A Study on Some Integer Sequences

Transcription

1 It. J. Cotemp. Math. Scieces, Vol. 3, 008, o. 3, A Study o Some Iteger Sequeces Serpil Halıcı Sakarya Uiversity, Departmet of Mathematics Esetepe Campus, Sakarya, Turkey shalici@sakarya.edu.tr Abstract. I this ote, some iteger sequeces are ivestigated ad the relatioships betwee them are obtaied. Especially, may similarities betwee eumeratio of geeralized iboacci sequeces ad Padova sequece are realized. By this similarities some ew sequeces are obtaied. Also, recurrece relatios that satisfied by the sequeces are studied. Mathematics Subject Classificatio: B37, B39, B0 Keywords: Cotiued ractio, Recurrece Relatio, iboacci Sequece. Itroductio Edouard ucas(84 89) made a deep study of sequeces which is called geeralized iboacci sequeces that begi the sum of the precedig two. The simplest such series; 0,,,,3,5,8,3,, is called the iboacci sequeces by ucas. The ext simplest series,,,3,4,7,,8, is the called the ucas umbers i his hoor. The iboacci rule of addig the latest two to get the ext is kept, but here we begi with ad. The positio of each umber i this sequeces is traditioally idicated by a subscript, so that 0 0,,, 3 () ad so o. Thus, iboacci sequece ca be defied as follows + - ; () which two boudary coditios; 0 0,. The ucas sequece is defied as follows where we write its members as - + -, > which two boudary coditios; 0,. ucas umbers have lots of properties similar to those of iboacci umbers.

2 04 S. Halıcı + If th iboacci umber, the is equal to a cotiued fractio.ideed, we apply the Euclidea algorithm to + ad. I effect we get a sequece of + equatios: [ ;,, ] If the value of cotiued fractio [ ;, ] is called δ, ad the th covergece of this fractio is [ ;,...,] + + δ lim[ ;,..., ] lim C lim lim C the, +,the positive root of this equatio δ + 5 ;,... which is kow as golde ratio[ 4 ]. is [ ] or ay positive real umber k, the k-iboacci sequece, say { k } N recurretly by k, + kk, + k,, is defied, (3) with iitial coditios 0 ;. If k, the classical iboacci sequece k, 0 k, is obtaied ad if k, the classical Pell sequece appears. Note that if k is a real variable x the iboacci polyomials defied by k x,, ad they correspod to the + ( x) x x ( x) + ( x) if 0, if, if f (4). Some Properties of iboacci Sequece Theorem.: If, 3 ad for 3, + ad 5 5 α + ve β the α β + (5) Proof: The proof by iductio is easy. Defiitio. ( Biet s ormula) Alteratively we ca defie the iboacci umbers for iteger by

3 A study o some iteger sequeces 05 (6) where α ad β are two solutios of ( z) z z α + α, β + β. The p. Note that α + β, αβ, + α ( α + ) β ( β + ) α β. Here the formula idirectly implies that, for 0, 0,, + which is a repeated correlatio[ 3 ] The th k- iboacci umber is give by k, r r r r (7) where r, r are the roots of the characteristic equatio r kr + ad r f r. iear Algebra. Defie matrices 0 M, P α β, 0 D α 0 β (8) Ad for iteger, defie v (9) Stadard computatios show that β - P β α - α α 0 0 β, D (0) The characteristic polyomial of M is p(z) ad a diagoal decompositio of M is M PDP [] 3. The Mv v +, ad two routie iductios show that v M v ad M PD P. We coclude that v PD P v yields followig equatio i vector form:

4 06 S. Halıcı Theorem.: or, the cosequtive ucas umbers are prime umbers i 4. theirselves, that meas (, + ) [ ] k k k Defiitio.: If p ( x) x ak x ak x... ax a0 polyom is a k degree moic polyom, the the compaio matrix is defied such as[ ]: A px ( ) a0 a a... ak ak () + Theorem.3 : et, 0, the A 0 With the help of this theorem, each required ucas umber ca easily be foud. or example 6 th ucas umber is: A Defiitio.3: The Padova sequece is the sequece of itegers P() defied by the iitial values p(0)p()p() () ad the recurrece relatio P()P( )+P( 3) (3) The first few values of P() are,,,,,3,4,5,7,9,,6,,8, [ ]. Whe the table writte for ucas sequece, is agai rewritte for iboacci sequece, the sum of the diagoal elemets sums up to Padova sequece. Now cosider the ifiite dimesioel matrix A ij where we defie recursively each elemets a ij as follows for i,j,[ 5 ]; a ij if, i j A ij a ij 0 if, i > j or j i+ (4) a ij a i-,j- + a i-,j- other cases.

5 A study o some iteger sequeces 07 Thus, we obtai the followig table: Table. The elemets of the A ij matrix i / j M The Results Accordig to the table above, the followig results could be writte: Corollary 3.: (a) or i,j ; ai, j j + of S() : j i (b) The sum of the diagoal elemets o the table are the elemets ( ) k+ S a, k. (5) k The first few values of S() are,,,,3,3,4,6,7,0,3,7,3, Note that the S() sequece also satisfies the recurrece relatios; S()P()+P( 3) ; P(0) P() P() S()P(-)+.P( 3) (6) S()P() -P( ) ; S(0) S(), S() (c) or, if the rows of the table are show by R(), the the sum of the elemets of R() could be foud accordig to this formula: R(). 3 (7) The elemets of this sequece are the elemets of Pisot sequece.

6 08 S. Halıcı Corollary 3.: or sequeces ad if >4, the -4. Proof: This could be verified by iductio. If 5, 5 4, 5 5, 5-4 the 5 5. et 4<k< be true. Namely, -4. We will show that Sice ad from hypothesis with the help of equatios - 4 ad - -, it is see that Corollary 3.3: et s cosider three differet iteger sequeces whose a few first terms are as followig: P:,,,3,4,7,,8, ; where P() P(-) + P(-3) P r :3,0,,3,,5,5,7,0,,7, ;wherep r ()P r (-)+P r (-3) (8) S:,,,3,3,4,6,7,0,3,7,3, ;where S() S(-) + S(-3) Betwee these sequeces, for >, S() P r ()+P(-) is valid. Here P r () is a Padova sequece the iitial terms of which are differet. Proof: et s assume 3 i order to prove by iductio. Sice, S(3)30, P r (3) 9, P(3-) P() rom equatio S(3) P r (3)+P() the validity of the equatio is obvius. et <k be true. That s for <k S()P r () +P(-), S(+)P r (+)+P(-). (9) The validity of this equatio should be show. Sice S(+) S(-) + S(-) from hypothesis S(-) P r (-)+P(-3). With the help of S(-)P r (-)+P(-4) the validity of this equatio is obvius. With the help of these tables several iteger sequeces could be established. Moreover, the relatios betwee the properties of the sequeces could be uderstood easily.

7 A study o some iteger sequeces 09 REERENCES [] [] I.Nive, H.S. Zuckerma, H.. Motgomery, A itroductio to the theory of umbers, 5 th Ed., Wiley, New York, 99. [] 3 M. Marcus ad H. Mic, A survey of matrix theory ad matrix iequalities, Ally ad Baco, Bosto, 964. [] 4 T. Koshy, iboacci ad ucas Numbers with Applicatios, Wiley, New York, 00. [] 5 Y.Türker Ulutaş ad N. Özgür, A Iterestig Table Cotaied Biomial Coefficiets, Iteratioal Joural of Cotemporary Mathematical Scieces, vol., Number 6, (006), Received: July 0, 007