On the Fibonacci-like Sequences of Higher Order

Transcription

1 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 Higher Order ISSN: 66-86X Florida, USA Deepika Jhala,*, GPS Rathore, Kira Sisodiya School of Studies i atheatics, Vikra Uiversity, Ujjai (Idia) College of Horticulture, adsaur (P) Idia * Author to who correspodece should be addressed; E-ail: jhaladeepika8@gailco Article history: Received 0 Deceber 0, Received i revised for 5 February 05, Accepted 0 April 05, Published April 05 Abstract: I this paper, we defie Fiboacci-like sequeces of higher order ad derived explicit forulas for solvig Fiboacci-like sequeces of higher order Forulas were validated for ay value of usig iductio Keywords: Fiboacci like sequeces, Fiboacci like sequece of higher order atheatics Subject Classificatio Code (00): B3, B99 Itroductio It is well-kow that the Fiboacci ad Lucas sequeces are ost proiet exaples of secod order recursive sequeces A Fiboacci sequece represets a sequece of ubers where the curret eber is calculated as a su of two previous cosecutive ubers [] For exaple, the sequece: 0,,,,3,5,8,3,,3,55,89, is a Fiboacci sequece, where F 0 0 ad F ad out of which the whole sequece ca be geerated usig the equatio: F F F ; Lucas sequece o the other had represets a sequece of ubers where the curret eber is also calculated as a su of two previous cosecutive ubers where the iitial uber is For exaple, the sequece: Copyright 05 by oder Scietific Press Copay, Florida, USA

2 It J oder ath Sci 05, 3(): ,,3,,,,8,9,,6,3,99, is a Lucas sequece, where L 0 ad L ad out of which the whole sequece ca be geerated usig the equatio: L L L ; ay authors have bee geeralized secod order recurrece sequeces by preservig the recurrece relatio ad alterig the first two ters of the sequece, while others have geeralized these sequeces by preservig the first two ters of sequece but alterig the recurrece relatio slightly Fiboacci sequece is a successio of ubers that are obtai through addig the two precedig ubers A derivative of this sequece is called Fiboacci like sequece Lucas, triboacci ad tetraacci sequeces are exaple of this sequece Geerally Fiboacci like sequece ca be expressed as [0] S S S ; Natividad [6], Derivig a Forula i solvig Fiboacci-Like sequece He foud issig ters i Fiboacci-Like sequece ad solved by stadard forula Levesque [3] defied Fiboacci sequece of th order ad derived a geeratig fuctio ad Biet s forula for Fiboacci sequece of th order iles [5] has give soe properties for the geeralized k-fiboacci ubers are defiig as for k, f f f f f f, 3 k f f f f f f 0, f where 0 3 k k Furtherore, Lee et al [], itroduced the geeralized k-fiboacci ubers ad derived the Biet forula for the geeralized Fiboacci sequece f Natividad [] established a forula for solvig the th ter of the Triboacci-Like sequece Sigh [9] defied a Tetraacci-Like sequece ad the preset the geeral forula for th ter of the Tetraacci-Like sequece I this paper, we will derive a geeral forula for fidig the th ter of Fiboacci-like sequece of seveth order (Heptaacci-like sequece), Fiboacci-like sequece of eighth order (Octaacci-like sequece) ad Fiboacci-like sequece of th order (-acci sequece) Fiboacci Like Sequeces of Higher Order Defiitio The sequeces Z, Z, Z3, Z,, Z i which Z Z Z 6 Z 5 Z Z 3 Z Z is a geeralizatio for the Fiboacci-like sequece of seveth order (Heptaacci -like sequece) This sequece follows the patter of Fiboacci sequece of seveth order (Heptaacci sequece) Copyright 05 by oder Scietific Press Copay, Florida, USA

3 It J oder ath Sci 05, 3(): The Heptaacci sequece S defied by the recurrece relatio S S S S S S S S for () 3 5 6, where S0 S S S3 S S5 S6 0, First few ters for Heptaacci sequece are 0, 0, 0, 0, 0, 0,,,,, 8, 6, 3, 6 (A0668) [] Defiitio The sequece,, 3,,, i which is a geeralizatio for the Fiboacci-like sequece of eighth order (Octaacci-like sequece) This sequece follows the patter of Fiboacci sequece of eighth order (Octaacci sequece) The Octaacci sequece Q defied by the recurrece relatio Q Q Q Q Q Q Q Q Q for () , where Q0 Q Q Q3 Q Q5 Q6 Q 0, First few ters for octaacci sequece are 0, 0, 0, 0, 0, 0, 0,,,,, 8, 6, 3, 6 (A096)[] Defiitio 3 The sequece H 3 5, i which H H H H H H H H H is geeralizatio for the Fiboacci-like sequece of th order (-acci-like sequece) This sequece follows the patter of Fiboacci sequece of th order [5] (-acci sequece) The -acci sequece I defied by the recurrece relatio where I0 I I I3 I I I I I I I 3 I I 5 I 6 I for, (3) 0, I [8], Natvidad derived forulae for Fiboacci-like sequeces of higher order Now i this sectio we will exted the work of Natvidad [8] o Fiboacci-like sequece of order seve, order eighth ad order th 3 ai Results Fro above discussios ad preliiaries, the followig theores are proposed ad proved Theore 3 For ay real ubers Z, Z, Z3, Z, Z5, Z6, Z the forula for fidig the th ter of geeralized Fiboacci-like sequece of seveth order (Heptaacci -like sequece) is i i 3 i i 5 i 6 i i i i i (3) Z S Z S Z S Z S Z S Z S Z S Z Copyright 05 by oder Scietific Press Copay, Florida, USA

4 It J oder ath Sci 05, 3(): where Z is the th ter of Heptaacci-like sequece, Z is the first ter, Z is the secod ter, Z is 3 the third ter, Z is the fourth ter, Z 5 is the fifth ter, Z 6 is the sixth ter, Z is the seveth ter ad S, S, S 3, S, S 5, S 6, S are the correspodig Fiboacci ubers of seveth order (Heptaacci ubers) Proof: Let the first seve ters of Heptaacci-like sequece be Z, Z, Z3, Z, Z5, Z6, Z The we will derive a explicit forula for Z give the first seve ters The sequece Z, Z, Z 3, Z,, Z is kow as geeralized Heptaacci sequece (Heptaacci-like sequece) We begi by coputig the uerical coefficiets for the first seve ters of the Heptaacci-like sequece Z, Z, Z3, Z,, Z Equatios were derived ad coefficiets are give for 8 Each coefficiet correspods to the Heptaacci uber We observe that that the coefficiet of Z correspod to S, Z correspod to S S, 3 Z 3 correspod to SS 3 S So we coclude that the th ter Z is equal to i i 3 i i 5 i 6 i i i i i S Z S Z S Z S Z S Z S Z S Z By usig atheatical iductio, the forula ca be validate for ay values of The forula ca be easily verify usig = 8, 9, 0 ad so o Let Pbe ( ) take as i i 3 i i 5 i 6 i i i i i Z S Z S Z S Z S Z S Z S Z S Z Now, we assue that Theore is true for soe iteger k 8 the is k k ki ki 3 ki ki 5 ki 6 k i i i i i P( k) : Z S Z S Z S Z S Z S Z S Z S Z We shall ow prove that Pk ( ) is true wheever is true that is (3) k k ki ki 3 ki ki 5 ki 6 k i i i i i P( k ) : Z S Z S Z S Z S Z S Z S Z S Z Now to verify, we will provide the assuptio of iplies the truth of Pk ( ) To do so, we will add Z, Z, Z, Z, Z ad Z to both side of Pkthe ( ) equatio (3) will becoe k k k 3 k k 5 k 6 (33) Z S Z S Z S Z S Z S Z S Z S Z Z Z Z Z Z Z Z ki k ki ki 3 ki ki 5 ki 6 k k k k3 k k5 k6 k i0 i i i k i But sice k k ki ki 3 ki ki 5 ki 6 k i i i i i Z S Z S Z S Z S Z S Z S Z S Z (3) Copyright 05 by oder Scietific Press Copay, Florida, USA

5 It J oder ath Sci 05, 3(): Substitutig ad rearragig the ters i equatio (3) we have Zk Ski Z Ski Ski Z Ski Ski Ski Z3 Ski Ski Ski Ski Z Ski Ski i i i3 i i3 i i i3 i i5 i i Ski Ski Ski Z5 Ski Ski Ski Ski Ski Ski Z6 Ski i i5 i6 i i3 i i5 i6 i i Now by the defiitio of Heptaacci sequece equatio () we have Z Z S Z S S Z S S S Z S S S S Z S S S S S Z k k k k k k k3 3 k k k3 k k k k3 k k5 5 S S S S S S Z S Z k k k3 k k5 k6 6 k Thus by atheatical iductio Pk ( ) is true, wheever is true (35) Hece i i 3 i i 5 i 6 i i i i i Z S Z S Z S Z S Z S Z S Z S Z Theore 3 For ay real ubers,, 3,, 5, 6,, the forula for fidig the th ter 8 of geeralized Fiboacci-like sequece of eighth order (Octaacci -like sequece) is where i i 3 i i 5 i 6 i 8 i i i i i i Q Q Q Q Q Q Q Q is the th ter of Octaacci -like sequece, is the first ter, (36) is the secod ter, is the third ter, is the fourth ter, 5 is the fifth ter, 6 is the sixth ter, is the seveth ter, 8 is the eighth ter ad Q, Q, Q 3, Q, Q 5, Q 6, Q, Q are the correspodig 8 Fiboacci ubers of eighth order (Octaacci ubers) Proof: Let the first seve ters of Octaacci-like sequece be,, 3,, 5, 6,, The 8 3 we will derive a explicit forula for give the first seve ters The sequece,, 3,,, is kow as geeralized Octaacci sequece (Octaacci-like sequece) We begi by coputig the uerical coefficiets for the first seve ters of the Octaacci-like seque0ce,, 3,,, Equatios were derived ad coefficiets are give for 9 Each coefficiet correspods to the Octaacci uber We observe that that the coefficiet of correspod toq, correspod toq Q, 3 correspod toq 3 Q 3 Q So we coclude that the th ter is equal to i i 3 i i 5 i 6 i 8 i i i i i i Q Q Q Q Q Q Q Q By usig atheatical iductio, the forula ca be validate for ay values of The forula ca be easily verify usig = 9, 0, ad so o Copyright 05 by oder Scietific Press Copay, Florida, USA

6 It J oder ath Sci 05, 3(): Let P ( ) be take as i i 3 i i 5 i 6 i 8 i i i i i i Q Q Q Q Q Q Q Q Now, we assue that Theore is true for soe iteger k 9 the k k ki ki 3 ki ki 5 ki 6 ki k 8 i i i i i i P( k) : Q Q Q Q Q Q Q Q We shall ow prove that Pk ( ) is true wheever is true that is k k ki ki 3 ki ki 5 ki 6 ki k 8 i i i i i i P( k ) : Q Q Q Q Q Q Q Q Now to verify, we will provide the assuptio of k k k 3 k k 5 k 6 k,,,,, ad to both side of is iplies the truth of Pk ( ) (3) (38) To do so, we will add the equatio (3) will becoe ki Qk Qk i Qk i 3 Qk i Qk i 5 Qk i 6 Qk i (39) i0 i i i k i i Q k 8 k k k3 k k5 k6 k Substitute the values of,,,,, ad ad rearragig the ters the equatio (39) becoes k k k 3 k k 5 k 6 k k Qk i Qk i Qk i Qk i Qk i Qk i 3 Qk i Qk i Qk i Qk i Qk i Qk i i i i3 i i3 i i i3 i i5 i i Qk i Qk i Qk i 5 Qk i Qk i Qk i Qk i Qk i Qk i 6 Qk i Qk i Qk i i i5 i6 i i3 i i5 i6 i i i3 i Qk i Qk i Qk i Qk i Qk i 8 i5 i6 i i8 i Now by the defiitio of Octaacci sequece equatio () Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q k k k k k k k3 3 k k k3 k k k k3 k k5 5 k k k3 k k5 k6 6 k k k3 k k5 k6 k k 8 Thus by atheatical iductio Pk ( ) is true, wheever is true (30) Hece i i 3 i i 5 i 6 i 8 i i i i i i Q Q Q Q Q Q Q Q Theore 33 For ay real ubers H 3,H 5,,H the forula for fidig the th ter of geeralized Fiboacci-like sequece of th order (-acci -like sequece) is 3 i i 3 i i i i i i H I H I H I H I H I H I H (3) Copyright 05 by oder Scietific Press Copay, Florida, USA

7 It J oder ath Sci 05, 3(): where H is the th ter of -acci -like sequece3,h,,h,h,h is the first, secod, third, forth,, (-) th, th ter ad I, I, I 3,I,,I (),I are the correspodig Fiboacci ubers of th order (-acci ubers) Proof: Let the first th ters of Octaacci-like sequece are H 3,H 5,,H The we will derive a explicit forula for H give the th ters The sequece H 3,H 5,,H is kow as geeralized -acci sequece (-acci like sequece) We begi by coputig the uerical coefficiets for the first ters of the -acci like sequece H 3,H 5,,H Equatios were derived ad coefficiets are give for Each coefficiet correspods to the -acci uber We observe that that the coefficiet of H correspod to I correspod to I I correspod to 3 3 I I3 I So we coclude that the th ter H is equal to 3 i i 3 i i i i i i I H I H I H I H I H I H By usig atheatical iductio, the forula ca be validate for ay values of The forula ca be easily verified usig,, 3 ad so o Let P ( ) be take as 3 i i 3 i i i i i i H I H I H I H I H I H I H Now, we assue that Theore is true for soe iteger k the is 3 k k ki ki 3 ki ki k i i i i P( k) : H I H I H I H I H I H I H We shall ow prove that Pk ( ) is true wheever is true that is (3) 3 k k ki ki 3 ki ki k i i i i P( k ) : H I H I H I H I H I H I H Now to verify, we will provide the assuptio of k k k 3 k k 5 k ( ) (33) iplies the truth of Pk ( ) To do so, we will add H, to both side of the equatio will becoe H I H I H I H I H I H I H H H H H H 3 (3) ki k ki ki 3 ki ki i0 i i i i k k k k3 k k5 k() Substitute the values of Hk k k 3k k 5, ad rearragig the ters by the defiitio k() of -acci sequece equatio (3) The equatio (3) becoes 3 k k ki ki 3 ki ki k i i i i H I H I H I H I H I H I H Copyright 05 by oder Scietific Press Copay, Florida, USA

8 It J oder ath Sci 05, 3(): Thus by atheatical iductio Hece Pk ( ) is true, wheever is true 3 i i 3 i i i i i i H I H I H I H I H I H I H Refereces [] Brow C, The Fiboacci Aalysis (Blooberg Professioal), Blooberg Press, 008 [] Koshy T, Fiboacci ad Lucas Nuber with Applicatio, Wiley, New York, 00 [3] Levesque C, O -TH Order Liear Recurreces, Fiboacci Quart 3()(985): 9-93 [] Lee G-Y, Lee S-G, Ki J-S, Shi HK, The Biet forula ad represetatios of k-geeralized Fiboacci ubers, Fiboacci Quart 39 ()(00): 58-6 [5] iles EP Jr, Geeralized Fiboacci ubers ad associated atrices, Aer ath othly, 6 (960): 5-5 [6] Natividad LR, Derivig a forula i solvig Fiboacci-like sequece, Iteratioal Joural of atheatics ad Scietific Coputig, () (0): 9- [] Natividad LR ad Policarpio PB, A ovel forula i solvig Triboacci-like sequece, Ge ath Notes, ()(03): 8-8 [8] Natividad L R, O Solvig Fiboacci-Like Sequeces of Fourth, Fifth ad Sixth Order, Iteratioal Joural of atheatics ad Scietific Coputig, 3() (03): 38-0 [9] Sigh B, Bhadouria P, Sikhwal O ad Sisodiya K, A Forula for Tetraacci-Like Sequece, Ge ath Notes, 0()(0): 36- [0] Sigh B, Sikhwal O, ad Bhatagar S, Fiboacci-Like Sequece ad its Properties, It J Cotep ath Scieces, 5(8)(00): [] Sloae N J A, The O-Lie Ecyclopedia of Iteger Sequeces wwwresearchattco/~jas/sequeces/ Copyright 05 by oder Scietific Press Copay, Florida, USA