21(2007) Adílson J. V. Brandão 1, João L. Martins 2

Transcription

1 (007) Recuece Foulas fo Fboacc Sus Adílso J. V. Badão, João L. Mats Ceto de Mateátca, Coputa cão e Cog cão, Uvesdade Fedeal do ABC, Bazl.adlso.badao@ufabc.edu.b Depataeto de Mateátca, Uvesdade Fedeal de Ouo Peto, Bazl. jats@ceb.ufop.b Resue E este atículo pesetaos ua ueva foula ecuete paa ua sua fta que voluca la secueca de Fboacc. Adeás, dcaos u algoto paa calcula la sua de ua see de potecas elacoadas a las sees de Fboacc, s el uso del teoea de dfeecacó téo po téo. Palabas claves: Secueca de Fboacc, sees de Fboacc. Abstact I ths pape we peset a ew foula fo a ecug fte aout volvg the Fboacc sequece. I addto, we dcated a algoth to calculate the su of a sees of powes elated to the Fboacc sees, wthout the use of theoe dffeetato te by te. Keywods: Fboacc sequece, Fboacc sees. 30

2 A. Badão, J. Mats (007) Itoducto The Fboacc sequece s oe of the ost faous uecal sequeces atheatcs. It s defed a ecusve way: the fst two tes ae gve ad the followg oes ae defed as the su of the two pecedg oes. Matheatcally speakg: F00,F,FF-F-,. The fst tes ae:,,, 3, 5, 8, 3,,.. Ths sequece coes fo the sgle pa of abbts pogey poble, whch was ealy poposed by Leoado de Psa (Fboacc) at the Lbe Abacc of 0. A tgug pot s that ths sequece appeas ay pobles fo Matheatcs as well as Botac, Cystallogaphy, Copute Scece, etc []. Cosde the followg fte su volvg the Fboacc sequece, whee s a eal ube, ad The questo heeby teposed s the followg: wth ts covegece teval, s thee a foula fo the su of the [.] sees? A aswe to ths questo s obtaed by vokg the te-by-te dffeetato theoe fo powe sees. Actually, such a equato s obtaed by usg D d/d opeato tes to the kow detty If we defe F S(, j) F a ecuece foula ca be obtaed by the followg way: S(,0) S(, j) D [ S(, j ) ], j,...,. [.3] Eaple. Usg the (.3) algoth, we ca calculate the uec sees su F. [.] F S [.4] 3 May authos have bee seekg to establsh a su foula fo [.] (see [], [3], [4],[5]). I ths atcle we state a su foula fo [.] that we beleve ay be cosdeed as a ew esult. Cosde ow the powe sees assocated to [.]: F. [. ] It s ot dffcult to deostate that the (.) coveges fo all ad all (/φ, /φ), whch φ ( 5) / s the golde ato, a wellkow costat assocated wth Fboacc s sequece []. I fact, f S(,0) /( ) the 3 S(,) S (,0) ( ) /( ). Hece, takg /3 S(, ), we get the su S 6/5 fo the (.4) sees. Eaple. Ty ow to copute the uec sees su below by usg the sae algoth: 50 F S 3 The [.3] algoth s poble s, fo each sgle step, hghe coputato cost ode to dffeetate a fucto. The eaple. pots out ths dffculty. I 3

3 A. Badão, J. Mats (007) ths pape we obta aothe ecuece foula to calculate the su of [.]. The atcle s hecefoth ogazed as follows: I the secod secto we peset ou a esult, a ecuece foula fo the [.] fte su ad we show that t ecoves soe esults o fte suato foulas volvg the Fboacc sequece. The thd secto was teded to goously poof ou foula. I the fouth secto we state a algoth to copute the su of [.] wthout the use of devatves. Fally, the ffth secto, we gve soe coets about the esults ad futue possbltes.. Fte sus Ou a esult ths secto s the theoe below: Theoe.. Let R, 0 be gve. The the followg fte ecoece foula holds F F ( ) ( ) F F. F [.] As cosequece of theoe. we obta ay closed foulas fo fte sus volvg Fboacc suato. I fact, takg (.) we obta the followg fte suato: F F F F [.] ( ) We beleve that (.) s a ew foula fo (.). Fo (.) we ca deve closed fo soe specal cases of. Fo stace, takg (.) we obta. [.3] F F F F ( ) It s well kow (see [4]) that F F. [.4] Thus, fo (.3) ad (.4) we coclude that Theefoe F F F F F F F. F F F 3 [.5] whch s the foula () that appeas []. Now, takg (.) we ca see that. F F F F ( ) that s F F F F F F. [.6] Thus, usg (.4) ad (.5) (.6), afte soe algebac apulato, we obta ( ) ( ) F F 3 F 8, 3 whch s the foula [7] []. I a aalogous way we ca ecove othe kow dettes takg dffeet values fo [.]. Actually, the ecuece foula [.] ca poduce a lot of dettes, sply choosg specal values to 3

4 A. Badão, J. Mats (007) ad. Fo stace, the foula [.] fo s Multplyg (3.) by - ad - we obta Takg [.7] we obta ( ) ( ) ( ) ( ) F F F F Sce that (see []) [.8] ( ) F ( ) F, [.9] ad usg (.9) (.8), we coclude that ( ) ( ) ( ) ( ) F F F F [.0] Afte soe splfcatos, (.0) becoes ( ) F ( ) ( ) F ( ) F. [.] F F ( ) F ( ) F ( ) F. ( ) ( ) ( ) 3. Poof of ou a esult Befoe povg Theoe. we eed to state soe aulay esults: Theoe 3.. Let o-egatve teges k be gve. Suppose that 0. The F F F F F F F [3.]. Poof. Cosde the su k k k k k k k k... (3.) S F k k F k k F k k F k3 k3 F - - F. -S -F k k F k k - F k k3 - - F F - - F. [3.3] - S -F k k - F k k3 - - F F - - F - - F. [3.4] Sug [3.], [3.3] ad [3.4], eebeg the defto of Fboacc sequece ad cacellg tes we have S - S - S F k k F k k - F k k - F - F - - F. [3.5] Usg aga the defto of Fboacc sequece we coclude that S - S - S F k k F k- k - F - F. that s, (3.) holds. Theoe 3.. Let R, 0 be gve. The the followg detty holds ( ( ) ) ( ) F F F ( ) F F Poof. Cosde the su F F F 3 F... F [3.6] It s easy to see that the su above ca be eaaged the followg way 3 (... 3 ) F F F F F 3 ( ) ( F F... F 3 ) 3 ( 3 ) ( F... F 3 )... ( ( ) ( ) ) ( F F ) ( ( ) ) ( F ). By usg the Lea 3. we ca wte the last su as 33

5 A. Badão, J. Mats (007) F F F0 F F ( ) ( 3 F F F ) F... ( ( ) ) ( F ) F F F Theefoe, the su ca be epessed by F ( ( ) ) ( F F ) ( ) ( F F ( ) ( 3 )... ( ) ( ) Afte cacelg soe tes we fally obta F ( ( ) ) ( F F ) ( ) F F whch s the desed esult. Poof of Theoe.. By usg a sutable chage of vaables we have ( ( ) ) ( F F ) ( ( ) ) θ ( θ ), ( ) F F ( ( ) ) ( ) ( ). θ θ F F ( ) F F ( ) F F The theoe follows by the esult above ad the Lea Powe Sees I ths secto we state a esult whch povdes a algoth to copute the su of [.] wthout the use of te-byte dffeetato theoe. Ths algoth s a cosequece of the theoe below: Theoe 4.. Let (/φ, /φ) be gve. The the followg ecuece foula holds F ( ) F F. [4.] Poof. Cosdeg the Theoe., t s suffcet to take [.] ad to eebe that l F 0, sce the sees (.) coveges fo all tege ad (/φ, /φ). By theoe 4., we ca obta the followg algoth ode to povde the su of (.): S(,0), j j (4.) j j S(, j) ( ) S(, j ) S(, j ), j,..., Ths algoth ca be pleeted a effcet way, stead of the epesve pocess usg the stada devatve opeato. It aswes, fo stace, the questo poposed the eaple.: 50 F 3 5. Fal Reaks S I ths atcle we state a ew ecuece foula fo a fte su elated to Fboacc sequece. Ths foula ecoves a lot of dettes fo Fboacc sus. Besdes ths, t ples a algoth to copute the su of Fboacc powe sees wthout the use of devatves. The schee used to obta ths esults ca be eteded to othes sees. The deas peseted 34

6 A. Badão, J. Mats (007) hee ae pat of a lage vestgato whch has bee developed coceg the sees a. [5.] whch {a } s a abtay sequece. I ths atcle {a } s the Fboacc sequece. Nevetheless, we ca eted ou esults fo othe sequece types (see [6], [7]). Fo eaple, f we take a, (5.) tus to the geealzed geoetc sees, [5.] whch coveges fo all (, ). Usg the sae deas developed the last secto, we ca fd out a ecuece foula fo such a sees:. ( ) Thee ae othe subjects stll ude vestgato by whch we seach to eted the esults heeby peseted fo othe sequeces such as, Lucas, Geealzed Fboacc s, Geealzed Lucas, Pell s, Tboacc s sequeces, etc. It should be obseved that [8], the autho studed a sees elated to (.), coveg Lucas ad Fboacc s geealzed sequeces. Howeve, the esults ae oly vald fo a postve atoal. Besdes, the eployed techque s qute dffeet fo ous. Addtoal efeeces coceg Fboacc ubes ad the golde ato ca be foud []. Refeeces R. A. Dulap (997). The Golde Rato ad Fboacc Nubes, Wold Scetfc. V. C. Has (965). O Idettes Ivolvg Fboacc Nubes. The Fboacc Quately 3.3, 4-8. A. Bousseau (967). Suato of k F k : Fte Dffeece Appoach. The k Fboacc G. Led (967). O a Ceta Kd of Fboacc Sus. The Fboacc Quately 5., N. Gauthe (998). Idettes fo Class of Sus Ivolvg Hoada s Geealzed Nubes {W}. The Fboacc Quately 36.4, J. L. Mats ad A. J. V. Badão (004). Ua classe de s ees ftas evolvedo teos de sequˆecas geealzadas. Bolet ı de la Asocaco Mate atca Veezolaa, J. L. Mats ad A. J. V. Badão (004). F oula de ecoˆeca paa a soa de s ees Iftas. Lectuas Mate atcas 5, 5-4. Peo Flppo (000). Evaluato of Ceta Ifte Sees Ivolvg Tes of Geealzed Sequeces. The Fboacc Quately 38.4,