Generalized Fibonacci Like Sequence Associated with Fibonacci and Lucas Sequences

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1 Turkih Joural of Aalyi ad Number Theory, 4, Vol., No. 6, Available olie at Sciece ad Educatio Publihig DOI:.69/tjat--6-9 Geeralized Fiboacci Like Sequece Aociated with Fiboacci ad Luca Sequece Yogeh Kumar Gupta,*, Mamta Sigh, Omprakah Sikhwal 3 School of Studie i Mathematic, Vikram Uiverity Ujjai, (M. P. Idia Departmet of Mathematical Sciece ad Computer applicatio, Budelkhad Uiverity, Jhai (U. P. 3 Departmet of Mathematic, Madaur Ititute of Techology, Madaur (M. P. Idia *Correpodig author: yogehgupta.88@rediffmail.com Received November, 4; Revied December, 4; Accepted December 8, 4 Abtract The Fiboacci equece, Luca umber ad their geeralizatio have may iteretig propertie ad applicatio to almot every field. Fiboacci equece i defied by the recurrece formula F = F- F-, ad F =, F =, where F i a th umber of equece. May author have bee defied Fiboacci patter baed equece which are popularized ad kow a Fiboacci-Like equece. I thi paper, Geeralized Fiboacci-Like equece i itroduced ad defied by the recurrece relatio B = B B, with B =,B =, where beig a fixed iteger. Some idetitie of Geeralized Fiboacci-Like equece aociated with Fiboacci ad Luca equece are preeted by Biet formula. Alo ome determiat idetitie are dicued. Keyword: Fiboacci equece, Luca equece, Geeralized Fiboacci-Like Sequece, Biet formula Cite Thi Article: Yogeh Kumar Gupta, Mamta Sigh, ad Omprakah Sikhwal, Geeralized Fiboacci Like Sequece Aociated with Fiboacci ad Luca Sequece. Turkih Joural of Aalyi ad Number Theory, vol., o. 6 (4: doi:.69/tjat Itroductio The Fiboacci ad Luca equece are well-kow example of ecod order recurrece equece. The Fiboacci umber are perhap mot famou for appearig i the rabbit breedig problem, itroduced by Leoardo de Pia i i hi book called Liber Abaci. A illutrate i the tome by Kohy [] the Fiboacci ad Luca umber are arguable two of the mot iteretig equece i all of mathematic. May idetitie have bee documeted i a exteive lit that appear i the work of Vajda [4], where they are proved by algebra mea, eve though combiatorial proof of may of thee iteretig idetitie. We itroduced Geeralized Fiboacci-Like Sequece ad ome idetitie Fiboacci umber, Luca umber' ad their geeralizatio have may iteretig Propertie ad applicatio to almot every field. The Fiboacci equece [] i a equece of umber tartig with iteger ad, where each ext term of the equece calculated a the um of the previou two. i.e., F = F- F -, ad F =, F =. (. The imilar iterpretatio alo exit for Luca equece. Luca equece [] i defied by the recurrece relatio, L = L- L-, ad L =, L = (. I thi paper, we preet variou propertie of the Geeralized Fiboacci-Like equece (GFLS aociated with Fiboacci ad Luca equece {B } defied by B = B B, ad B =,B =. (.3 The Biet' formula for Fiboacci equece i give by R R F = = RR (.4 where R = Golde ratio =.68 ad R = Golde ratio = -.68 Similarly, the Biet' formula for Luca equece i give by L =R R =.. Prelimiary Reult Geeralized Fiboacci-Like Sequece We eed to itroduce ome baic reult of Geeralized Fiboacci-Like equece aociated with Fiboacci ad Luca equece {B } i defied by recurrece relatio:

2 34 Turkih Joural of Aalyi ad Number Theory B = B B, (. With iitial coditio B = ad B =. The aociated iitial Coditio B ad B are the um of iitial coditio of geeralized Fiboacci-Like equece repectively. i.e. F L = Bad F L = B (. The few term of above equece are,, 3, 4, 37, ad o o. The relatio betwee Fiboacci equece ad Geeralized Fiboacci-Like Sequece ca be writte a B = F L,. The recurrece relatio (. ha the characteritic equatio x = x which ha two root R = ad R =. Now otice a few thig about R ad R R R =, RR = ad RR =. Uig thee two root, we obtai Biet recurrece relatio R R B = R R = 3. Geeratig Fuctio Now we tate derive geeratig fuctio of geeralized Fiboacci-Like equece ( x Bx = = ( xx (3. Let' apply power erie to equece { B } x 3 x = Bx = Let ( ( Where B i th term of equece { B }. Thi i called geeratig erie of Geeralized Fiboacci - Like Sequece { B }. Now multiplyig the geeratig erie ( x x Bx = = Bx Bx Bx = = = = B Bx B x = Bx Bx Bx = = = B ( B B x ( B BB x = B B = ( x x B B = = ( x ( x = = x ( Therefore, ( Hece x x Bx = ( x. = ( x Bx =. = ( xx 4. Propertie of Geeralized Fiboacci- Like Sequece Depite it imple appearace the Geeralized Fiboacci-Like equece {B } cotai a wealth of ubtle ad faciatig propertie [4,6,9,]. Sum of Firt term: Theorem (4.. Let B be the th Fiboacci-Like umber, the Sum of the firt term of geeralized Fiboacci- Like equece i ( B B B3 B = Bk = B ( 3 (4. Proof: we kow that the follow relatio hold: B = B3 B B = B4 B3 B3 = B B4 B = B B B = B B ( Sice B = B B 3 Term wie additio of all above equatio, we obtai ( B B B3 B = B B ( = B 3 Sum of Firt term with eve idice Theorem (4.. Let B be the th Fiboacci-Like equece, the Sum of the firt term with eve idice i B B4 B6 B = Bk = B ( (4.

3 Turkih Joural of Aalyi ad Number Theory 3 Sum of Firt term with quare idice: Theorem (4.3. Let B be the th Fiboacci-Like equece, the Sum of the quare of firt term i ( B B B3... B = Bk = BB (4.3 Sum of Firt term with odd idice: Theorem (4.4. Let B be the th Fiboacci-Like equece, the Sum the firt term with odd idice i ( B B B B B 3 7. = Bk = B B (4.4 Now we tate ad prove ome ice idetitie imilar to thoe obtaied for Fiboacci ad Luca equece [,,4,].. Some Idetitie Geeralized Fiboacci- Like Sequece I thi ectio, ome idetitie of Geeralized Fiboacci-Like equece are preeted which ca be eaily derived by Explicit um formula uig geeratig fuctio ad Biet formula. Author [,6] have bee decribed uch type idetitie. Explicit Sum Formula: Theorem (.. The explicit um formula for Geeralized Fiboacci-Like equece i give by For poitive iteger, Prove that B m= ( m Bm = (. Proof: By equatio (., it follow that B = B B = = B B3 B4 = Hece B = ( B B ( B B ( B3 B4 ( B4 B ( B B 6 =.. =.. ( = B B B.. ( B B B B m m m=. Theorem (. The explicit um formula for Geeralized Fiboacci-Like equece i give by For poitive iteger, ( B k B k = (. Theorem (.3. For every poitive iteger, prove that Bm B Bm B ( Bm B m, = (.3 Proof: Let be fixed ad we Proved by iductig o m. Whe m =, the BB BB = ( BB ( ( = ( ( ( B B = ( B ( ( B = ( B ( B = ( B Which i true. Whe m=, the B B B B B B B = ( B B BB BB = ( BB B( B B = ( BB ( 3 ( = ( ( 3 ( 3( B = ( 3 B ( 3 B = ( 3 B B B B which alo i true. Now aume that idetity i true for m = k, the by aumptio ( = (.4 BB k BB k BB k k B ( k B Bk B BkB k = (. Addig equatio (.4 ad (., we get BB k Bk B BB k k Bk B Bk Bk B Bk Bk B = ( BB k k ( B k ( ( = ( ( BB k k B k = ( Bk B Bk B Bk Bk Which i preciely our idetity whe k = m Hece Bm B Bm B = ( Bm Bm,. Theorem (.4. For every poitive iteger, prove that B B B = (.6 Proof: we hall have proved thi idetity by iductio matched over. For =, B = B B B = B B = ( = ( = which i alo true for =. Whe = tha B = B B B = B3 B ( 3 = 4 ( ( 3 = 3

4 36 Turkih Joural of Aalyi ad Number Theory which i alo true for =. For = k Bk = Bk Bk For = k which i alo true. Now aume that idetity i true for =,, 3...k ad We o that it hold: For = k, the by aumptio B( k = B( k B( k Bk = Bk 3 Bk = ( Bk Bk Bk = B k Which i alo true, for = k Hece, the reult i true for all. Theorem (.. For every poitive iteger, prove that ( F = B B,. (.7 Proof: we hall Prove thi idetity by iductio over, for = ( = ( = ( = (. = F F = B B Now uppoe that idetity hold for =k-, = k- tha, ( Fk Bk Bk3 = (.8 ( Fk3 Bk Bk4 = (.9 O addig equatio (.8 & (.9 we get, ( Fk ( Bk3 = ( Bk Bk ( Bk3 Bk4 ( ( Fk Bk3 = Bk Bk ( F = B B which i true for = k, k k k ( F = B B,. Theorem(. 6. For every poitive iteger, B3 B6 B9.. B3 = B3 ( 3 (. Proof. By uig Biet formula, we have B3 B6 B9.. B R R 3 3 R R = ( R R ( R R ( 3 3 R R R R R R R R = R R R R R R R R R R R R R R R R = 3 3 R R R R R R 3 3 R R 3 3 R R R R = 3 3 R R R R ( 3 3 ( 3 3 R R = R R R R R R = ( B3 B = B3 ( 3 Thi i complete the proof. Theorem (.7. For every poitive iteger B B8 B.. B3 = [ B3 4 37] Proof. By uig Biet formula, we have B B8 B.. B3 8 8 R R R R = ( R R R R R R R R ( 3 3 ( 8 3 R R R R = 8 3 R R R R 8 3 R R R R 8 3 R R R R 3 3 R R R R = R R R R R R R R (.

5 Turkih Joural of Aalyi ad Number Theory R R R R = 3 3 R R R R ( B B [ B 3 7] ( ( R R = R R R R R R 3 = = 3 4 Thi i complete the proof. Theorem (.8. For poitive iteger, prove that ( ( B B B, = (. Thi ca be derived ame a theorem (.4 Theorem (.9. For poitive iteger, prove that B ( ( B, = (.3 Thi ca be derived ame a theorem (.4. Theorem (.. For every iteger, prove that B = F L, (.4 Thi ca be derived ame a theorem (.4 Theorem (.. For every iteger, prove that F L = B, (. Thi ca be derived ame a theorem (.4. Theorem (.. For every iteger, prove that ( F F = B B, (.6 Thi ca be derived ame a theorem ( Coectio Formulae I thi ectio, coectio formulae of Geeralized Fiboacci-Like equece aociated with Fiboacci ad Luca equece, iductio method are preeted. Theorem (6.. For poitive iteger, Prove that F B B, 3 = (6. Proof: We hall prove thi idetity by iductio. It i eay to how that for = 3 F = F3 = F = F =. = = B B. Now uppoe the idetity hold = k-, = k-. The, Fk Bk Bk3. = (6. Fk3 Bk3 Bk4. = (6.3 O addig equatio (6. ad (6.3, we get ( ( i.e. F F = B B B B ( Fk Fk3 = BkBk F = B B k k3 k k3 k3 k4 k k k Which i preciely our idetity whe = k. Hece F - = B - - B -, 3. Theorem (6.. For poitive iteger, Prove that L B B, = (6.4 Proof: We hall Prove thi idetity by iductio over. for = L = bl= L =. = = B B. Now uppoe the idetity hold for = k-, = k-. The, L k B k B k = (6. L k 3 B k B k 3 = (6.6 Addig equatio (6. ad (6.6, we get ( = ( ( i.e. Lk Lk3 Bk Bk Bk Bk3 Lk = Bk Bk Which i true for = k, Hece L - = B - B -,. Theorem (6.3. For poitive iteger, prove that ( L B F, = (6.7 Theorem (6.4. For poitive iteger, prove that ( L B B, = (6.8 Theorem (6.. For poitive iteger, prove that B3 L F3, 3. = (6.9 Theorem (6.6. For poitive iteger, prove that F B B,. = (6. 7. Some Determiat Idetitie There i a log traditio of uig matrice ad determiat to tudy Fiboacci umber. Problem o determiat of Fiboacci equece ad Luca equece are appeared i variou iue of Fiboacci Quarterly. T. Kohy [] explaied two chapter o the ue of matrice ad determiat. May determiat idetitie of geeralized Fiboacci equece are dicued i [4,6] ad []. I thi ectio ome determiat idetitie of Geeralized Fiboacci-Like equece are preeted. Etrie of determiat are atifyig the recurrece relatio of Geeralized Fiboacci-Like equece ad other equece. Theorem (7.. Let be a poitive iteger. The B F B F = B F [ FB BF ]

6 38 Turkih Joural of Aalyi ad Number Theory Proof: Let Ad B F B F B F = (7. aume B a, B =b, B =ab F PF, qf, p q = (7. = = = (7.3 Now ubtitutig the value of equatio (7. & (7.3 i (7., we get a p = b q a b p q Applyig R R R ab pq = b q a b p q Applyig R R R3. ab pq = b ( a b q ( p q a b p q ab pq = a p a b p q [ pb aq] = (7.4 Agai ubtitutig the value of the equatio (7. ad (7.3 i (7.4. We get = [ FB BF ]. B F Hece B F = [ FB BF ]. B F Similarly we ca derive followig idetitie: Theorem (7.. For every iteger, prove that B B B B B B 3 ( B 3 B B B B = (7. Theorem (7.3. For ay iteger, prove that B L B L ( LB BL B L = (7.6 Theorem (7.4. For every poitive iteger, prove that B B B B B B B B B = (7.7 Theorem (7.. For every poitive iteger, prove that B B B B B B = B B B B B B (7.8 The idetitie from (7. to (7.4 ca be proved imilarly. 8. Cocluio I thi paper, Geeralized Fiboacci-Like equece i itroduced. Some tadard idetitie of geeralized Fiboacci-Like equece aociated with Fiboacci ad Luca equece have bee obtaied ad derived uig Biet formula. Alo ome determiat idetitie have bee etablihed ad derived. Ackowledgemet We would like to thak the aoymou referee for umerou helpful uggetio. Referece [] A. F. Horadam: A Geeralized Fiboacci Sequece, America Mathematical Mothly, Vol. 68. (, 96, [] A. F. Horadam: Baic Propertie of a Certai Geeralized Sequece of Number, The Fiboacci Quarterly, Vol. 3 (3, 96, [3] A.T. Bejami ad D. Walto, Coutig o Chebyhev polyomial, Math. Mag. 8, 9, 7-6. [4] B. Sigh, O. Sikhwal ad S. Bhatagar: Fiboacci-Like Sequece ad it Propertie, It. J. Cotemp. Math. Sciece, Vol. (8,, [] B. Sigh, Omprakah Sikhwal, ad Yogeh Kumar Gupta, Geeralized Fiboacci-Luca Sequece, Turkih Joural of Aalyi ad Number Theory, Vol., No.6. (4, [6] B. Sigh, S. Bhatagar ad O. Sikhwal: Fiboacci-Like Sequece, Iteratioal Joural of Advaced Mathematical Sciece, (3 (3 4-. [7] D. V. Jaiwal: O a Geeralized Fiboacci equece, Labdev J. Sci. Tech. Part A 7, 969, [8] M. Edo ad O. Yayeie: A New Geeralizatio of Fiboacci equece ad Exteded Biet Formula, Iteger Vol. 9, 9, [9] M. E. Waddill ad L. Sack: Aother Geeralized Fiboacci equece, The Fiboacci Quarterly, Vol. (3, 967, 9-. [] M. Sigh, Y. Gupta, O. Sikhwal, Geeralized Fiboacci-Luca Sequece it Propertie, Global Joural of Mathematical Aalyi, (3 4, [] M. Sigh, Y. Gupta, O. Sikhwal, Idetitie of Geeralized Fiboacci-Like Sequece. Turkih Joural of Aalyi ad Number Theory, vol., o. (4: 7-7. doi:.69/tjat --3. [] S. Falco ad A. Plaza: O the Fiboacci K- Number, Chao, Solutio & Fractal, Vol. 3 (, 7, [3] Sigh, M., Sikhwal, O., ad Gupta, Y., Geeralized Fiboacci- Luca Polyomial, Iteratioal Joural of Advaced Mathematical Sciece, ( (4, 8-87 [4] S. Vajda, Fiboacci & Luca Number, ad the Golde Sectio, Theory ad Applicatio, Elli Horwood Ltd., Chicheter, 989. [] T. Kohy, Fiboacci ad Luca Number with Applicatio, Wiley-Iterciece Publicatio, New York (. [6] Y. Gupta, M. Sigh, ad O. Sikhwal, Geeralized Fiboacci-Like Polyomial ad Some Idetitie Global Joural of Mathematical Aalyi, (4 (9-8.