It. J. Cotem. Mth. iees Vol. 8 03 o. 9 4-46 HIKARI Lt www.m-hikri.om Geerliztio of Fioi euee i Cse of Four euees jy Hre Govermet College Meleswr M. P. Ii Bijer igh hool of tuies i Mthemtis Vikrm Uiversity Ujji M.P. Ii Gureer Kur Chhr hool of tuies i Mthemtis Vikrm Uiversity Ujji M.P. Ii gureer_khuj@yhoo.om Mjeet igh Teeth M.B. Khls College Iore M.P. Ii Coyright 03 jy Hre et l. This is oe ess rtile istriute uer the Cretive Commos Attriutio Liese whih ermits urestrite use istriutio reroutio i y meium rovie the origil work is roerly ite. Astrt. I this er we geerlize the Fioi seuee i se of four seuees. eletig two more rel umers to e fixe we erive ew ietities. Fmily we efie sustitutio oertio over these shemes erive results. Mthemtis ujet Clssifitio: B39 Keywors: Fioi seuee. INTRODUCTION Atssov gve the ew iretio for geerlizig the Fioi seuee i [3 4]. Atssov geerlizes the Fioi seuee i se of three seuees [5].
4 jy Hre et l. I this er we geerlize the Fioi seuee i se of four seuees. Let C C C 3 C 4 C 5 C 6 C 7 C 8 e fixe rel umers. Usig C to C 8 we shll ostrut ew shemes whih re of the Fioi tye re lle 4- F seuees [ ]. The form is C C C C C C C C 0 0 0 3 0 4 5 6 7 8 x x x 3 x 4 3 4 0 (.) Where 3 4 x + x + x + x + is y ermuttio of 3 4 y y y y is y ermuttio of. + + + +. GENERALIZATION OF THE FIBONACCI EQUENCE Here we shll erive ew sheme of fioi tye s efie i (.) whih is the geerliztio of 3-F seuees we ll it s 4-F seuees. The umer of ifferet shemes is oviously 576. Firstly let us exli the ottio use i this er with the hel of followig sheme: C C C C C C C C 0 0 0 3 0 4 5 6 7 8 + + + + ( 0 ) To mke the reresettio of this sheme simler similr to the reresettio of system of lier eutio i mtrix from we oly elimite susrits symols use write s follows: (.)
Geerliztio of Fioi seuee 43 To miti ooritio etwee (.) (.) rememer tht the elemets i their first olum re lwys i the sme orer while the elemets i the seo thir olum e ermute withi tht olum resetively. Also ote tht every elemet must e use i eh olum. For further isussio of roerties of these shemes first let us efie some of them. imilrly we efie ll the remiig shemes. We ow efie oertio lle sustitutio over these 4-F seuees reresete s [ ] where { }. This oertio simly iterhges ll ourrees of i eh olum. For exmle usig (.) we hve
44 jy Hre et l. [ ] (.) The followig the rule to miti the orer i first olum we rrive t (.3) whih orresos to the sheme 0 0 0 3 0 4 5 6 0 7 8 C C C C C C C C + + + + Where C C C3 C 4 C5 C 6 C 7 C8 re rel umers. O the sis of ove geerliztio we rove the followig imortt theorems: Theorem: If re two shemes mog ll ossile shemes the re euivlet uer the oertio of sustitutio of legth whih mes mthemtilly where reresets the euivlee. Proof: For y two 4-F seuees if [ ] the [ ] { }. For exmle if
Geerliztio of Fioi seuee 45 [ ] o Theorem: If l is sheme hvig ll olums ietil the ll ossile sustitutio oertio of y legth o oes ot lter the struture of. Proof: By ssumtio of the theorem we hve Here ll the sustitutio oertio of y legth (34) simly reles eh elemet y its suessor lst oe reles first. It mes tht } { s r ueul to eh other gives ] [ ] [ ] [ s r r 3. IMPORTANT REULT A oservtio of ll 576 ossile shemes gives followig results: Result -: Oservig ll the ossile shemes for 4-F seuees we get the result tht there re 5 trivil 4-F seuees. A 4-F seuees is efie to e trivil if t lest oe of the resultig seuees is Fioi seuee. Remiig ll shemes re essetil geerliztio of the Fioi seuee.
46 jy Hre et l.. O y trivil seuee returs either the sme trivil seuee or y other trivil seuee. Result -: At lest oe of the sustitutio oertio { } 4. CONCLUION There re severl wys of geerliztio of Fioi seuee there is ew iretio of geerlizig the Fioi seuee. I this er we geerlize Fioi seuee i se of four seuees. The er shows tht how the ermuttio withi the elemets of 4-F Fioi seuee hges the result usig sustitutio oertio.these results oul e geerlize to the se of five or more seuees. REFERENCE. C.Z Ji H.M. Liu T.M. Wg -Alogs of Geerlize Fioi Lus Polyomils. Fioi Qurterly (Fe 007): 6-34. J.A. Lee & J.. Lee ome Proerties of the Geerliztio of the Fioi euee. Fioi Qurterly 5. (987):-7. 3. K Atssov L Atssov & D. sselov A New Persetive to the Geerliztio of the Fioi euee. Fioi Qurterly 3. (985) : -8 4. K. Atssov O New Geerliztio of the Fioi euee. Fioi Qurterly 4.4 (986): 36-365. 5. Krssimir T. Atssov O Geerliztio of the Fioi euee i the se of three seuees Fioi Qurterly (Fe 989): 7-0. Reeive: Jury 03