P. Z. Chinn Department of Mathematics, Humboldt State University, Arcata, CA
|
|
- Gabriella Brooks
- 5 years ago
- Views:
Transcription
1 RISES, LEVELS, DROPS AND + SIGNS IN COMPOSITIONS: EXTENSIONS OF A PAPER BY ALLADI AND HOGGATT S. Heubach Departmet of Mathematics, Califoria State Uiversity Los Ageles 55 State Uiversity Drive, Los Ageles, CA sheubac@calstatela.edu P. Z. Chi Departmet of Mathematics, Humboldt State Uiversity, Arcata, CA 955 phyllis@math.humboldt.edu R. P. Grimaldi Departmet of Mathematics, Rose-Hulma Istitute of Techology Terre Haute, IN ralph.grimaldi@rose-hulma.edu (Submitted March 00, Revised Jauary 00). INTRODUCTION A compositio of cosists of a ordered sequece of positive itegers whose sum is. A palidromic compositio (or palidrome) is oe for which the sequece reads the same forwards ad backwards. We derive results for the umber of + sigs, summads, levels (a summad followed by itself), rises (a summad followed by a larger oe), ad drops (a summad followed by a smaller oe) for both compositios ad palidromes of. This geeralizes a paper by Alladi ad Hoggatt [], where summads were restricted to be oly s ad s. Some results by Alladi ad Hoggatt ca be geeralized to compositios with summads of all possible sizes, but the coectios with the Fiboacci sequece are specific to compositios with s ad s. However, we will establish a coectio to the Jacobsthal sequece [8], which arises i may cotexts: tiligs of a 3 x board [], meets betwee subsets of a lattice [3], ad alteratig sig matrices [4], to ame just a few. Alladi ad Hoggatt also derived results about the umber of times a
2 particular summad occurs i all compositios ad palidromes of, respectively. Geeralizatios of these results are give i []. I Sectio we itroduce the otatio that will be used, methods to geerate compositios ad palidromes, as well as some easy results o the total umbers of compositios ad palidromes, the umbers of + sigs ad the umbers of summads for both compositios ad palidromes. We also derive the umber of palidromes ito i parts, which form a elarged Pascal s triagle. Sectio 3 cotais the harder ad more iterestig results o the umbers of levels, rises ad drops for compositios, as well as iterestig coectios betwee these quatities. I Sectio 4 we derive the correspodig results for palidromes. Ulike the case of compositios, we ow have to distiguish betwee odd ad eve. The fial sectio cotais geeratig fuctios for all quatities of iterest.. NOTATION AND GENERAL RESULTS We start with some otatio ad geeral results. Let C, P = the umber of compositios ad palidromes of, respectively C +, P + = the umber of + sigs i all compositios ad palidromes of, respectively C, s P s = the umber of summads i all compositios ad palidromes of, respectively C (x) = the umber of compositios of edig i x C (x, y) = the umber of compositios of edig i x + y r, l, d = the umber of rises, levels, ad drops i all compositios of, respectively r, l, d = the umber of rises, levels, ad drops i all palidromes of, respectively. We ow look at ways of creatig compositios ad palidromes of. Compositios of + ca be created from those of by either appedig + to the right
3 ed of the compositio or by icreasig the rightmost summad by. This process is reversible ad creates o duplicates, hece creates all compositios of +. To create all palidromes of, combie a middle summad of size m (with the same parity as, 0 m ) with a compositio of m o the left ad its mirror image o the right. Agai, the process is reversible ad creates o duplicates (see Lemma of []). We will refer to these two methods as the Compositio Creatio Method (CCM) ad the Palidrome Creatio Method (PCM), respectively. Figure illustrates the PCM Figure : Creatig palidromes of =6ad = We ca ow state some basic results for the umber of compositios, palidromes, + sigs ad summads. Theorem. C = for, C 0 :=.. P k = P k+ = k for k C + =( ) for, C + 0 := P + k+ = kk for k 0, P + k =(k )k for k, P + 0 := C s =( +),for, C s 0 :=. 3
4 6. P s k+ =(k +) k for k 0, P s k =(k +) k for k, P s 0 :=. Proof:. The umber of compositios of ito i parts is ( ) i (see Sectio.4 i [5]). Thus, for, C = ( ) =. i. Usig the PCM as illustrated i Figure, it is easy to see that P k = P k+ = C i =+(++ + k )= k. i=0 3. A compositio of with i summads has i + sigs. Thus, the umber of + sigs ca be obtaied by summig accordig to the umber of summads i the compositio: ( ) C + ( )! = (i ) = (i ) i i= (i )!( i)! ( ) = ( ) =( ). () i= i 4. The umber of + sigs i a palidrome of k+ is twice the umber of + sigs i the associated compositio, plus two + sigs coectig the two compositios with the middle summad. P + k+ = = (C i +C i + )= ( i +(i ) i ) (i +) i = k k, where the last equality is easily proved by iductio. For palidromes of k, thesame reasoig applies, except that there is oly oe + sig whe a compositio of k is combied with its mirror image. Thus, P + k = k (C i +C i + )+(C k +C k + )= (C i +C i + ) C k = k k k =(k ) k. 4
5 5. & 6. The umber of summads i a compositio or palidrome is oe more tha the umber of + sigs, ad the results follows by substitutig the previous results ito C S = C + + C ad P S = P + + P. Part 4 of Theorem 4 could have bee proved similarly to part, usig the umber of palidromes of ito i parts, deoted by P i. These umbers exhibit a iterestig patter which will be proved i Lemma Figure : Palidromes with i parts Lemma P j j k =0ad Pk = P j k = P j k = ( ) k j for j =,..., k, k. Proof: The first equality follows from the fact that a palidrome of a odd umber has to have a odd umber of summads. For the other cases we will iterpret the palidrome as a tilig where cuts are placed to create the parts. Sice we wat to create a palidrome, we look oly at oe of the two halves of the tilig ad fiish the other half as the mirror image. If =k, to create j parts we select (j ) = j positios out of the possible (k ) = k cuttig positios. 5
6 If =k, the we eed to distiguish betwee palidromes havig a odd or eve umber of summads. If the umber of summads is j, the there caot be a cut directly i the middle, so oly k = k cuttig positios are available, out of which we select (j ) = j. If the umber of summads is j, the the umber of palidromes correspods to the umber of compositios of k, withhalftheumber of summads (=j), which equals ( k j ) 3. LEVELS, RISES AND DROPS FOR COMPOSITIONS We ow tur our attetio to the harder ad more iterestig results for the umbers of levels, rises ad drops i all compositios of. Theorem 3. l = 36 ((3 +) +8( ) ) for ad l 0 =0.. r = d = 9 ((3 5) ( ) ) for 3 ad r 0 = r = r =0. Proof:. I order to obtai a recursio for the umber of levels i the compositios of, we look at the right ed of the compositios, as this is where the CCM creates chages. Applyig the CCM, the levels i the compositios of + are twice those i the compositios of, modified by ay chages i the umber of levels that occur at the right ed. If a is added, a additioal level is created i all the compositios of that ed i, i.e., a total of C () = C additioal levels. If the rightmost summad is icreased by, oe level is lost if the compositio of eds i x + x, ad oe additioal level is created if the compositio of eds i x +(x ). Thus, l k+ = l k + C k C k (x, x)+ C k (x, x ) x= x= 6
7 = l k + k C k x + C k (x ) x= x= = l k + k ( k 3 + k )+( k 4 + +) = l k +( k k 3 + k 4 +) = l k + k, 3 while l k = l k + k C k C k (x, x)+ C k (x, x ) x= x= = l k + k 3 ( k 4 + k )+( k ) = l k +( k 3 k 4 + k 5 + ) + = l k + k +. 3 Altogether, for all, l =l + +( ) 3. () The homogeeous ad particular solutios, l (h) ad l (p), respectively, are give by l (h) = c ad l (p) = A ( ) + B. Substitutig l (p) ito Eq. () ad comparig the coefficiets for powers of ad -, respectively, yields A = ad B =. Substitutig l 9 = l (h) + l (p) = c + 9 ( ) + ito Eq. () ad usig the iitial coditio l =yieldsc =, givig the 36 equatio for l for 3. (Actually, the formula also holds for ).. It is easy to see that r = d, sice for each opalidromic compositio there is oe which has the summads i reverse order. For palidromic compositios, the symmetry matches each rise i the first half with a drop i the secod half ad vice versa. Sice C + = r + l + d, it follows that r = C+ l.
8 Table shows values for the quatities of iterest. I Theorem 4 we will establish the patters suggested i this table C l r = d Table : Values for C +, l ad r Theorem 4. r + = r + l ad more geerally, r = i= l i for 3.. C + = r + r C + =4 (l + l )=4 (r r ). 4. l r = a,wherea is the th term of the Jacobsthal sequece. Proof:. The first equatio follows by substitutig the formulas of Theorem 3 for r ad l ad collectig terms. The geeral formula follows by iductio.. This follows from part, sice C + = r + l + d ad r = d. 3. The first equality follows by substitutig the formula i Theorem 3 for l ad l. The secod equality follows from part. 4. The sequece of values for f = l r is give by,, 3, 5,,, 43,... This sequece satisfies several recurrece relatios, for example f =f +( ) or f = f, both of which ca be verified by substitutig the formulas give i Theorem 3. These recursios defie the Jacobsthal sequece (A00045 i [8]), ad compariso of the iitial values shows that f = a. 8
9 4. LEVELS, RISES AND DROPS FOR PALINDROMES We ow look at the umbers of levels, rises ad drops for palidromes. Ulike the case for compositios, there is o sigle formula for the umber of levels, rises ad drops, respectively. Here we have to distiguish betwee odd ad eve values of, as well as look at the remaider of k whe divided by 3. Theorem 5 For k,. l k = 9 ( )k + ( k 53 + ) k lk+ = 9 ( )k + ( k + ) k r k = d k = 9 ( )k ( k 58 ) k k 0 mod (3) k mod (3) 4 k mod (3) 4 k 0 mod (3) 6 k mod (3) k mod (3) r k+ = d k+ = 9 ( )k ( k ) k k 0 mod (3) k mod (3) k mod (3) k 0 mod (3) 3 k mod (3) k mod (3) Proof: We use the PCM, where a middle summad m =l or m =l + (l 0) is combied with a compositio of k l ad its mirror image, to create a palidrome of = k or =k +, respectively. The umber of levels i the palidrome is twice the umber of levels of the compositio, plus ay additioal levels created whe the compositios are joied with the middle summad. We will first look at the case where (ad thus m) iseve. Ifl = m =0, a compositio of k is joied with its mirror image, ad we get oly oe additioal level. If l>0, the we get two additioal levels for a compositio edig i m, for m =l k l. Thus, k/3 lk = l k l + C k + l=0 l= 9 C k l (l) =s + k + s. (3)
10 Sice l 0 = l = 0, the first summad reduces to s = 8 { (3i +) i +8( ) i} = i + i i + 4 ( ) i i= 9 i= 3 i= 9 i=0 = 9 (k ) + ( ) d x i + 3 dx i= x = 9 (( )k +) = 9 k + { (k +) k k+} ( )k = ( k 9 ( )k + 3 ) k. (4) 9 To compute s,otethatc (i) =C (i ) =... = C i+ () = C i+ = i for i<ad C () =. The latter case oly occurs whe k =3l. Letk := 3j + r, where r =,, 3. (This somewhat ucovetioal defiitio allows for a uified proof.) Thus, with I A deotig the idicator fuctio of A, k/3 j s = C k l (l) = 3j+r l l + I {r=3} l= l= j ( = r ) 3 j l + I{r=3} = r (3 ) j + I {r=3} l= = k r { k +6 k 0mod(3) + I {r=3} = k r k r mod (3), for r =,. Combiig Equatios (3), (4) ad (5) ad simplifyig gives the result for l k. (5) For =k +, we make a similar argumet. Agai, each palidrome has twice the umber of levels of the associated compositio, ad we get two additioal levels wheever the compositio eds i m, form =l + k l. Thus, lk+ = l=0 l k l + (k )/3 l=0 C k l (l +)=:s + s 3. With a argumet similar to that for s,we derive k+ 4 k 0mod(3) s 3 = k+ +6 k mod(3) k+ k mod(3) Combiig Equatios (4) ad (6) ad simplifyig gives the result for l k+. Fially, the results for r ad d follow from the fact that r = d = P + l. (6) 0
11 5. GENERATING FUNCTIONS Let G a (x) = k=0 a k x k be the geeratig fuctio of the sequece {a } 0.We will give the geeratig fuctios for all the quatities of iterest. Theorem 6. G C (x) = x x ad G P (x) = +x x.. G C + (x) = x ( x) ad G P + (x) = x +x 3 +x 4 ( x ). 3. G C s (x) = 3x+3x ( x) ad G P s (x) = +x x +x 4 ( x ). 4. G l (x) = x ( x) (+x)( x) ad G r (x) =G d (x) = x 3 (+x)( x). 5. G l (x) = x (+3x+4x +x 3 x 4 4x 5 6x 6 ) (+x )(+x+x )( x ) ad G r (x) =G d (x) = x4 (+3x+4x +4x 3 +4x 4 ) (+x )(+x+x )( x ). Proof:. &. The geeratig fuctios for {C } 0, {P } 0 ad {C + } 0 are straightforward usig the defiitio ad the formulas of Theorem. We derive G P + (x), as it eeds to take ito accout the two differet formulas for odd ad eve. From Theorem, we get G P + (x) = = P + k xk + P + k xk k= k= (k ) k x k + (k ) k x k () k= k= Separatig each sum i Eq. () ito terms with ad without a factor of k, ad recombiig like terms across sums leads to G P + (x) = +x 4 = +x 4 4xk(x ) k (x + x ) (x ) k k= k= d ( ) dx x x+x x = x +x 3 +x 4 ( x ). 3. Sice C s = C + C +, G Cs (x) =G C (x)+g C + (x); likewise for G P s (x).
12 4. The geeratig fuctio for l ca be easily computed usig Mathematica or Maple, usig either the recursive or the explicit descriptio. The relevat Mathematica commads are <<DiscreteMath RSolve GeeratigFuctio[{a[+]==a[]+(/3)*^(-)+(-/3)*(-)^(-), a[0]==0,a[]==0},a[],,z][[,]] PowerSum[((/36) + (/))*^ + (/9)*(-)^,{z,,}] ( Furthermore, G r (x) =G d (x) = GC + (x) G l (x) ),sicer = d = C+ l. 5. I this case we have six differet formulas for l, depedig o the remaider of with respect to 6. Let G i (x) deote the geeratig fuctio of { l6k+i } k=0. The, usig the defiitio of the geeratig fuctio ad separatig the sum accordig to the remaider (similar to the computatio i part ), we get G l (x) =G 0 (x 6 )+x G (x 6 )+x G (x 6 )+ + x 5 G 5 (x 6 ). The fuctios G i (x) ad the resultig geeratig fuctio (x) are derived usig G l the followig Mathematica commads: <<DiscreteMath RSolve g0[z_]=powersum[(/6)((6()+53)* ^(3)+08+8(-)^()),{z,,}] g[z_]=powersum[(/63)((63+)* ^(3)-36+4(-)^),{z,,}] g[z_]=powersum[(/63)((6+95)* ^(3)-8-4(-)^),{z,,0}] g3[z_]=powersum[(/63)((6+86)* ^(3)+54-4(-)^),{z,,0}] g4[z_]=powersum[(/63)((5+4)* ^(3)-36+4(-)^),{z,,0}] g5[z_]=powersum[(/63)((5+56)* ^(3)-8+4(-)^),{z,,0}] gefu[z]:= g0[z^6]+z g[z^6]+z^ g[z^6]+z^3 g3[z^6]+z^4 g4[z^6]+z^5 g5[z^6] Fially, G r (x) =G d (x) = ( GP + (x) G l (x) ),sice r = d = P + l.
13 ACKNOWLEDGEMENTS The authors would like to thak the aoymous referee for his thorough readig ad for helpful suggestios which have led to a improved paper. Refereces [] K. Alladi & V.E. Hoggatt, Jr. Compositios with Oes ad Twos. Fiboacci Quarterly 3.3 (95): [] P. Z. Chi, R. P. Grimaldi & S. Heubach. The Frequecy of Summads of a Particular Size i Palidromic Compositios. To appear i Ars Combiatoria. [3] D. E. Dayki, D. J. Kleitma & D. B. West. Number of Meets betwee two Subsets of a Lattice. Joural of Combiatorial Theory, A6 (99): [4] D. D. Frey & J. A. Sellers. Jacobsthal Numbers ad Alteratig Sig Matrices. Joural of Iteger Sequeces, 3 (000): # [5] R. P. Grimaldi. Discrete ad Combiatorial Mathematics, 4 th Editio. Addiso- Wesley Logma, Ic., 999. [6] R. P. Grimaldi. Compositios with Odd Summads. Cogressus Numeratium 4 (000): 3-. [] S. Heubach. Tilig a m-by- Area with Squares of Size up to k-by-k with m 5. Cogressus Numeratium 40 (999): [8] Sloae s Olie Iteger Sequeces. jas/sequeces AMS Classificatio Number: 05A99 3
FLOOR AND ROOF FUNCTION ANALOGS OF THE BELL NUMBERS. H. W. Gould Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 FLOOR AND ROOF FUNCTION ANALOGS OF THE BELL NUMBERS H. W. Gould Departmet of Mathematics, West Virgiia Uiversity, Morgatow, WV
More informationCALCULATION OF FIBONACCI VECTORS
CALCULATION OF FIBONACCI VECTORS Stuart D. Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithaca.edu ad Dai Novak Departmet of Mathematics, Ithaca College
More informationNICK DUFRESNE. 1 1 p(x). To determine some formulas for the generating function of the Schröder numbers, r(x) = a(x) =
AN INTRODUCTION TO SCHRÖDER AND UNKNOWN NUMBERS NICK DUFRESNE Abstract. I this article we will itroduce two types of lattice paths, Schröder paths ad Ukow paths. We will examie differet properties of each,
More informationRecursive Algorithms. Recurrences. Recursive Algorithms Analysis
Recursive Algorithms Recurreces Computer Sciece & Egieerig 35: Discrete Mathematics Christopher M Bourke cbourke@cseuledu A recursive algorithm is oe i which objects are defied i terms of other objects
More informationMatrix representations of Fibonacci-like sequences
NTMSCI 6, No. 4, 03-0 08 03 New Treds i Mathematical Scieces http://dx.doi.org/0.085/tmsci.09.33 Matrix represetatios of Fiboacci-like sequeces Yasemi Tasyurdu Departmet of Mathematics, Faculty of Sciece
More informationA GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS. Mircea Merca
Idia J Pure Appl Math 45): 75-89 February 204 c Idia Natioal Sciece Academy A GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS Mircea Merca Departmet of Mathematics Uiversity
More informationSequences of Definite Integrals, Factorials and Double Factorials
47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry Daa-Picard Departmet of Applied Mathematics Jerusalem College of Techology
More informationThe r-generalized Fibonacci Numbers and Polynomial Coefficients
It. J. Cotemp. Math. Scieces, Vol. 3, 2008, o. 24, 1157-1163 The r-geeralized Fiboacci Numbers ad Polyomial Coefficiets Matthias Schork Camillo-Sitte-Weg 25 60488 Frakfurt, Germay mschork@member.ams.org,
More informationYou may work in pairs or purely individually for this assignment.
CS 04 Problem Solvig i Computer Sciece OOC Assigmet 6: Recurreces You may work i pairs or purely idividually for this assigmet. Prepare your aswers to the followig questios i a plai ASCII text file or
More informationSOME TRIBONACCI IDENTITIES
Mathematics Today Vol.7(Dec-011) 1-9 ISSN 0976-38 Abstract: SOME TRIBONACCI IDENTITIES Shah Devbhadra V. Sir P.T.Sarvajaik College of Sciece, Athwalies, Surat 395001. e-mail : drdvshah@yahoo.com The sequece
More informationMath 155 (Lecture 3)
Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationSequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018
CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical
More informationUniversity of Twente The Netherlands
Faculty of Mathematical Scieces t Uiversity of Twete The Netherlads P.O. Box 7 7500 AE Eschede The Netherlads Phoe: +3-53-4893400 Fax: +3-53-48934 Email: memo@math.utwete.l www.math.utwete.l/publicatios
More information~W I F
A FIBONACCI PROPERTY OF WYTHOFF PAIRS ROBERT SILBER North Carolia State Uiversity, Raleigh, North Carolia 27607 I this paper we poit out aother of those fasciatig "coicideces" which are so characteristically
More informationAppendix to Quicksort Asymptotics
Appedix to Quicksort Asymptotics James Alle Fill Departmet of Mathematical Scieces The Johs Hopkis Uiversity jimfill@jhu.edu ad http://www.mts.jhu.edu/~fill/ ad Svate Jaso Departmet of Mathematics Uppsala
More informationThe Random Walk For Dummies
The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli
More informationThe log-concavity and log-convexity properties associated to hyperpell and hyperpell-lucas sequences
Aales Mathematicae et Iformaticae 43 2014 pp. 3 12 http://ami.etf.hu The log-cocavity ad log-covexity properties associated to hyperpell ad hyperpell-lucas sequeces Moussa Ahmia ab, Hacèe Belbachir b,
More informationCombinatorially Thinking
Combiatorially Thiig SIMUW 2008: July 4 25 Jeifer J Qui jjqui@uwashigtoedu Philosophy We wat to costruct our mathematical uderstadig To this ed, our goal is to situate our problems i cocrete coutig cotexts
More informationDIVISIBILITY PROPERTIES OF GENERALIZED FIBONACCI POLYNOMIALS
DIVISIBILITY PROPERTIES OF GENERALIZED FIBONACCI POLYNOMIALS VERNER E. HOGGATT, JR. Sa Jose State Uiversity, Sa Jose, Califoria 95192 ad CALVIN T. LONG Washigto State Uiversity, Pullma, Washigto 99163
More informationGAMALIEL CERDA-MORALES 1. Blanco Viel 596, Valparaíso, Chile. s: /
THE GELIN-CESÀRO IDENTITY IN SOME THIRD-ORDER JACOBSTHAL SEQUENCES arxiv:1810.08863v1 [math.co] 20 Oct 2018 GAMALIEL CERDA-MORALES 1 1 Istituto de Matemáticas Potificia Uiversidad Católica de Valparaíso
More informationPell and Lucas primes
Notes o Number Theory ad Discrete Mathematics ISSN 30 532 Vol. 2, 205, No. 3, 64 69 Pell ad Lucas primes J. V. Leyedekkers ad A. G. Shao 2 Faculty of Sciece, The Uiversity of Sydey NSW 2006, Australia
More informationChapter 4. Fourier Series
Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,
More informationWeek 5-6: The Binomial Coefficients
Wee 5-6: The Biomial Coefficiets March 6, 2018 1 Pascal Formula Theorem 11 (Pascal s Formula For itegers ad such that 1, ( ( ( 1 1 + 1 The umbers ( 2 ( 1 2 ( 2 are triagle umbers, that is, The petago umbers
More informationAMS Mathematics Subject Classification : 40A05, 40A99, 42A10. Key words and phrases : Harmonic series, Fourier series. 1.
J. Appl. Math. & Computig Vol. x 00y), No. z, pp. A RECURSION FOR ALERNAING HARMONIC SERIES ÁRPÁD BÉNYI Abstract. We preset a coveiet recursive formula for the sums of alteratig harmoic series of odd order.
More informationarxiv: v1 [math.nt] 10 Dec 2014
A DIGITAL BINOMIAL THEOREM HIEU D. NGUYEN arxiv:42.38v [math.nt] 0 Dec 204 Abstract. We preset a triagle of coectios betwee the Sierpisi triagle, the sum-of-digits fuctio, ad the Biomial Theorem via a
More informationHoggatt and King [lo] defined a complete sequence of natural numbers
REPRESENTATIONS OF N AS A SUM OF DISTINCT ELEMENTS FROM SPECIAL SEQUENCES DAVID A. KLARNER, Uiversity of Alberta, Edmoto, Caada 1. INTRODUCTION Let a, I deote a sequece of atural umbers which satisfies
More informationAccepted in Fibonacci Quarterly (2007) Archived in SEQUENCE BALANCING AND COBALANCING NUMBERS
Accepted i Fiboacci Quarterly (007) Archived i http://dspace.itrl.ac.i/dspace SEQUENCE BALANCING AND COBALANCING NUMBERS G. K. Pada Departmet of Mathematics Natioal Istitute of Techology Rourela 769 008
More informationw (1) ˆx w (1) x (1) /ρ and w (2) ˆx w (2) x (2) /ρ.
2 5. Weighted umber of late jobs 5.1. Release dates ad due dates: maximimizig the weight of o-time jobs Oce we add release dates, miimizig the umber of late jobs becomes a sigificatly harder problem. For
More informationSeunghee Ye Ma 8: Week 5 Oct 28
Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value
More informationA Note on the Symmetric Powers of the Standard Representation of S n
A Note o the Symmetric Powers of the Stadard Represetatio of S David Savitt 1 Departmet of Mathematics, Harvard Uiversity Cambridge, MA 0138, USA dsavitt@mathharvardedu Richard P Staley Departmet of Mathematics,
More informationRandom Models. Tusheng Zhang. February 14, 2013
Radom Models Tusheg Zhag February 14, 013 1 Radom Walks Let me describe the model. Radom walks are used to describe the motio of a movig particle (object). Suppose that a particle (object) moves alog the
More informationOptimally Sparse SVMs
A. Proof of Lemma 3. We here prove a lower boud o the umber of support vectors to achieve geeralizatio bouds of the form which we cosider. Importatly, this result holds ot oly for liear classifiers, but
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationLegendre-Stirling Permutations
Legedre-Stirlig Permutatios Eric S. Egge Departmet of Mathematics Carleto College Northfield, MN 07 USA eegge@carleto.edu Abstract We first give a combiatorial iterpretatio of Everitt, Littlejoh, ad Wellma
More informationRecurrence Relations
Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The
More informationSequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet
More informationCSE 1400 Applied Discrete Mathematics Number Theory and Proofs
CSE 1400 Applied Discrete Mathematics Number Theory ad Proofs Departmet of Computer Scieces College of Egieerig Florida Tech Sprig 01 Problems for Number Theory Backgroud Number theory is the brach of
More informationLet us consider the following problem to warm up towards a more general statement.
Lecture 4: Sequeces with repetitios, distributig idetical objects amog distict parties, the biomial theorem, ad some properties of biomial coefficiets Refereces: Relevat parts of chapter 15 of the Math
More informationTHE N-POINT FUNCTIONS FOR INTERSECTION NUMBERS ON MODULI SPACES OF CURVES
THE N-POINT FUNTIONS FOR INTERSETION NUMBERS ON MODULI SPAES OF URVES KEFENG LIU AND HAO XU Abstract. We derive from Witte s KdV equatio a simple formula of the -poit fuctios for itersectio umbers o moduli
More informationA sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as
More information4.3 Growth Rates of Solutions to Recurrences
4.3. GROWTH RATES OF SOLUTIONS TO RECURRENCES 81 4.3 Growth Rates of Solutios to Recurreces 4.3.1 Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer.
More informationExact Horadam Numbers with a Chebyshevish Accent by Clifford A. Reiter
Exact Horadam Numbers with a Chebyshevish Accet by Clifford A. Reiter (reiterc@lafayette.edu) A recet paper by Joseph De Kerf illustrated the use of Biet type formulas for computig Horadam umbers [1].
More informationThe log-behavior of n p(n) and n p(n)/n
Ramauja J. 44 017, 81-99 The log-behavior of p ad p/ William Y.C. Che 1 ad Ke Y. Zheg 1 Ceter for Applied Mathematics Tiaji Uiversity Tiaji 0007, P. R. Chia Ceter for Combiatorics, LPMC Nakai Uivercity
More informationSums Involving Moments of Reciprocals of Binomial Coefficients
1 2 3 47 6 23 11 Joural of Iteger Sequeces, Vol. 14 (2011, Article 11.6.6 Sums Ivolvig Momets of Reciprocals of Biomial Coefficiets Hacèe Belbachir ad Mourad Rahmai Uiversity of Scieces ad Techology Houari
More informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
More informationOn Random Line Segments in the Unit Square
O Radom Lie Segmets i the Uit Square Thomas A. Courtade Departmet of Electrical Egieerig Uiversity of Califoria Los Ageles, Califoria 90095 Email: tacourta@ee.ucla.edu I. INTRODUCTION Let Q = [0, 1] [0,
More information1 Generating functions for balls in boxes
Math 566 Fall 05 Some otes o geeratig fuctios Give a sequece a 0, a, a,..., a,..., a geeratig fuctio some way of represetig the sequece as a fuctio. There are may ways to do this, with the most commo ways
More informationarxiv: v1 [math.fa] 3 Apr 2016
Aticommutator Norm Formula for Proectio Operators arxiv:164.699v1 math.fa] 3 Apr 16 Sam Walters Uiversity of Norther British Columbia ABSTRACT. We prove that for ay two proectio operators f, g o Hilbert
More informationQ-BINOMIALS AND THE GREATEST COMMON DIVISOR. Keith R. Slavin 8474 SW Chevy Place, Beaverton, Oregon 97008, USA.
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 2008, #A05 Q-BINOMIALS AND THE GREATEST COMMON DIVISOR Keith R. Slavi 8474 SW Chevy Place, Beaverto, Orego 97008, USA slavi@dsl-oly.et Received:
More informationON SOME DIOPHANTINE EQUATIONS RELATED TO SQUARE TRIANGULAR AND BALANCING NUMBERS
Joural of Algebra, Number Theory: Advaces ad Applicatios Volume, Number, 00, Pages 7-89 ON SOME DIOPHANTINE EQUATIONS RELATED TO SQUARE TRIANGULAR AND BALANCING NUMBERS OLCAY KARAATLI ad REFİK KESKİN Departmet
More informationA Combinatoric Proof and Generalization of Ferguson s Formula for k-generalized Fibonacci Numbers
Jue 5 00 A Combiatoric Proof ad Geeralizatio of Ferguso s Formula for k-geeralized Fiboacci Numbers David Kessler 1 ad Jeremy Schiff 1 Departmet of Physics Departmet of Mathematics Bar-Ila Uiversity, Ramat
More informationECE-S352 Introduction to Digital Signal Processing Lecture 3A Direct Solution of Difference Equations
ECE-S352 Itroductio to Digital Sigal Processig Lecture 3A Direct Solutio of Differece Equatios Discrete Time Systems Described by Differece Equatios Uit impulse (sample) respose h() of a DT system allows
More informationCHAPTER I: Vector Spaces
CHAPTER I: Vector Spaces Sectio 1: Itroductio ad Examples This first chapter is largely a review of topics you probably saw i your liear algebra course. So why cover it? (1) Not everyoe remembers everythig
More informationSubject: Differential Equations & Mathematical Modeling-III
Power Series Solutios of Differetial Equatios about Sigular poits Subject: Differetial Equatios & Mathematical Modelig-III Lesso: Power series solutios of differetial equatios about Sigular poits Lesso
More informationMAT 271 Project: Partial Fractions for certain rational functions
MAT 7 Project: Partial Fractios for certai ratioal fuctios Prerequisite kowledge: partial fractios from MAT 7, a very good commad of factorig ad complex umbers from Precalculus. To complete this project,
More informationOn the Linear Complexity of Feedback Registers
O the Liear Complexity of Feedback Registers A. H. Cha M. Goresky A. Klapper Northeaster Uiversity Abstract I this paper, we study sequeces geerated by arbitrary feedback registers (ot ecessarily feedback
More informationsubcaptionfont+=small,labelformat=parens,labelsep=space,skip=6pt,list=0,hypcap=0 subcaption ALGEBRAIC COMBINATORICS LECTURE 8 TUESDAY, 2/16/2016
subcaptiofot+=small,labelformat=pares,labelsep=space,skip=6pt,list=0,hypcap=0 subcaptio ALGEBRAIC COMBINATORICS LECTURE 8 TUESDAY, /6/06. Self-cojugate Partitios Recall that, give a partitio λ, we may
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationA Simplified Binet Formula for k-generalized Fibonacci Numbers
A Simplified Biet Formula for k-geeralized Fiboacci Numbers Gregory P. B. Dresde Departmet of Mathematics Washigto ad Lee Uiversity Lexigto, VA 440 dresdeg@wlu.edu Zhaohui Du Shaghai, Chia zhao.hui.du@gmail.com
More informationSome remarks for codes and lattices over imaginary quadratic
Some remarks for codes ad lattices over imagiary quadratic fields Toy Shaska Oaklad Uiversity, Rochester, MI, USA. Caleb Shor Wester New Eglad Uiversity, Sprigfield, MA, USA. shaska@oaklad.edu Abstract
More informationAN ALMOST LINEAR RECURRENCE. Donald E. Knuth Calif. Institute of Technology, Pasadena, Calif.
AN ALMOST LINEAR RECURRENCE Doald E. Kuth Calif. Istitute of Techology, Pasadea, Calif. form A geeral liear recurrece with costat coefficiets has the U 0 = a l* U l = a 2 " ' " U r - l = a r ; u = b, u,
More informationA LIMITED ARITHMETIC ON SIMPLE CONTINUED FRACTIONS - II 1. INTRODUCTION
A LIMITED ARITHMETIC ON SIMPLE CONTINUED FRACTIONS - II C. T. LONG J. H. JORDAN* Washigto State Uiversity, Pullma, Washigto 1. INTRODUCTION I the first paper [2 ] i this series, we developed certai properties
More informationMATH 324 Summer 2006 Elementary Number Theory Solutions to Assignment 2 Due: Thursday July 27, 2006
MATH 34 Summer 006 Elemetary Number Theory Solutios to Assigmet Due: Thursday July 7, 006 Departmet of Mathematical ad Statistical Scieces Uiversity of Alberta Questio [p 74 #6] Show that o iteger of the
More informationk-generalized FIBONACCI NUMBERS CLOSE TO THE FORM 2 a + 3 b + 5 c 1. Introduction
Acta Math. Uiv. Comeiaae Vol. LXXXVI, 2 (2017), pp. 279 286 279 k-generalized FIBONACCI NUMBERS CLOSE TO THE FORM 2 a + 3 b + 5 c N. IRMAK ad M. ALP Abstract. The k-geeralized Fiboacci sequece { F (k)
More informationOn Some Identities and Generating Functions for Mersenne Numbers and Polynomials
Turish Joural of Aalysis ad Number Theory, 8, Vol 6, No, 9-97 Available olie at htt://ubsscieubcom/tjat/6//5 Sciece ad Educatio Publishig DOI:69/tjat-6--5 O Some Idetities ad Geeratig Fuctios for Mersee
More informationSection 5.1 The Basics of Counting
1 Sectio 5.1 The Basics of Coutig Combiatorics, the study of arragemets of objects, is a importat part of discrete mathematics. I this chapter, we will lear basic techiques of coutig which has a lot of
More informationBINOMIAL PREDICTORS. + 2 j 1. Then n + 1 = The row of the binomial coefficients { ( n
BINOMIAL PREDICTORS VLADIMIR SHEVELEV arxiv:0907.3302v2 [math.nt] 22 Jul 2009 Abstract. For oegative itegers, k, cosider the set A,k = { [0, 1,..., ] : 2 k ( ). Let the biary epasio of + 1 be: + 1 = 2
More informationMath 475, Problem Set #12: Answers
Math 475, Problem Set #12: Aswers A. Chapter 8, problem 12, parts (b) ad (d). (b) S # (, 2) = 2 2, sice, from amog the 2 ways of puttig elemets ito 2 distiguishable boxes, exactly 2 of them result i oe
More informationMathematics review for CSCI 303 Spring Department of Computer Science College of William & Mary Robert Michael Lewis
Mathematics review for CSCI 303 Sprig 019 Departmet of Computer Sciece College of William & Mary Robert Michael Lewis Copyright 018 019 Robert Michael Lewis Versio geerated: 13 : 00 Jauary 17, 019 Cotets
More informationApplied Mathematics Letters. On the properties of Lucas numbers with binomial coefficients
Applied Mathematics Letters 3 (1 68 7 Cotets lists available at ScieceDirect Applied Mathematics Letters joural homepage: wwwelseviercom/locate/aml O the properties of Lucas umbers with biomial coefficiets
More informationSome identities involving Fibonacci, Lucas polynomials and their applications
Bull. Math. Soc. Sci. Math. Roumaie Tome 55103 No. 1, 2012, 95 103 Some idetities ivolvig Fiboacci, Lucas polyomials ad their applicatios by Wag Tigtig ad Zhag Wepeg Abstract The mai purpose of this paper
More informationComplex Numbers Solutions
Complex Numbers Solutios Joseph Zoller February 7, 06 Solutios. (009 AIME I Problem ) There is a complex umber with imagiary part 64 ad a positive iteger such that Fid. [Solutio: 697] 4i + + 4i. 4i 4i
More informationSEQUENCES AND SERIES
Sequeces ad 6 Sequeces Ad SEQUENCES AND SERIES Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives rise to what is called a sequece. Sequeces
More informationProblem 4: Evaluate ( k ) by negating (actually un-negating) its upper index. Binomial coefficient
Problem 4: Evaluate by egatig actually u-egatig its upper idex We ow that Biomial coefficiet r { where r is a real umber, is a iteger The above defiitio ca be recast i terms of factorials i the commo case
More informationThe Riemann Zeta Function
Physics 6A Witer 6 The Riema Zeta Fuctio I this ote, I will sketch some of the mai properties of the Riema zeta fuctio, ζ(x). For x >, we defie ζ(x) =, x >. () x = For x, this sum diverges. However, we
More informationBeurling Integers: Part 2
Beurlig Itegers: Part 2 Isomorphisms Devi Platt July 11, 2015 1 Prime Factorizatio Sequeces I the last article we itroduced the Beurlig geeralized itegers, which ca be represeted as a sequece of real umbers
More informationOn the Fibonacci-like Sequences of Higher Order
Article Iteratioal Joural of oder atheatical Scieces, 05, 3(): 5-59 Iteratioal Joural of oder atheatical Scieces Joural hoepage: wwwoderscietificpressco/jourals/ijsaspx O the Fiboacci-like Sequeces of
More information#A51 INTEGERS 14 (2014) MULTI-POLY-BERNOULLI-STAR NUMBERS AND FINITE MULTIPLE ZETA-STAR VALUES
#A5 INTEGERS 4 (24) MULTI-POLY-BERNOULLI-STAR NUMBERS AND FINITE MULTIPLE ZETA-STAR VALUES Kohtaro Imatomi Graduate School of Mathematics, Kyushu Uiversity, Nishi-ku, Fukuoka, Japa k-imatomi@math.kyushu-u.ac.p
More informationOn Generalized Fibonacci Numbers
Applied Mathematical Scieces, Vol. 9, 215, o. 73, 3611-3622 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.215.5299 O Geeralized Fiboacci Numbers Jerico B. Bacai ad Julius Fergy T. Rabago Departmet
More informationBinomial transform of products
Jauary 02 207 Bioial trasfor of products Khristo N Boyadzhiev Departet of Matheatics ad Statistics Ohio Norther Uiversity Ada OH 4580 USA -boyadzhiev@ouedu Abstract Give the bioial trasfors { b } ad {
More informationOn the Jacobsthal-Lucas Numbers by Matrix Method 1
It J Cotemp Math Scieces, Vol 3, 2008, o 33, 1629-1633 O the Jacobsthal-Lucas Numbers by Matrix Method 1 Fikri Köke ad Durmuş Bozkurt Selçuk Uiversity, Faculty of Art ad Sciece Departmet of Mathematics,
More information1. By using truth tables prove that, for all statements P and Q, the statement
Author: Satiago Salazar Problems I: Mathematical Statemets ad Proofs. By usig truth tables prove that, for all statemets P ad Q, the statemet P Q ad its cotrapositive ot Q (ot P) are equivalet. I example.2.3
More informationEnd-of-Year Contest. ERHS Math Club. May 5, 2009
Ed-of-Year Cotest ERHS Math Club May 5, 009 Problem 1: There are 9 cois. Oe is fake ad weighs a little less tha the others. Fid the fake coi by weighigs. Solutio: Separate the 9 cois ito 3 groups (A, B,
More informationAn enumeration of flags in finite vector spaces
A eumeratio of flags i fiite vector spaces C Rya Viroot Departmet of Mathematics College of William ad Mary P O Box 8795 Williamsburg VA 23187 viroot@mathwmedu Submitted: Feb 2 2012; Accepted: Ju 27 2012;
More informationHarmonic Number Identities Via Euler s Transform
1 2 3 47 6 23 11 Joural of Iteger Sequeces, Vol. 12 2009), Article 09.6.1 Harmoic Number Idetities Via Euler s Trasform Khristo N. Boyadzhiev Departmet of Mathematics Ohio Norther Uiversity Ada, Ohio 45810
More informationAbsolutely Harmonious Labeling of Graphs
Iteratioal J.Math. Combi. Vol. (011), 40-51 Absolutely Harmoious Labelig of Graphs M.Seeivasa (Sri Paramakalyai College, Alwarkurichi-6741, Idia) A.Lourdusamy (St.Xavier s College (Autoomous), Palayamkottai,
More informationMathematical Induction
Mathematical Iductio Itroductio Mathematical iductio, or just iductio, is a proof techique. Suppose that for every atural umber, P() is a statemet. We wish to show that all statemets P() are true. I a
More informationFibonacci numbers and orthogonal polynomials
Fiboacci umbers ad orthogoal polyomials Christia Berg April 10, 2006 Abstract We prove that the sequece (1/F +2 0 of reciprocals of the Fiboacci umbers is a momet sequece of a certai discrete probability,
More informationMath 2784 (or 2794W) University of Connecticut
ORDERS OF GROWTH PAT SMITH Math 2784 (or 2794W) Uiversity of Coecticut Date: Mar. 2, 22. ORDERS OF GROWTH. Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really
More informationCLOSED FORM FORMULA FOR THE NUMBER OF RESTRICTED COMPOSITIONS
Submitted to the Bulleti of the Australia Mathematical Society doi:10.1017/s... CLOSED FORM FORMULA FOR THE NUMBER OF RESTRICTED COMPOSITIONS GAŠPER JAKLIČ, VITO VITRIH ad EMIL ŽAGAR Abstract I this paper,
More informationarxiv: v4 [math.co] 5 May 2011
A PROBLEM OF ENUMERATION OF TWO-COLOR BRACELETS WITH SEVERAL VARIATIONS arxiv:07101370v4 [mathco] 5 May 011 VLADIMIR SHEVELEV Abstract We cosier the problem of eumeratio of icogruet two-color bracelets
More informationMATH 304: MIDTERM EXAM SOLUTIONS
MATH 304: MIDTERM EXAM SOLUTIONS [The problems are each worth five poits, except for problem 8, which is worth 8 poits. Thus there are 43 possible poits.] 1. Use the Euclidea algorithm to fid the greatest
More informationGENERALIZATIONS OF ZECKENDORFS THEOREM. TilVIOTHY J. KELLER Student, Harvey Mudd College, Claremont, California
GENERALIZATIONS OF ZECKENDORFS THEOREM TilVIOTHY J. KELLER Studet, Harvey Mudd College, Claremot, Califoria 91711 The Fiboacci umbers F are defied by the recurrece relatio Fi = F 2 = 1, F = F - + F 0 (
More informationarxiv: v3 [math.nt] 24 Dec 2017
DOUGALL S 5 F SUM AND THE WZ-ALGORITHM Abstract. We show how to prove the examples of a paper by Chu ad Zhag usig the WZ-algorithm. arxiv:6.085v [math.nt] Dec 07 Keywords. Geeralized hypergeometric series;
More informationSome Explicit Formulae of NAF and its Left-to-Right. Analogue Based on Booth Encoding
Vol.7, No.6 (01, pp.69-74 http://dx.doi.org/10.1457/ijsia.01.7.6.7 Some Explicit Formulae of NAF ad its Left-to-Right Aalogue Based o Booth Ecodig Dog-Guk Ha, Okyeo Yi, ad Tsuyoshi Takagi Kookmi Uiversity,
More informationMath 113 Exam 3 Practice
Math Exam Practice Exam will cover.-.9. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for you
More informationProduct measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014.
Product measures, Toelli s ad Fubii s theorems For use i MAT3400/4400, autum 2014 Nadia S. Larse Versio of 13 October 2014. 1. Costructio of the product measure The purpose of these otes is to preset the
More informationCounting Well-Formed Parenthesizations Easily
Coutig Well-Formed Parethesizatios Easily Pekka Kilpeläie Uiversity of Easter Filad School of Computig, Kuopio August 20, 2014 Abstract It is well kow that there is a oe-to-oe correspodece betwee ordered
More information