methods. We denote by x2, n the negative zero of Hn or Gn and by x1, n 6= 1,the positive zero of Gn(z). In this work, we prove that

x2,n = −1 +∑m≥1

cmnm

,

where c1 = −b, c2 = − b2

2+ b

2+ b

√5

10, c3 = − b3

6+ b2

√5

10+ 3b2

5− 3b

10− b

√5

10, . . . , with

b = − log 52.

Moreover the coefficients cm satisfy |cm| ≤ 54

(112

)m.

Also we prove that the zero x1, n, is given by the following hypergeometric series:

x1, n = ϕ2

(1−

∑k≥1

1

kϕ(2n+1)k

((n+ 1)k − 2

k − 1

)).


Cookie Monster Meets the Fibonacci NumbersMmmmmm — Theorems!

Ben Kaufman, Brian McDonald and Madeleine Weinstein

A beautiful theorem of Zeckendorf states that every positive integer can be writtenuniquely as a sum of non-consecutive Fibonacci numbers. Once this has been shown,it’s natural to ask how many Fibonacci numbers are needed. Lekkerkerker proved thatthe average number of such summands needed for integers in [Fn, Fn+1) is n/(φ2 + 1),where φ is the golden mean. We present a combinatorial proof of this through thecookie problem and differentiating identities, and further prove that the fluctuationsabout the mean are normally distributed and the distribution of gaps between sum-mands is exponentially decreasing. We discuss generalizations of this idea, both toother recurrences as well as extensions to subintervals in [Fn, Fn+1).

The Infinite Fibonacci Tree and Other TreesGenerated by Rules

Clark Kimberling and Peter Moses

Suppose that I is a subset of a set U and that C is a collection of operations f definedin U. Create a set S by these rules: every element of I is in S, and if x is in S, thenf(x) is in S for all f in C for which f(x) is defined. Then S “grows” in successivegenerations. If I consists of a single number r then S can be regarded as a tree withroot r. We examine several examples, including these: (1) 1 ∈ S, and if x ∈ S thenx + 1 ∈ S and 1/x ∈ S; (2) 1 ∈ S, and if x ∈ S then x + 1 ∈ S and 2x ∈ S; (3)1 ∈ S, and if x ∈ S then x + 1 ∈ S, and if x 6= 0 then −1/x ∈ S; (4) 1 ∈ S, andif x ∈ S then x + 1 ∈ S and

√−1x ∈ S, and if x 6= 0 then 1/x ∈ S. The first of

these examples is the infinite Fibonacci tree, in which every positive rational numberoccurs as a node.

29

Fibonacci and Phi: Puzzles, Pictures and Proof

Ron Knott

A tour of some of the motivations and newer features on my Fibonacci web site, withpuzzles to illustrate the many and varied mathematical places in which the Fibonaccinumbers appear, new “look-and-see” pictures of seed-heads to show the relationshipbetween phi and the Fibonacci numbers, with illustrations and animations I use whenpresenting motivational talks to maths specialists, schools and the general public.

Higher-Order Identities for Fibonacci Numbers

Takao Komatsu

We give some expressions for ∑j1+···+jr=nj1,...,jr≥1

Fj1 · · ·Fjr

and2r−3∑l=0

(2r − 3

l

) ∑j1+···+jr=n−2l

j1,...,jr≥1

Fj1 · · ·Fjr ,

where r ≥ 2 and n ≥ 3r − 5.

Computing Arbitrary Integer Powers for CertainType of Band Matrices with Fibonacci Numbers

Takao Komatsu

We discuss certain family of 4k-square (k = 1, 2, . . . ) band matrices whose band widthis 2k + 1, and the entries in their arbitrary powers for matrices are expressed by thegeneralized Fibonacci numbers only.


Facts and Conjectures about Factorizationsof Fibonacci and Lucas Numbers

Jeffrey C. Lagarias

Edouard Lucas Memorial Lecture

Fibonacci and Lucas numbers have highly structured primality and divisibility prop-erties, many of them worked out by Edouard Lucas. I will discuss work in his 1877book: “Recherches sur plusieurs ouvrages de Leonard de Pise et sur diverses ques-tions d’arithmetique superieure.” Fibonacci and Lucas numbers have since been usedas a testbed for primality and factorization algorithms, and I will survey later thingsfound out about their divisibility and factorization. Finally I will discuss the recentappearance of Fibonacci factorizations as a (conjectural) bogeyman in the theory ofinteger orbits of thin groups.

Generalizations of Balancing Numbers

Kalman Liptai

A positive integer n is called a balancing number if

1 + 2 + · · ·+ (n− 1) = (n+ 1) + (n+ 2) + · · ·+ (n+ r)

for some positive integer r. We study some generalizations of balancing numberswhich are called (a, b)-type balancing numbers, cobalancing numbers, multiplyingbalancing numbers or (k, l)-power numerical center. We give effective finiteness theo-rems for the polynomial values of (a, b)-balancing numbers. Specially, we investigatethe power values of (a, b)-type balancing numbers. We mention new results concerninggeneralizations of balancing numbers.

31

A New Computation of Viswanath’s Constant

Karyn McLellan

This talk will extend some ideas from both Viswanath’s and Rittaud’s work on randomFibonacci sequences. We can think of these sequences as forming a binary tree T .Viswanath has shown that almost all random Fibonacci sequences grow exponentiallyat the rate 1.13198824 . . .. We will discuss a new computation of Viswanath’s constantwhich uses a reduction R of the tree T developed by Rittaud. Further, we considerthe growth rate of the expected value of the nth term in a sequence, using the binarytrees R and T , and a Pascal-like array of numbers.

Mind the Gap: Distribution of Gapsin Generalized Zeckendorf Decompositions

Steven J. Miller

Zeckendorf proved that any integer can be decomposed into a unique sum of non-adjacent Fibonacci numbers, Fn. Using continued fractions, Lekkerkerker showedthat the average number of summands in a decomposition of an integer in [Fn, Fn+1)is essentially n/(φ2 + 1), where φ is the golden ratio. Miller-Wang generalized this byadopting a combinatorial perspective, proving that for any positive linear recurrenceof the form An = c1An−1 + c2An−2 + · · · + cLAn+1−L, the number of summands indecompositions for integers in [An, An+1) converges to a Gaussian distribution asn→∞.We prove that the probability of a gap larger than the recurrence length converges todecaying geometrically, with decay rate equal to the largest eigenvalue of the char-acteristic polynomial of the recurrence, and that the distribution of the smaller gapsdepends on the coefficients of the recurrence (which we analyze through the combi-natorial perspective). These results hold both for the average over all m ∈ [An, An1),as well as holding almost surely for the gap measure associated to individual m. Thetechniques work for related systems as well, and can also be used to determine the dis-tribution of the longest gap between summands (which is similar to the distribution ofthe longest gap between heads in tosses of a biased coin), as well as for far-differencerepresentations (where positive and negative summands are allowed).


Two Analogues of Stern’s Diatomic Sequence

Sam Northshield

Stern’s diatomic sequence (a1 = 1, a2n = an, a2n+1 = an +an+1) has many similaritiesto the Fibonacci sequence. Among them is a Binet type formula:

an+1 =n∑

k=0

σs2(k)σs2(n−k) (1)

where σ = (1 +√−3)/2, and s2(n) is the number of ones in the binary expansion of

n. We study the sequence formed by replacing, in formula (1), s2(n) by sF (n), thenumber of ones in the Zeckendorf representation of n.Also, we study the sequence defined by b1 = 0, b2n = bn, b2n+1 = bn ⊕ bn+1 wherem⊕ n = m+ n+

√4mn+ 1.

33

Fibonacci Unimodal Maps

Tomasz Nowicki

Palis’ conjecture states that for a typical dynamical system a typical behavior oftrajectories can be either converging to one of finitely many attracting periodic pointsor will describe one of the finitely many invariant measures absolutely continuous wrtthe Lebesgue measure.This conjecture was proven in the case of unimodal maps, to be more precise for thereal quadratic family.The unimodal maps are classified by their topological invariant, the kneading se-quence, i.e. the coding of the trajectory of the critical point into Left/Right symbolswhich say where the iterates land with respect to the critical point.This invariant provides the understanding how new branches of iterates are created.One topological type of unimodal maps is called Fibonacci maps as the new branchcreation happens at Fibonacci times, i.e. at the iterates fSn , where Sn is the Fibonaccisequence. The study of this type was crucial to provide the background of the proofthat the behavior which is neither periodic nor measure creating is not typical amongthe quadratic family.


Balancing-like Sequences Associated with IntegralStandard Deviations of Consecutive Natural Numbers

Gopal Krishna Panda

It is well known that the variance of first n natural numbers is sn = (n2 − 1)/12and is a natural number if n is odd but not a multiple of 3. It is interesting toexplore those N for which the standard deviation sN = σ is a natural number.This leads us to consider the Pell’s equation N2 − 12σ2 = 1 whose fundamentalsolution is N1 = 7 and σ1 = 2 and hence all solutions are given by Nk + 4

√3σk =

(7 + 4√

3)k; k = 1, 2, · · · . Putting k = 1, 2, · · · we infer that the standard deviationsof first 7, 97 and 1351 natural numbers are respectively 2, 28 and 390. The sequences{Nk} and {σk} satisfy the recurrence relation xk+1 = 14xk − xk−1, k ≥ 2. Thesequence {σk} is a balancing-like sequence and the corresponding Lucas-balancinglike sequence is {Nk}. Further, N2

k = 12σ2k + 1 and the sequence {Nk} and {σk}

enjoy certain interesting properties: σk+1 = 7σk + 2Nk, Nk+1 = 7Nk + 24σk, σk+l =σkNl +Nkσl, Nk+l = NkNl +12σkσl, σk−l.σk+l = σ2

k−σ2l , Nk−l.Nk+l +1 = N2

k +N2l and

2(σ1+σ3+· · ·+σ2l−1) = σ2l . For each natural number k, (Nk+1)/2 is a perfect square

and Mk =√

(Nk + 1)/2 satisfies the binary recurrence Mk+1 = 4Mk −Mk−1. Foreach, Mk divides σk and Lk = σk/Mk satisfies a recurrence relation identical to thatof Mk. Further, M2k = Nk,M2k+1 = σk+1− σk, L2k = 2σk and L2k+1 = (σk+1 + σk)/2.The sequence {Lk}∞k=1 is also balancing-like sequence, {Mk}∞k=1 is the correspondingLucas-balancing-like sequence and are related by M2

k = 3L2k + 1. These sequences

have several fascinating properties.

35

Trigonometric Pseudo Fibonacci Sequence

C. N. Phadte and S. P. Pethe

In this paper, we establish some results about second order non-homogeneous re-currence relation containing extended trigonometric function. Earlier [1], we provedsome properties of recurrence relation

Gn+2 = gn+1 + gn + Atn, n = 0, 1, . . .

with g0 = 0, g1 = 1, where both A 6= 0 and t 6= 0, and also t 6= α, β, where α, β arethe roots of x2 − x− 1 = 0.Using the properties of generalized circular functions and Elmore’s method, we definea new sequence {Hn} which is the extension of Pseudo Fibonacci Sequence, given byrecurrence relation

Hn+2 = pHn+1qHn +RtnNr,0(t∗x),

where Nr,0(tx) is extended circular function. We state and prove some properties forthis extended Pseudo Fibonacci Sequence {Hn}.Reference:[1]. Phadte, C. N., Pethe, S. P. On Second Order Non-Homogeneous RecurrenceRelation, Annales Mathematica et Informatica, 41 (2013) pp 205-210.


Symmetries of Fibonacci Points Modulo m

Marc Renault

Given a modulus m, we examine the set of all points (Fi, Fi+1) ∈ Z2m where F is

the usual Fibonacci sequence. We graph the set in the fundamental domain [0,m −1] × [0,m − 1], and observe that as m varies, sometimes the graph appears as arandom scattering of points, but often it shows striking symmetry. We prove that inexactly three cases (m = 2, 3, or 6) the graph shows symmetry by reflection acrossthe line y = x. We prove that symmetry by rotation occurs exactly when the terms0,−1 appear half-way through a period of F (mod m). We prove that symmetry bytranslation can occur in essentially one way, and we provide conditions equivalent tothe graph having symmetry by translation.

An Extension of Some Results Due to Jarden

Anthony G. Shannon, Charles K. Cook, and Rebecca A. Hillman

This paper defines some generalized Fibonacci and Lucas sequences which satisfyarbitrary order linear recurrence relations and which answer a problem posed byJarden in 1966 about generalizing an elegant result for a connection between evenand odd subscripted Fibonacci and Lucas numbers.

Second Type of Pulsated Fibonacci Sequence

Anthony G. Shannon and Krassimir T. Atanassov

In this note we define a new type of the notion of a “pulsated” Fibonacci sequence.Some examples are given.

37

On Primes in Lucas Sequences

Lawrence Somer

Consider the Lucas sequence u(a, b) = {un(a, b)} and the companion Lucas sequencev(a, b) = {vn(a, b)} which both satisfy the second order recursion relation

wn+2 = awn+1 − bwn

with initial terms u0 = 0, u1 = 1, and v0 = 2, v1 = a, respectively. We give bothnecessary and sufficient tests and also necessary tests for the primality of |un| and |vn|.For those tests which are only necessary, we show that these tests are not sufficient bymeans of a simple criterion using the Legendre symbol. These results are specializedto the Fibonacci numbers {Fn} and to the Lucas numbers {Ln}. In particular, wegeneralize a result of Drobot giving criteria for Fp not to be prime, where p is a prime,to the Lucas numbers {Ln}.


Counting Affine Equivalence Classesin Prime Power Dimension

Pante Stanica

A Boolean function f : Fn2 → F2 is called rotation symmetric if its algebraic normal

form (polynomial representation) is invariant under a cyclic permutations of indices.A monomial rotation symmetric function (MRS) of degree d is a rotation symmetricfunction generated by a single monomial x1xj2xj3 . . . xjd , that is,

f(x1, . . . , xn) = x1xj2xj3 . . . xjd + x2xj2+1xj3+1 . . . xjd+1 + · · · ,

where indices are taken Mod n (a Mod n is the unique integer b ∈ {1, 2, . . . , n} suchthat b ≡ a (mod n)).There are many papers dealing with the following (related) questions:

(A) Given two multivariable functions f, g, does there exist a permutation σ on thevariables such that f ◦ σ = g?

(B) How many classes of (such) equivalent functions are there?

Using a new method previously developed by the author, D. Canright and J. H.Chung, we are able to completely count the number of equivalence classes for degreethree, four and five MRS in prime power dimension... and more.

39

A Generalization of the Jacobsthal Function

Marton Szikszai

Let m be a positive integer. The ordinary Jacobsthal function j(m) is the least valuesuch that in every set of j(m) consecutive integers there exists an integer which iscoprime to m. This number theoretic function is in connection with the problem ofso-called prime coverings of sets; the gaps between consecutive primes; and with theproblem of the least prime in arithmetic progressions.A possible generalization of this function can be the following. Let B = (Bn)∞n=0 be asequence of arbitrary integers and let jB(m) be defined as the least number such thatin every set of jB(m) consecutive terms of B one can find a term which is coprimeto m. In this talk we investigate jB(m) in different families of well-known integersequences, like the Fibonacci sequence or the Lucas sequences.

When is the Number of CompositionsGiven by a Recurrence Sequence?

William Webb and Nathan Hamlin

Let un denote the number of compositions of n subject to given set of constraints. Formany constraints these numbers satisfy a linear recurrence equation. For example,each of the numbers which count compositions of n into either (a) parts 1 and 2, (b)parts greater than 1, (c) odd parts, all result in the Fibonacci sequence.We investigate more generally when the sequence un is a recurrence sequence. Inparticular we prove that if the constraint is that all parts of the composition mustbe in a set S, then un is a recurrence sequence if and only if S = S1 ∪ S2 where S1

is a finite set, and S2 = {k : k ≡ r1, · · · , rh (mod m), k ≥ mk0} for some integers0 ≤ k ≤ m, and k0 ≥ 0.


A Wiki for Fibonacci Identities

William Webb

Plans are hatching for the Fibonacci Association to develop an on-line tool (inspiredby the OEIS). This brief discussion will bring Fibonacci friends up to date on theplans as well as invite wide participation.

On Convolutions of Linear Recurrence Sequences

Michael D. Weiner, Daniel Birmajer, and Juan B. Gil

For arbitrary homogeneous linear recurrence sequences with fixed coefficients, we use arepresentation in terms of partial Bell polynomials to discuss combinatorial formulasfor multifold convolutions. In particular, we will discuss convolutions of arbitraryn-step Fibonacci sequences and Jacobsthal polynomials.

Symmetries of Stirling Number Series

Paul Young

We consider Dirichlet series generated by weighted Stirling numbers and demonstratea symmetry of such series which is reminiscent of a duality relation of negative-order poly-Bernoulli numbers. These series are connected to several types of zetafunctions and this symmetry plays a prominent role. We do not know whether thereare combinatorial explanations for this symmetry, as there are for the related poly-Bernoulli identity.

Chapter 4

List of Participants

Emrah AkyarAnadolu University, Eskisehir, [email protected]

Handan AkyarAnadolu University, Eskisehir, [email protected]

Jane AndersonRochester, NY, [email protected]

Peter G. AndersonRochester Institute of Technology, Rochester, NY, [email protected]

Suman BalasubramanianDePauw University, Greencastle, IN, [email protected]

41

42 CHAPTER 4. LIST OF PARTICIPANTS

Christian BallotUniversite Caen, Ifs, [email protected]

Arthur BenjaminHarvey Mudd College, Claremont, CA, [email protected]

Gerald BergumBrookings, SD, [email protected]

Mirac Cetin FirengizBaskent University, Ankara, Turkey, Ankara, [email protected]

Charles CookUniv of So Carolina Sumter, Sumter, SC, [email protected]

Curtis CooperUniversity of Central Missouri, Warrensburg, MO, [email protected]

Janet CooperRockhurst University, Kansas City, MO, [email protected]

Marjorie CooperJoplin, MO, USA(none)

43

Ashley DeFazioFibonacci Association Subscription Manager, Sun City, AZ, [email protected]

Daryl DeFordDartmouth College, Hanover, NH, [email protected]

Karl DilcherDalhousie University, Halifax, NS, [email protected]

Patrick DynesClemson University, Clemson, CA, [email protected]

Xixi EdelsbrunnerWilliams College, Williamstown, MA, [email protected]

Larry EricksenMillville, NJ, [email protected]

Rigoberto FlorezThe Citadel, The Military College of South Carolina, Charleston, SC, [email protected]

Josep FreixasTechnical University of Catalonia, Manresa, [email protected]


Heather GaddyWake Forest University, Winstron-Salem, NC, [email protected]

Dale GerdemannU. Tuebingen, Moessingen, [email protected]

George GrossmanCentral Michigan University, Mount Pleasant, MI, [email protected]

Susan GurneyNew York, NY, [email protected]

Jaime GutierrezUniversity of Cantabria, Santander, [email protected]

Ashley HamlinPullman, WA, [email protected]

Nathan HamlinWashington State University, Pullman, WA, [email protected]

Heiko HarborthTU Braunschweig, Braunschweig, [email protected]

45

Russell HendelTowson University, Baltimore, MD, [email protected]

Dahbia Boukari HernaneUniv. of Sciences & Tech. Houari Boumediene, Algiers, [email protected]

Mohand Ouamar HernaneUniv. of Sciences & Tech. Houari Boumediene, Algiers, [email protected]

Marjorie Bicknell JohnsonSanta Clara, CA, [email protected]

Virginia JohnsonColumbia College, Greenwood, SC, [email protected]

Ben KaufmanWilliams College, Williamstown, MA, [email protected]

Clark KimberlingUniversity of Evansville, Evansville, IN, [email protected]

Margaret KimberlingEvansville, IN, [email protected]


Ron KnottUniversity of Surrey, Guildford, Bolton, [email protected]

Takao KomatsuHirosaki University, Hirosaki, [email protected]

Jeffrey LagariasUniversity of Michigan, Ann Arbor, MI, [email protected]

Kalman LiptaiEsterhazy Karoly College, Eger, [email protected]

Lynda MartinRubery, Birmingham, [email protected]

Brian McDonaldUniversity of Rochester, Rochester, NY, [email protected]

Karyn McLellanSt. Francis Xavier University, Halifax, NS, [email protected]

Steven J. MillerWilliams College, Williamstown, MA, [email protected]

47

Peter MosesWorcestershire, [email protected]

Sam NorthshieldSUNY-Plattsburgh, Plattsburgh, NY, [email protected]

Tomasz NowickiIBM, Yorktown Heights, NY, [email protected]

Gopal Krishna PandaNational Institute of Technology, Rourkela, Rourkela, OR, Indiagkpanda [email protected]

Jesse L. PatsolicWake Forest University, Winston-Salem, NC, [email protected]

Chandrakant PhadteG.V.Ms College of Commerce & Eco., Nasik, [email protected]

Marc RenaultShippensburg University, Shippensburg, PA, [email protected]

Anthony ShannonUniversity of Technology, Sydney, Sydney, [email protected]


Lawrence SomerCatholic University of America, Prague, Czech [email protected]

Pantelimon StanicaNPS, Applied Mathematics, Seaside, CA, [email protected]

Paul StockmeyerThe College of William and Mary, Williamsburg, VA, [email protected]

Marton SzikszaiUniversity of Debrecen, Debrecen, [email protected]

Kimsy TorManhattan College, Riverdale, NY, [email protected]

Caroline Turnage-ButterbaughWilliams College & North Dakota State U., Oxford, MS, [email protected]

William WebbWashington State University, Moscow, ID, [email protected]

Michael WeinerPenn State Altoona, Altoona, PA, [email protected]

49

Madeleine WeinsteinHarvey Mudd College, Claremont, CA, [email protected]

Fatma YesilAnkara, [email protected]

Paul YoungCollege of Charleston, Charleston, SC, [email protected]

Contentspga/conference.pdf · History of our Association and Quarterly The Fibonacci Association is...

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Transcript of Contentspga/conference.pdf · History of our Association and Quarterly The Fibonacci Association is...