The k-nacci triangle and applications

Transcription

1 Kuhapataakul & Aataktpasal, Coget Mathematcs 7, : 9 PURE MATHEMATICS RESEARCH ARTICLE The k-acc tragle ad applcatos Katapho Kuhapataakul * ad Porpawee Aataktpasal Receved: March 7 Accepted: May 7 Frst Publshed: May 7 *Correspodg author: Katapho Kuhapataakul, Faculty of Scece, Departmet of Mathematcs, Kasetsart Uversty, Bagkok 9, Thalad E-mal: fsckpkk@kuacth Revewg edtor: Lsha Lu, Qufu Normal Uversty, Cha Addtoal formato s avalable at the ed of the artcle Abstract: A geeralzato of the classcal Fboacc umbers F s the k-geeralzed Fboacc umbers F k for k whose frst k terms are,,, ad each term afterward s the sum of the precedg k terms I ths artcle, we frst troduce the k-acc tragle to derve a explct formula of the th k-geeralzed Fboacc umber Secod, we also troduce the k-geeralzed Pascal tragle for dervg the formula of the k-geeralzed Fboacc umbers Subects: Scece; Mathematcs & Statstcs; Advaced Mathematcs Keywords: k-geeralzed Fboacc umbers; Fboacc umbers; Pascal tragle; k-acc tragle; bomal coeffcet AMS Mathematcs subect classfcatos: B7; B9 Itroducto For fxed k, the k-geeralzed Fboacc sequece or, for smplcty, the k-acc sequece {F k } k s defed as F k F k Fk Fk k, wth the tal codtos F k k Fk k Fk ad F k Such a sequece s also called k-step Fboacc sequece or the Fboacc k-sequece Clearly for k, we obta the well-kow Fboacc umbers F F, for k, the trboacc umbers F T, for k, the tetraacc umbers F, ad for k, the petaacc umbers F I geeral case, the frst k o-zero terms F k are powers of two, amely F k, F k, F k, F k,, F k k k, whle the ext term the above sequece s F k k k ABOUT THE AUTHORS Katapho Kuhapataakul s a assstat professor at departmet of mathematcs, faculty of Scece, Kasetsart Uversty, Bagkok, Thalad He got PhD 9 Hs research terests are areas of Number Theory ad Algebra Porpawee Aataktpasal s a master studet of assstat professor Katapho Kuhapataakul PUBLIC INTEREST STATEMENT Fboacc umbers ad ther geeralzatos have may terestg propertes ad applcatos to almost every felds of scece ad art Pascals tragle has bee explored for lks to the Fboacc sequece as well as to geeralzed sequeces I ths paper, the authors gve some coectos of the coeffcets the multomal expaso wth the geeralzed Fboacc umbers They also costruct the k-geeralzed Pascal tragle to derve the formula of the th k-geeralzed Fboacc umbers 7 The Authors Ths ope access artcle s dstrbuted uder a Creatve Commos Attrbuto CC-BY lcese Page of

2 Kuhapataakul & Aataktpasal, Coget Mathematcs 7, : 9 The Fboacc umbers ad ther geeralzatos have may terestg propertes ad applcatos to almost every felds of scece ad art eg see Debart, ; Koshy, ; Vada, 989 It s well-kow that the Fboacc umber F ca be derved by the summg of elemets o the rsg dagoal les Pascal s tragle F, where x s the largest teger ot exceedg x, see Koshy, chapter Wog ad Maddocks 97 geeralzed the Pascal s tragle ad showed that sums of elemets o the rsg dagoal les ther tragle gve the trboacc umber T some authors called ths tragle that the trboacc tragle, eg see Allad ad Hoggatt 977, Kuhapataakul There s yet aother tragular array that yelds the varous trboacc umbers Feberg 9 used the tromal expasos of x x for ad showed that the rsg dagoal sums of ths tromal coeffcet array also yeld the trboacc umbers Kuhapataakul ad Sukrua have show the -trboacc tragle smlar to Pascal s tragle ad derved a explct formula for the trboacc umbers The -trboacc tragle s a array of the shape where They showed that the sums of all elemets the -trboacc tragle gve the th trboacc umber The expaso for the trboacc umber T terms of bomal coeffcets, see Kuhapataakul, Kuhapataakul ad Sukrua, Shao 977, as the followg T Phlppou ad Muwaf showed 98 that the F k ca be wrtte the form F k r,, r k r r kr k r r k! r! r k! Let α, α,, α k be the roots of x k x k x The followg Bet-lke formula for F k appears Dresde ad Du : F k k α k α α I ths artcle, we exted the result of Kuhapataakul ad Sukrua o the -trboacc tragle to the k-acc tragle for, ad derve the th k-geeralzed Fboacc umbers We also costruct the k-geeralzed Pascal s tragle to derve the th k-geeralzed Fboacc umbers Page of

3 Kuhapataakul & Aataktpasal, Coget Mathematcs 7, : 9 Fgure The tromal coeffcets array The k-acc tragle Defe the symbol C q, as the coeffcet of x the multomal expaso of x x x q for q ad,, e q x x x q C q, x, wth C, s the bomal coeffcet ad C q for > q Usg the classcal bomal coeffcet, we get that, C q q,, q q or C q, q q q q We gve the arrays of the coeffcets of x the multomal expaso for q,, to show the Fgures, respectvely, see also Sloae as A797, A887, A Fgure The quadromal coeffcets array Fgure The petaomal coeffcets array Page of

4 Kuhapataakul & Aataktpasal, Coget Mathematcs 7, : 9 Some well-kow propertes of the multomal coeffcet arrays: Every row s symmetrc about a vertcal le through the mddle, e C q, C q, q Ay teror umber each row, the excepto of the frst two rows, ca be obtaed from the precedg row, e q C q C q,, l l Next, we troduce the k-acc tragle for a postve teger Defto Let ad k be postve tegers The k-acc tragle for s a array that each elemet row ad colum s products of, ad as show put t: k : t t t C t, k C t, C t, C t, t k C t, C t, C t, t C t, C t, k C t, C t, C t, t For clarty, we gve the examples of the k-acc tragle for k,, The -acc tragle for Page of

5 Kuhapataakul & Aataktpasal, Coget Mathematcs 7, : The -acc tragle for The -acc tragle for Note that the -acc tragle for s the -trboacc tragle whch s defed by Kuhapataakul ad Sukrua, see Kuhapataakul ad Sukrua It s terestg that sums of all elemets the k-acc tragle for gve the th k-acc umber We wll gve some examples Example I the -acc tragle for a Substtutg, we get 7 7 ad the sums of all elemets s equal to 8, whch s the th tetraacc umber b Substtutg, we get Page of

6 Kuhapataakul & Aataktpasal, Coget Mathematcs 7, : ad sums of all elemets s equal to, whch s the th tetraacc umber Example I the -acc tragle for a Substtutg, we get 7 ad sums of all elemets s equal to, whch s the th petaacc umber b Substtutg, we get Page of

7 Kuhapataakul & Aataktpasal, Coget Mathematcs 7, : 9 ad sums of all elemets s equal to 9, whch s the th petaacc umber We wll fact prove that the sums of all elemets the k-acc tragle for gve the th k-acc umber Let k Deote S as the sums of elemets the th row of k-acc tragle for a postve teger, that s, S k, The S ca wrte the recurret relato Lemma Let, be postve tegers The S k S k S k S Proof Set C, :, ad we see that C, whe > k We have k k S k C, k k k k C, C, k k k k S k C, C, k k k k C, C, k k k k k S k C, C, k k k k C, C, k k S k S k C, k k k C, C, S k S k S, as desred Now, we state a explct formula for F k by summg all row sums of the k-acc tragle for ad prove the followg theorem Theorem Let k ad S as defed ths secto The F k S Proof We wll prove by ducto o We see that F k S, F k S, F k S S, Page 7 of

8 Kuhapataakul & Aataktpasal, Coget Mathematcs 7, : 9 ad F k k S k S k S k k Assume that holds for all tegers,,,, k By the ductve hypothess ad Lemma, we get F k Fk Fk Fk k S S S S S S S k S S S S k k S k S S S S k S k S k Thus, the proof s complete We ca rewrte Equato terms of bomal coeffcets usg Equato F k k, The k-geeralzed Pascal s tragle Throughout the secto, the teger k wll be fxed Defto Let, be tegers wth Defe k,,,, where,,, ad, for < or > Defto Deote the k-geeralzed Pascal s tragle as follows: Well-kow examples of, for k, are wrote terms of bomal coeffcets Page 8 of

9 Kuhapataakul & Aataktpasal, Coget Mathematcs 7, : 9 C, ad C, It s to see that the -geeralzed Pascal s tragle s the classcal Pascal s tragle ad the -geeralzed Pascal s tragle s the geeralzed Pascal s tragle whch s defed by Wog ad Maddocks 97 For coveece, we arrage the elemets of the k-geeralzed Pascal s tragle to form a leftustfed tragular array as follows: The k-geeralzed Pascal s tragle,,,,,,,,,,,,,,,,,,,,,,,,, For clarty, we also gve the followg examples of the k-geeralzed Pascal s tragles for k, The -geeralzed Pascal s tragle The -geeralzed Pascal s tragle Observe that the sum of elemets o each rsg dagoal le The, -geeralzed Pascal s tragles gve the -geeralzed Fboacc umbers F ad the -geeralzed Fboacc umbers, respectvely The followg table provdes further formato: F Page 9 of

10 Kuhapataakul & Aataktpasal, Coget Mathematcs 7, : 9 F F We coecture that the sums of elemets o each rsg dagoal le the k-geeralzed Pascal s tragle gves the k-geeralzed Fboacc umber Theorem Let k be fxed, ad let be o-egatve teger The F k, Proof For k, C, s a bomal coeffcet, the Equato becomes the Equato Suppose k We wll prove ths result by ducto o It s easy to see that, for m k, m m, m F k m Now, we assume Equato holds for > ad prove that t holds for Usg the defto of F k ad the ductve hypothess, we get that F k Fk Fk Fk k,, Thus, the proof of result s complete k,,,,, k, k, k, Next, we gve a alteratve defto of, terms of bomal coeffcets We beg provde the followg lemma whch wll be used the proof Lemma Let be o-egatve teger The l l l l l l l l l Page of

11 Kuhapataakul & Aataktpasal, Coget Mathematcs 7, : 9 l l l l l l l l l Proof Usg the Pascal s detty, we get the parts ad that l l l l l l l l l l l l l l l, l l l ad l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l Theorem Let k be fxed, ad let, be tegers wth The, k k k k k Proof For, t s to see that Equato holds Assume Equato s true for ad We wll show holds for Set N: k, usg Lemma, we get k, k, k k k k N k Usg Lemma ad the above equato, we get Page of

12 Kuhapataakul & Aataktpasal, Coget Mathematcs 7, : 9 k, k, k, k k k k k k k k N k Repeatg Lemma, we obta, k, k, k k N k By the recurrece ad Pascal s detty, we obta k,,, k, k k N k k k k k k N k k k k N Ths makes Equato holds for Therefore, Equato s true Now we preset some examples of the explct formulas for the trboacc, tetraacc, ad petaacc umbers F T ; F l ; l l F l l l m l m l m We ca rewrte, the oly oe summato as follows, k k k k Hece, we wrte the detty the bomal coeffcets Corollary For fxed k, we have F k k k k k Ackowledgemets The authors are grateful to the aoymous referee for hs/ her helpful commets The frst author would lke to thak the Professor Motvato PM uder Faculty of Scece, Kasetsart Uversty for facal support Fudg Ths work was supported by Faculty of Scece, Kasetsart Uversty Author detals Katapho Kuhapataakul E-mal: fsckpkk@kuacth Porpawee Aataktpasal E-mal: g7@kuacth Faculty of Scece, Departmet of Mathematcs, Kasetsart Uversty, Bagkok 9, Thalad Ctato formato Cte ths artcle as: The k-acc tragle ad applcatos, Katapho Kuhapataakul & Porpawee Aataktpasal, Coget Mathematcs 7, : 9 Refereces Allad, K, & Hoggatt, Jr, V E 977 O trboacc umbers ad related fuctos The Fboacc Quarterly,, Page of

13 Kuhapataakul & Aataktpasal, Coget Mathematcs 7, : 9 Debart, L A short hstory of the Fboacc ad golde umbers wth ther applcatos Iteratoal Joural of Mathematcal Educato Scece ad Techology,, 7 7 Dresde, G P, & Du, Z A smplfed Bet formula for k-geeralzed Fboacc umbers Joural of Iteger Sequeces, 7, Artcle 7 Feberg, M 9 New slats The Fboacc Quarterly,, 7 Koshy, T Fboacc ad Lucas umbers wth applcatos New York, NY: Joh Wley ad Sos Kuhapataakul, K Some coectos betwee a geeralzed trboacc tragle ad a geeralzed Fboacc sequece The Fboacc Quarterly,, Kuhapataakul, K, & Sukrua, L -trboacc tragles ad applcato Iteratoal Joural of Mathematcal Educato Scece ad Techology,, 8 7 Phlppou, A N, & Muwaf, A A 98 Watg for the Kth cosecutve success ad the Fboacc sequece of order K The Fboacc Quarterly,, 8 Shao, A G 977 Trboacc umbers ad Pascal s pyramd The Fboacc Quarterly,, 8 7 Sloae, N J A The o-le ecyclopeda of teger sequeces Retreved from Vada, S 989 Fboacc ad Lucas umbers, ad the golde secto Chchester: Ells Horwood Lmted Wog, C K, & Maddocks, T W 97 A geeralzed Pascal s tragle The Fboacc Quarterly,, 7 The Authors Ths ope access artcle s dstrbuted uder a Creatve Commos Attrbuto CC-BY lcese You are free to: Share copy ad redstrbute the materal ay medum or format Adapt remx, trasform, ad buld upo the materal for ay purpose, eve commercally The lcesor caot revoke these freedoms as log as you follow the lcese terms Uder the followg terms: Attrbuto You must gve approprate credt, provde a lk to the lcese, ad dcate f chages were made You may do so ay reasoable maer, but ot ay way that suggests the lcesor edorses you or your use No addtoal restrctos You may ot apply legal terms or techologcal measures that legally restrct others from dog aythg the lcese permts Coget Mathematcs ISSN: -8 s publshed by Coget OA, part of Taylor & Fracs Group Publshg wth Coget OA esures: Immedate, uversal access to your artcle o publcato Hgh vsblty ad dscoverablty va the Coget OA webste as well as Taylor & Fracs Ole Dowload ad ctato statstcs for your artcle Rapd ole publcato Iput from, ad dalog wth, expert edtors ad edtoral boards Reteto of full copyrght of your artcle Guarateed legacy preservato of your artcle Dscouts ad wavers for authors developg regos Submt your mauscrpt to a Coget OA oural at wwwcogetoacom Page of