Iteratioal Joural of Pure a Applie Mathematics Volume 9 No. 5 8 3357-3364 ISSN: 34-3395 (o-lie versio) url: http://www.acapubl.eu/hub/ http://www.acapubl.eu/hub/ A Remark o Relatioship betwee Pell Triagle a Zeckeorf Triagle P. Balamuruga A. Vekatachalam Departmet of Mathematics M. Kumarasamy College of Egieerig (Autoomous) Karur- 6393 Iia. pbalamuruga986@gmail.com Abstract: This article surfaces few of the typical of which is metioe to as a Zeckeorf triagle that is collectio of Pell umber proucts of the Pell sequece. It asces Pell umber relate to the Fiboacci umbers Pascal array. AMS Subject Classificatio: B37. Keywors: Pell umbers Fiboacci umbers Lucas umbers Recurrece relatios Kroecker elta a Zeckeorf escriptio.. Itrouctio I [] Charles K.Cook. A.G.Shao stuie Geeralizatio of ifamous ietities the Fiboacci a Lucas Sequeces respectively by a F F F L F F. The aim of this paper is primarily to collect a relate a umber of kow Lucas - relate triagle i the writte works the popular of that is the presece from the Lucas umbers with the iagoals from the Pascal triagle [] [] [3] [4]. This research article aim to fi the sequeces erive from iagoal a row sums a the partial cetral colum sums of this triagle. To geeralize a result which coects the Fiboacci a Lucas umbers amely L F F. Arisig out of this is a Pascal-type array which is relate to the Fiboacci umbers i the same way that the Pascal array is relate to the Lucas umbers. 3357
Iteratioal Joural of Pure a Applie Mathematics The Pell sequece ( ) is the sequece of positive umbers fulfillig the recurrece = + with the iitial coitios = a =. The Pell sequece cotais lot of uexpecte possessios but the preset article eals with relates to etermiats of matrices oly. The etermiats of matrix that is make eve to the Pell sequece. = For ay.. () = = = 5 = Where = a. = = = ------------- () I () a () The specifie matrices are the particular cases of triiagoal matrix what is a square matrix = of the orer with etries = >. ( ) = Our aim is the coectio of etermiats of particular triiagoal matrices accompayig Pell sequece.we show that matrix i () may be altere ito a matrix which etermiat is make eve to Pell Sequece also. 3358
Iteratioal Joural of Pure a Applie Mathematics Theorem:. Let ( ) ( ) for all sequece of complex umbers with property = for ay. Let { ( ) = 3 } are the Pell sequece from triiagoal matrices of the form = = = + = h ie) ( ) = the ( ) = Proof: By mathematical iuctio o. the positive statemet possessios for = = as () = = () = = 5 = (3) = = = Suppose that the positive statemet possessios for all 3. The we prove positive statemet is true for +. et ( + ) = ( ) ( ) = ( ) ( ) = ( ) + et ( ) = + = Corrollary:. Settig = = a = = = i the positive statemet () we get lot of uexpecte square matrices which etermiats are equal to the pell umber usig theorem ()however there are whole umbers matrices of the oly type for etries = ± = where. 3359
Iteratioal Joural of Pure a Applie Mathematics Corollary:.3 Let ( ) = = 3 are the sequeces of = triiagoal matrix of the particular form = ( ) = ± h ( ) = ( ) ( ) ( ) ( ) ( ) = The Zeckeorf Triagle It is appropriate to call it the Zekeorf triagle rather tha reamig the Pell triagle as it as Pell umbers alog its left a right eges. It come ito sight below i a left - correcte arragemet o accout of this creates several uerstaable that the colums are Lucas umber proucts of the umbers i the Lucas sequece tha whe the triagle is presete i a isosceles format. 5 4 5 9 4 5 4 9 7 58 6 6 58 7 69 4 45 44 45 4 69 48 338 35 348 348 35 338 48 985 86 845 84 84 84 845 86 985 378 97 4 8 3 3 8 4 97 378 574 4756 495 4896 49 49 49 4896 495 4756 574 The sequeces as metioe above i the colum are specifie cases of the geeralize Pell a Fiboacci sequeces P that fulfill the Lucas partial recurrece relatio [3] m P m Pm Pm m 3 () 336
Iteratioal Joural of Pure a Applie Mathematics The sequeces of iagoal row a partial colum sums are labele as r c respectively. It is observe to tur as follows 3 4 5 6 7 8 9 { } 7 6 44 4 64 63 557 { r } 4 4 44 3 376 5 888 783 { c } 6 348 3 83 6895 488 83433 We have forme the { c } triagle i [4] viz from the cetral colum of the origial isosceles from of the { 5 c } { z z3 z53 z74 z9...} {4544 84...} () I which z i j are the members of the isosceles form of the Zeckeorf triagle. The correspoig SequecesF L } a{p} are Fiboacci Lucas a Pell Sequeces that are relate by the equatios { 5F L L 5L 5F 5F (3) (4) The outsie of the triagle it ca otice that we get the recurrece relatios P j (5) (6) i which } is the Kroecker elta a L } are the Lucas umbers; Similarly { i j { r r c L r (mo ) r (mo ) (mo ) (7) (8) (9) () a c c S c () 336
Iteratioal Joural of Pure a Applie Mathematics Here S be a layer susceptibility series form square lattices []: { S } = {3376347643687443557353} () Refereces []. P.Balamuruga a R.Srikath(5).A Remark o Relatioship betwee Lucas Triaglea Zeckeorf Triagle. Iteratioal Joural of Applie Egieerig Research Volume Number 3 pp. 578-5788. []. A. G Shao (). A ote o some Diagoal Row a partial Colum sums of a Zeckeorf triagle. Notes o umber theory a Discrete Mathematics 6 () 33 36. [3]. Boareko Boris A. (993) Geeralize Pascal Triagles a pyramis: Their Fractals Graphs a Pyramis. (Traslate by Richar C. Bolliger.) Sata ClaraCA: The Fiboacci Associatio. [4]. Cook Charles A.G Shao.(6) Geeralize Fiboacci a Lucas sequeces with pascal - type of Arrays. Notes o Number Theory a Discrete mathematics (4) 9. [5]. Griffiths. Marti. () Digit proportios i Zeckeorf Represetatios The Fiboacci Quarterly. 48() 68 74. [6]. Hoggatt V.E. Jr. Marjorie Bickell Johso. (977) Fiboacci Covolutio Sequeces. The Fiboacci Quarterly5() 7. [7]. Y.Bugla F.Luca M.Migotte a S.Sikasik.(8) Fiboacci umbers at most oe away from a perfact Power. Elem. Math.636575. [8]. Mohame Taoufiq Damir B.Farge F.Luca A. Tall.(4) Fiboacci umbers with prime sums of Complemetary ivisors. The Electroic joural of itegers 4 ( - 9). [9]. Pavel Trojovsky (5) O A Sequece of Triiagoal matrices whose etermiats are Fiboocci Numbers F+. Iteratioal Joural of Pure a Applie Mathematics (3) 57 53. []. N. D. Cahil J. R. D Errio a J. P. Spece Complex Factorizatios of the Fiboacci a Lucas Numbers Fib. Quart. 4 No. (3) 3 9. 336
Iteratioal Joural of Pure a Applie Mathematics []. N. D. Cahil D. A. Naraya Fiboacci a Lucas Numbers as triiagoal matrix etermiats Fib. Quart. 4 No. 3 (4) 6. []. K. Kaygisiz A. Sahi Determiat a Permaet of Hesseberg Matrix a Fiboacci Type Numbers Ge. Math. Notes 9 No. () 3. [3]. E. Kilic D. Tasci Negatively subscripte Fiboacci a Lucas umbers a their complex factorizatios Ars combi. 96 () 75 88. [4]. T. Koshy Fiboacci a Lucas Numbers with applicatios Joh Wiley & Sos. [5]. G. Y. Lee S. G. Lee A oto o geeralize Fiboacci umbers Fib. Quart. 33 No. 3 (995) 73 78. [6]. G. Y. Lee J. S. Kim The liear algebra of the k Fiboacci marix Liear Algabra Appl. 373 (3) 75 87. [7]. A. Nalli H. Civciv A geeralizatio of matrix etermiats Fiboacci a Lucas umbers Chaos Solitos Fractals 4 No. (9) 355 36. [8]. A. A. Ocal N. Tuglu a E. Altiisik O the represetatio of k geeralize Fiboacci a Lucas umbers Appl Math. Comput. 7 (5) 584 596. [9]. G. Strag Liear Algebra a its applicatios Brooks/Cole 3 r eitio 988. []. F. Yilmaz T. Sogabe a ote o symmetric k triiagoal matrix family a Fiboacci umbers It. J. Pure a Appl. Math. 96 No. (4) 89 98. 3363
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