On Generalized Fibonacci Numbers

Transcription

1 Applied Mathematical Scieces, Vol. 9, 215, o. 73, HIKARI Ltd, O Geeralized Fiboacci Numbers Jerico B. Bacai ad Julius Fergy T. Rabago Departmet of Mathematics ad Computer Sciece College of Sciece Uiversity of the Philippies Baguio Baguio City 26, Philippies Copyright c 215 Jerico B. Bacai ad Julius Fergy T. Rabago. This article is distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Abstract We provide a formula for the th term of the k-geeralized Fiboaccilike umber sequece usig the k-geeralized Fiboacci umber or k- acci umber, ad by utilizig the ewly derived formula, we show that the limit of the ratio of successive terms of the sequece teds to a root of the equatio x+x k = 2. We the exted our results to k-geeralized Horadam kgh ad k-geeralized Horadam-like kghl umbers. I dealig with the limit of the ratio of successive terms of kgh ad kghl, a lemma due to Z. Wu ad H. Zhag [8] shall be employed. Fially, we remark that a aalogue result for k-periodic k-ary Fiboacci sequece ca also be derived. Mathematics Subject Classificatio: 11B39, 11B5 Keywords: k-geeralized Fiboacci umbers, k-geeralized Fiboacci-like umbers, k-geeralized Horadam umbers, k-geeralized Horadam-like umbers, covergece of sequeces 1 Itroductio A well-kow recurrece sequece of order two is the widely studied Fiboacci sequece {F } =1, which is defied recursively by the recurrece relatio F 1 = F 2 = 1, F +1 = F + F

2 3612 Jerico B. Bacai ad Julius Fergy T. Rabago Here, it is covetioal to defie F =. I the past decades, may authors have extesively studied the Fiboacci sequece ad its various geeralizatios cf. [2, 3, 4, 6, 7]. We wat to cotribute more i this topic, so we preset our results o the k-geeralized Fiboacci umbers or k-acci umbers ad of its some geeralizatios. I particular, we derive a formula for the k-geeralized Fiboacci-like sequece usig k-acci umbers. Our work is motivated by the followig statemet: Cosider the set of sequeces satisfyig the relatio S = S 1 + S 2. Sice the sequece {S } is closed uder term-wise additio resp. multiplicatio by a costat, it ca be viewed as a vector space. Ay such sequece is uiquely determied by a choice of two elemets, so the vector space is two-dimesioal. If we deote such sequece as S, S 1, the the Fiboacci sequece F =, 1 ad the shifted Fiboacci sequece F 1 = 1, are see to form a caoical basis for this space, yieldig the idetity: S = S 1 F + S F 1 2 for all such sequeces {S }. For example, if S is the Lucas sequece 2, 1, 3, 4, 7,..., the we obtai S := L = 2F 1 + F. Oe of our goals i this paper is to fid a aalogous result of the equatio 2 for k-geeralized Fiboacci umbers. The result is sigificat because it provides a explicit formula for the th term of a k-acci-like resp. k- geeralized Horadam ad k-geeralized Horadam-like sequeces without the eed of solvig a system of equatios. By utilizig the formula, we also show that the limit of the ratio of successive terms of a k-acci sequece teds to a root of the equatio x + x k = 2. We the exted our results to k-geeralized Horadam ad k-geeralized Horadam-like sequeces. We also remark that a aalogue result for k-periodic k-ary Fiboacci sequeces ca be derived. 2 Fiboacci-like sequeces of higher order We start off this sectio with the followig defiitio. Defiitio 2.1. Let N {} ad k N\{1}. Cosider the sequeces {F k } = ad { } = havig the followig properties: ad F k = = {, < k 1; 1, = k 1; 3 k F k i, > k 1, G k, k 1; k Gk i, > k 1, 4

3 O Geeralized Fiboacci umbers 3613 ad for some [, ]. The terms F k ad satisfyig 3 ad 4 are called the th k-geeralized Fiboacci umber or th k-step Fiboacci umber cf. [7], ad th k-geeralized Fiboacci-like umber, respectively. For {F k } =, some famous sequeces of this type are the followig: k ame of sequece first few terms of the sequece 2 Fiboacci, 1, 1, 2, 3, 5, 8, 13, 21, 34,... 3 Triboacci,, 1, 1, 2, 4, 7, 13, 24, 44, 81,... 4 Tetraacci,,, 1, 1, 2, 4, 8, 15, 29, 56, 18,... 5 Petaacci,,,, 1, 1, 2, 4, 8, 16, 31, 61, 12,... By cosiderig the sequeces {F k } = ad { } = we obtai the followig relatio. Theorem 2.2. Let F k ad be the th k-geeralized Fiboacci ad k- geeralized Fiboacci-like umbers, respectively. The, for all atural umbers k, k 3 = F k 1 + j= F k 1 j + F k. 5 Proof. We prove this usig iductio o. Let k be fixed. Equatio 5 is obviously valid for < k. Now, suppose 5 is true for r k where r N. The, r+1 = = r+1 i k 3 F k r+1 i k 3 = F k r j= F k r+1 i j= F k r+1 1 j F k r+1 i 1 j + F k r+1. Remark 2.3. Usig the formula obtaied by G. P. B. Dresde cf. [2, Theorem 1], we ca ow express explicitly i terms of as follows: = k 3 Ai; kαi 2 + Ai; kα 2 j i j= + Ai; kα 1 i,

4 3614 Jerico B. Bacai ad Julius Fergy T. Rabago where Ai; k = α i 1[2 + k + 1α i 2] 1 ad α 1, α 2,..., α k are roots of x k x 1 =. Aother formula of Dresde for F k cf. [2, Theorem 2] ca also be used to express explicitly i terms of. More precisely, we have = Roud [ k 3 Akαi 2 ] + [ ] Roud Akα 2 j i j= + Roud [ Akαi 1 ], where Ak = α 1[2 + k + 1α 2] 1 for all 2 k ad for α the uique positive root of x k x 1 =. Extedig to Horadam umbers I 1965, A. F. Horadam [5] defied a secod-order liear recurrece sequece {W a, b; p, q} =, or simply {W } = by the recurrece relatio W = a, W 1 = b, W +1 = pw + qw 1, 2. The sequece geerated is called the Horadam s sequece which ca be viewed easily as a certai geeralizatio of {F }. The th Horadam umber W with iitial coditios W = ad W 1 = 1 ca be represeted by the followig Biet s formula: W, 1; p, q = α β 2, α β where α ad β are the roots of the quadratic equatio x 2 px q =, i.e. α = p+ p 2 + 4q/2 ad β = p p 2 + 4q. We exted this defiitio to the cocept of k-geeralized Fiboacci sequece ad we defie the k-geeralized Horadam resp. Horadam-like sequece as follows: Defiitio 2.4. Let q i N for i {1, 2,..., }. For k, the th k- geeralized Horadam sequece, deoted by {U k,..., 1; q 1,..., q k } =, or simply {U k } =, is a sequece whose th term is obtaied by the recurrece relatio U k = q 1 U k 1 + q 2 U k q k U k k = q i U k i, 6 with iitial coditios U k i = for all i < k 1 ad U k = 1. Similarly, the k-geeralized Horadam-like sequece, deoted by {V k a,..., a ; q 1,..., q k } = } =, has the same recurrece relatio give by equatio 6 but or simply {V k with iitial coditios V k i at least oe of them is ot zero. = a i for all i k 1 where a i s N {} with

5 O Geeralized Fiboacci umbers 3615 It is easy to see that whe q 1 = = q k = 1, the U k,..., 1; 1,..., 1 = F k ad V k a,..., a ; 1,..., 1 = G k. Usig Defiitio 2.4 we obtai the followig relatio, which is a aalogue of equatio 5. Theorem 2.5. Let U k ad V k be the th k-geeralized Horadam ad th k-geeralized Horadam-like umbers, respectively. The, for all k, V k k 3 = q k V k U k 1 + V k j= q k +j U k 1 j + V k U k. 7 Proof. The proof uses mathematical iductio ad is similar to the proof of Theorem 2.2. Covergece properties I the succeedig discussios, we preset the covergece properties of the sequeces {F k } =, { } =, {U k } =, ad {V k } =. First, it is kow e.g. i [7] that lim F k /F k 1 = α, where α is a k-acci costat. This costat is the uique positive real root of x k x 1 = ad ca also be obtaied by solvig the zero of the polyomial x k 2 x 1. Usig this result, we obtai the followig: Theorem 2.6. lim Gk / 1 = α, 8 where α the uique positive root of x k x 1 =. Proof. The proof is straightforward. Lettig i / 1 we have lim Gk / 1 = lim = lim = = α. Gk F k 1 + k 3 F k 2 + k 3 + k 3 F k 2 F k 1 + k 3 α 1 + k 3 j= F k 1 j j= F k 2 j F k 1 j j= + k 3 F k 1 j= j= α j F k 2 j F k 1 + α + j= α j F k + F k 1 F k F k 1 +

6 3616 Jerico B. Bacai ad Julius Fergy T. Rabago Now, to fid the limit of U k /U k 1 resp. V k /V 1 k as we eed the followig results due to Wu ad Zhag [8]. Here, it is assumed that the q i s satisfy the iequality q i q j 1 for all j i, where 1 i, j k with 2 k N. Lemma 2.7. [8] Let q 1, q 2,..., q k be positive itegers with q 1 q 2 q k 1 ad k N\{1}. The, the polyomial fx = x k q 1 x q 2 x k 2 q x q k, 9 i has exactly oe positive real zero α with q 1 < α < q 1 + 1; ad ii its other k 1 zeros lie withi the uit circle i the complex plae. Lemma 2.8. [8] Let k 2 ad let {u } = be a iteger sequece satisfyig the recurrece relatio give by u = q 1 u 1 + q 2 u q u k+1 + q k u k, > k, 1 where q 1, q 2,..., q k N with iitial coditios u i N {} for i < k ad at least oe of them is ot zero. The, a formula for u may be give by u = cα + Od, 11 where c >, d > 1, ad q 1 < α < q is the positive real zero of fx. We ow have the followig results. Theorem 2.9. Let {U } = be the iteger sequece satisfyig the recurrece relatio 6 with iitial coditios U k i = for all i < k 1, 2 k N ad U k = 1 with q 1 q 2 q k 1. The, U k = cα + Od, 12 where c >, d > 1, ad α q 1, q is the positive real zero of fx. Furthermore, lim /U k 1 = α. 13 U k Proof. Equatio 12 follows directly from Lemmas 2.7 ad 2.8. To obtai 13, we simply use 12 ad take the limit of the ratio U k /U k 1 as ; that is, we have the followig maipulatio: lim U k /U k cα + Od 1 = lim cα 1 + Od 1 = cα + lim Od /α 1 c + lim Od 1 /α 1 = α.

7 O Geeralized Fiboacci umbers 3617 Cosequetly, we have the followig corollary. Corollary 2.1. Let {V } = be a iteger sequece satisfyig 6 but with iitial coditios V k i = a i for all i where a i s N {} with atleast oe of them is ot zero. Furthermore, assume that q 1 q 2 q k 1, where 2 k N the lim /V k 1 = α, 14 Vk where q 1 < α < q is the positive real zero of fx. Proof. The proof uses Theorem 2.5 ad the argumets used are similar to the proof of Theorem 2.6. Remark Observe that whe q i = 1 for all i =, 1,..., k i Corollary 2.1, the lim r : = lim F k /F k 1 = α, where 1 < α < 2. Ideed, the limit of the ratio r is 2 as icreases. k-periodic Fiboacci Sequeces I [3], M. Edso ad O. Yayeie gave a geeralizatio of Fiboacci sequece. He called it geeralized Fiboacci sequece {F a,b } = which he defied it by usig a o-liear recurrece relatio depedig o two real parameters a, b. The sequece is defied recursively as { F a,b =, F a,b 1 = 1, F a,b af a,b 1 + F a,b 2, if is eve, = bf a,b 1 + F a,b 15 2, if is odd. This geeralizatio has its ow Biet-like formula ad satisfies idetities that are aalogous to the idetities satisfied by the classical Fiboacci sequece see [3]. A further geeralizatio of this sequece, which is called k-periodic Fiboacci sequece has bee preseted by M. Edso, S. Lewis, ad O. Yayeie i [4]. A related result cocerig to two-periodic terary sequece is preseted i [1] by M. Alp, N. Irmak ad L. Szalay. We expect that aalogous results of 5, 7, ad 12 ca easily be foud for these geeralizatios of Fiboacci sequece. For istace, if we alter the startig values of 15, say we start at two umbers A ad B ad preserve the recurrece relatio i 15, the we obtai a sequece that we may call 2-periodic Fiboacci-like sequece, which is defied as follows: = A, 1 = B, = The first few terms of {F a,b { a 1 + 2, b 1 + 2, if is eve, if is odd. } = ad { } = are as follows: 16

8 3618 Jerico B. Bacai ad Julius Fergy T. Rabago F a,b A 1 1 B 2 a ab + A 3 ab + 1 ab + 1B + ba 4 a 2 b + 2a a 2 b + 2aB + ab + 1A 5 a 2 b 2 + 3ab + 1 a 2 b 2 + 3ab + 1B + ab 2 + 2bA 6 a 3 b 2 + 4a 2 b + 3a a 3 b 2 + 4a 2 b + 3aB + a 2 b 2 + 3ab + 1A 7 a 3 b 3 + 5a 2 b 2 + 6ab + 1 a 3 b 3 + 5a 2 b 2 + 6ab + 1B + a 2 b 3 + 4ab 2 + 3bA Suprisigly, by lookig at the table above, G a,b ca be obtaied usig F a,b ad F b,a. More precisely, we have the followig result. Theorem Let F a,b ad be the th terms of the sequeces defied i 15 ad 16, respectively. The, for all N, the followig formula holds = 1 F a,b + F b,a Proof. The proof is by iductio o. Evidetly, the formula holds for =, 1, 2. We suppose that the formula also holds for some 2. Hece, we have 1 = 1 F a,b 1 + F b,a 2, = 1 F a,b + F b,a 1. Suppose that is eve. The case whe is odd ca be prove similarly. So we have +1 = a = a = 1 provig the theorem F a,b af a,b + F b,a 1 + F a,b 1 = 1 F a,b +1 + F b,a, F a,b 1 + af b,a 1 + F b,a 2 F b,a 2 The sequece { } = has already bee studied i [3], Sectio 4. The authors [3] have related the two sequeces {F a,b } = ad { } = usig the formula = 1 F a,b + 2 /2 b F a,b a

9 O Geeralized Fiboacci umbers 3619 Notice that by simply comparig the two idetities 17 ad 18, we see that F b,a 1 = 2 /2 b F a,b 1, N. a The covergece property of {F a,b +1 /F a,b } = has also bee discussed i [3], Remark 2. It was show that, for a = b, we have F a,b +1 F a,b α a = a + a as. 19 Usig 17 ad 19, we ca also determie the limit of the sequece { +1 / } as teds to ifiity, ad for a = b, as follows: +1 lim = lim = 1 F a,b +1 + F b,a 1 F a,b G a,a F 1 lim a,a +1 + F b,a 1 F a,a + G a,a G a,a 1 + G a,a F lim a,a 1 F a,a = lim 1 F a,a +1 + G a,a F a,a G a,a G a,a 1 F a,a = αa 1 G a,a 1 + G a,a G a,a 1 + aα 1 G a,a + G a,a F a,a 1 = α a. For the case a b, the ratio of successive terms of {F a,b } does ot coverge. However, it is easy to see that F a,b 2 F a,b 2 1 α b, F a,b 2+1 F a,b 2 α a a,b F +2, ad F a,b α + 1, where α = ab + a 2 b 2 + 4ab/2 cf. [3]. Kowig all these limits, we ca ivestigate the covergece property of the sequeces { 2 / 2 1}, { ad { }. Notice that F a,b = F b,a for every {1, 3, 5,...}. So lim +2 / F a,b 2 1 F a,b = lim = lim G + F b,a a,b F b,a F b,a F b,a 2 2 F a,b 2 2 F a,b 2 1 = lim G a,b F b,a F a,b = αa aα 1 = α a. 2+1/ 2 }, F a,b 2 2 F b,a 2 1 Similarly, it ca be show that as. 2+1/ 2 α/b ad +2 / α + 1

10 362 Jerico B. Bacai ad Julius Fergy T. Rabago The recurrece relatios discussed above ca easily be exteded ito subscripts with real umbers. For istace, cosider the piecewise defied fuctio x : = A, 1 = B, x = a x 1 + Ga,b x 2, if x is eve, b x 1 + Ga,b x 2, if x is odd. 2 Obviously, the properties of 16 will be iherited by 2. For example, suppose = 2, 1 = 3, a =.2, ad b =.3. The, G.2,.3 x = G.2,.3 1 F.2,.3 x + G.2,.3 F.3,.2 x 1. Also, lim x G.2,.3 2 x G.2,.3 2 x 1 = , lim G.2,.3 2 x +1 x G.2,.3 2 x = , lim G.2,.3 x +2 x G.2,.3 x = If a = b =.1, the the ratio of successive terms of {G.1,.1 } with G.1,.1 = 2 ad G.1,.1 1 = 3 coverges to See Figure 1 for the plots of these limits. Figure 1: G = 2, G 1 = 3, a =.2, b =.3; G = 2, G 1 = 3, a =.1, b =.1. Now, we may take the geeralized Fiboacci sequece 15 a bit further by cosiderig a 3-periodic terary recurrece sequece related to the usual

11 O Geeralized Fiboacci umbers 3621 Triboacci sequece: T a,b,c = T a,b,c =, T a,b,c 1 =, T a,b,c 2 = 1, at a,b,c 1 bt a,b,c 1 ct a,b,c 1 + T a,b,c 2 + T a,b,c 2 + T a,b,c 2 + T a,b,c 3, if mod 3, + T a,b,c 3, if 1 mod 3, + T a,b,c 3, if 2 mod Suppose we defie the sequece {U a,b,c } = satisfyig the same recurrece equatio as i 21 but with arbitrary iitial coditios U a,b,c, U a,b,c 1, ad U a,b,c 2. The, the sequeces {U a,b,c } = ad {T a,b,c } = are related as follows: U a,b,c = U a,b,c T b,c,a 1 + U a,b,c 1 T b,c,a 1 + T c,a,b 2 + U a,b,c 2 T a,b,c, 2. Remark I geeral, the k-periodic k-ary sequece {F a 1,a 2,...,a k }: = } related to k-acci sequece: {F k F k = F k 1 = = F k k 2 =, a 1 F k 1 + k j=2 Fk j, a F k 2 F k 1 + k j=2 = Fk j, a k F k 1 + k j=2 Fk j, Fk = 1, if mod k, if 1 mod k,. if 1 mod k. k 22 ad the sequece {G a 1,a 2,...,a k }:= { } defied i the same recurrece equatio 22 but with arbitrary iitial coditios, 1,..., are related i the followig fashio: k 3 = F k;1 1 + j= F k;j+1 1 j + Fk, 23 where k; j:= a j+1, a j+2,..., a k, a 1, a 2,..., a j, j =, 1,..., k 1. This result ca be prove by mathematical iductio ad we leave this to the iterested reader. Refereces [1] Alp, M., Irmak, N., ad Szalay, L., Two-periodic terary recurreces ad their biet-formula, Acta Math. Uiv. Comeiaae, Vol. LXXXI, 2 212, pp

12 3622 Jerico B. Bacai ad Julius Fergy T. Rabago [2] Dresde, G. P. B., A Simplifed Biet Formula for k-geeralized Fiboacci Numbers, J. Iteger Sequeces, 19, 213. [3] Edso, M., Yayeie, O., A New Geeralizatio of Fiboacci Sequece ad Exteded Biets Formula, Itegers, 9 # A48 29, pp [4] Edso, M., Lewis, S., Yayeie, O., The k-periodic Fiboacci sequece ad a exteded Biet s formula, Itegers, 11 # A32 211, pp [5] Horadam, A. F., Basic properties of certai geeralized sequece of umbers, Fiboacci Quarterly, , pp [6] Koshy, T., Fiboacci ad Lucas Numbers with Applicatios, Joh Wiley, New York, 21. [7] Noe, Toy; Piezas, Tito III; ad Weisstei, Eric W. Fiboacci -Step Number. From MathWorld A Wolfram Web Resource. [8] Wu, Z., Zhag, H., O the reciprocal sums of higher-order sequeces, Adv. Diff. Equ., 213, 213: Received: February 16, 215; Published: May 2, 215