Notes o Nube Theoy ad Discete Matheatics it ISSN Olie ISSN 67 87 Vol. 4 8 No. DOI:.746/td.8.4..- Geealized golde atios ad associated ell sequeces A. G. Shao ad J. V. Leyedees Waae College The Uivesity of New South Wales NSW Austalia e-ail: tshao8@gail.co aculty of Sciece The Uivesity of Sydey NSW 6 Austalia e-ail: jeavalde@gail.co Received: 6 May 8 Accepted: August 8 Abstact: This pape cosides geealizatios of the golde atio based o a extesio of the ell ecuece elatio. These iclude elated patial diffeece equatios. It develops geealized ell ad Copaio-ell ubes ad shows how they ca yield elegat geealizatios of iboacci ad Lucas idetities. This sheds light o the foat of the oigial idetities such as the Siso foula to distiguish what is sigificat ad substatial fo what is icidetal o accidetal. Keywods: Golde atio iboacci ubes Lucas ubes ell ubes Copaio-ell ubes Siso s idetity Biet foula Recuece elatios Diffeece equatios ythagoea tiples. Matheatics Subject Classificatio: B9. Itoductio Each of the equatios: x x 4 x x x x 4x has two of its oots ( ) ± because each polyoial has x x as facto. The positive oot of (.) is ueically equal to φ the Golde Ratio ow sice the tie of Euclid []. Siply φ is the atio of two lie segets (ajo/io) ad siultaeously the atio of the total lie (.)
legth ove the ajo lie seget (that is both atios ae equal). o otatioal coveiece we let φ ψ. utheoe φ is essetial to a ultitude of atual pattes fo suflowe floets to the shapes of galaxies ad has bee studied extesively ove the cetuies. I 9 acioli published a thee volue teatise o the Divie opotio that is φ. The atio has bee used i the ovel Goldpoit Geoety []. We coside hee soe othe popeties of φ to geealize soe wellow secod ode ecusive sequeces. The Golde Ratio ad iboacci ubes The powes of φ ae elated to the eleets of the iboacci sequece {} []: ϕ ϕ (.) ψ ϕ. (.) These powe elatios wee ow to Eule (77 78) ad de Moive (667 74) ad ediscoveed by Biet (786 86) oe of the discovees of the foula fo the geeal te of the iboacci sequece [4]: ϕ ψ. (.) We ote the coespodig foula fo the geeal te of the Lucas sequece: so that Thus so that L ϕ ψ L. ϕ ψ (.4) ϕ ψ li ϕ (.) which ca be used i calculatig iboacci ubes fo lage eve though φ is iatioal ad is atioal. Nevetheless the decial pattes fo the latte ca have a vey log iteval befoe epeatig; fo exaple whee the last digits of > have a peiodicity of whee is the ube of ed digits Thus fo the peiodicity of the thee ed digits is. This ca lead to elatively lage decial epeat pattes fo the golde atio: 7 84 97 8.6884487864746994996788844477448 Stahov [] used Table below to pove vaious esults fo ϕ ± ψ. 4
4 6 7 ϕ 4 7 8 8 9 ψ 4 7 8 8 9 Table. Stahov s powes of the golde atio The appeaace of the Lucas ad iboacci ubes i the ueatos is o accidet. Stahov s cotext was a foudatio fo copute aithetic [cf. 6] but his Table also suggests Q 8 extedig fo istace i which {} ad {Q} ae the classical ell { } ad Copaio-ell { 6 4 4 } sequeces espectively which we shall attept i the ext sectio. Hee we set ad L 4 (.6) L 4 β (.7) i a ae aalogous to that of [7]. Z hee is associated with geealized ell {} ad Copaio-ell sequeces {Q} [8] which satisfy the liea hoogeeous secod ode ecuece elatio: (.8) with iitial coditios (fo otatioal coveiece) Q Q to iclude taditioal iboacci Lucas ell ad Copaio-ell sequeces ad thei geealizatios as we see i Tables ad. I Hoada s otatio [9] these ae { ( ; )} {L ( ; )}. These ae extesios of []. The cases {w (a b; q)} ae diffeet but siila. Exaples of (.6) ad.7 iclude β 6 6 β 8 8 β 6 8 6 8 β. Thus Equatios (.6) (.7) ad (.8) geeate the taditioal Biet foulas fo the geeal tes []; that is
β 4 ad β Q. We see the that as a geealizatio of (.): β β Q 4 as. o exaples fo Tables ad : ad so o..6.6.68; 6 4 8.47.44.44; 6 4.8.8.7 6 4 4 6 8 9 7 69 9 6 89 4 4 7 7 9 47 6 7 64 79 6 6 7 8 4 868 7 7 7 49 8 9949 Table. Soe values of 4 6 4 7 8 6 4 4 8 98 6 9 9 98 4 4 8 76 64 778 7 4 77 77 96 6 6 8 4 44 8886 478 7 7 64 99 87 498 Table. Soe values of 6
7 owes of the coefficiet i the ecuece elatio We ca exted the ecuece elatio (.8) by fo exaple cosideig powes of the coefficiet. (.) with iitial coditios as befoe so that fo istace {} {} {} {} the odiay ell ubes. This is a extesio of []. We shall specifically focus o {} hee; that is. (.) o istace 4 6 9 7 69 9 8 6 68 889 7 8 96 7969 4 8 678 96 Table 4. Soe values of If we wish to exted to a abitay ode atix fo odificatios of the va de Laa ad ell adova sequeces [] we ca use atices. ) ( ad R ) ( with a coespodig ) ( Q so that ) ( Q
ad ( ).. The last two atices ca obviously be used to geeate popeties aalogous to well-ow iboacci sequece popeties. It would also see that fo aalogues of the Siso idetity: det () ( ) (.) det () ( ) (.4) det () ( ) so that i which det (4) ( ) 9 det ( ) ( ) w w w w. (.) Equatio (.) is a fist ode o-hoogeeous diffeece equatio with a stadad solutio of w. We ote that (.) cofis Siso s idetity but (.4) ad followig show that it is pat of a bigge pictue. Abitay ode extesios ca also be defied by S S S (.6) with iitial coditios S. o istace whe ad we get the sequeces set out i Table : 4 6 7 6 7 4 9 889 69 977 4 49 786 79 Table. Soe values of S o this we ca see that thid ode aalogues of the Siso idetity have the foats det () det () det () det (4) ( ) ( ) ( ) ( ) 8 8 94 4 66.
4 Cocludig coets It ca be eadily cofied that thee ae ay elegat aalogues of well-ow iboacci ad Lucas ubes. o istace (allowig fo the otatioal vaiatios i the iitial values ad L (4.) ( ) (4.) as a vaiatio of Siso s idetity. Soewhat siila ideas fo extesios ay be foud i [4]. It is also of iteest to exploe soe of the popeties of Tables ad. o exaple Equatio (.8) is a patial diffeece equatio with as the vaiable ad we have so fa cosideed the ow sequeces of these tables. If howeve we ae the vaiable (that is coside the colu sequeces) the it ca be eadily cofied by calculatio fo Table that fo it holds that but is thee a patial diffeece equatio i (o ) fo all? It should be oted that the iboacci sequece equatio fo the Golde Ratio aily {ϕ} [] is (4.) with a stuctual vaiable fo Modula Rig Theoy cotasts with Equatio (.8) whee is lied to the ell sequeces ad is the ultiplied copoet. The oigial Golde Ratio aily sud is a ϕ a (4.4) with (a 4 ) 4 Z4 a odula ig i cotast with the suds costucted hee with types of geealized ell ubes. Sice 4 whe is odd the su of squaes ust equal 4 o a. This povides a li betwee the two golde atio failies ad also with piitive ythagoea tiples [6]. Refeeces [] Livio M. () The Golde Ratio. Golde Boos New Yo. [] Ataassov K. Ataassova V. Shao A. & Tue J. () New Visual espectives o the iboacci Nubes. Wold Scietific New Yo. [] Hoggatt V. E. J. (969) iboacci ad Lucas Nubes. Houghto-Miffli Bosto. 9
[4] Tee G. J. () Russia easat Multiplicatio ad Egyptia Divisio i Zecedof Aithetic. Austalia Matheatical Society Gazette 67 76. [] Stahov O. (997) Copute Aithetic based o iboacci Nubes ad Golde Sectio: New Ifoatio ad Aithetic Copute oudatios. Viitsa: Uaiia Acadey of Egieeig Scieces Ch.. [6] Gaha R. L. Kuth D. E. & atashi O. (99) Cocete Matheatics: A oudatio fo Copute Sciece. Addiso-Wesley Readig MA. [7] Shao A. G. & Leyedees J. V. () The Golde Ratio faily ad the Biet equatio. Notes o Nube Theoy ad Discete Matheatics () 4. [8] alco S. (8) Soe ew foulas o the K-iboacci ubes. Joual of Advaces i Matheatics 4() 749 744. [9] Hoada A.. (96) Basic popeties of a cetai geealized sequece of ubes. The iboacci Quately () 6 76. [] Coo C. K. & Shao A. G. (6) Geealized iboacci ad Lucas sequeces with ascal-type aays. Notes o Nube Theoy ad Discete Matheatics (4) 9. [] Shao A. G. Adeso. G. & Hoada A.. (6) opeties of Codoie ei ad va de Laa Nubes. Iteatioal Joual of Matheatical Educatio i Sciece ad Techology 7(7) 8 8. [] Shao A. G. & Hoada A.. (4) Geealized ell ubes ad polyoials. I: edic T. Howad (ed.) Applicatios of iboacci Nubes Volue 9. Dodecht/Bosto/ Lodo: Kluwe 4. [] Deveci Ö. & Shao A. G. (7) ell adova-ciculat sequeces ad thei applicatios. Notes o Nube Theoy ad Discete Matheatics () 4. [4] Shao A. G. & Leyedees J. V. (8) The iboacci Nubes ad Itege Stuctue. Nova Sciece ublishes New Yo. [] Leyedees J. V. & Shao A. G. () The Golde Ratio aily ad geealized iboacci Nubes. Joual of Advaces i Matheatics () 7. [6] Leyedees J. V. & Ryba J. M. (99) ellia sequeces deived fo ythagoea tiples. Iteatioal Joual of Matheatical Educatio i Sciece ad Techology 6(6) 9 9.