Recurrent Formulas of the Generalized Fibonacci Sequences of Third & Fourth Order
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1 Natioal Coferece o 'Advaces i Computatioal Mathematics' 7-8 Sept.03 :- 49 Recurret Formulas of the Geeralized Fiboacci Sequeces of hird & Fourth Order A. D.Godase Departmet of Mathematics V.P.College Vaijapur Auragabad (Maharashtra Idia) ashokgodse0@gmail.com Abstract Coupled Fiboacci sequeces ivolve two sequeces of itegers i which the elemets of oe sequece are part of the geeralizatio of the other ad vice versa. K.. Ataassov was first itroduced coupled Fiboacci sequeces of secod order i additive form. here are 8 differet schemes of geeralizatio for the riboacci sequeces i the case of two sequeces & there are 6 differet schemes of geeralizatio for the etraacci sequeces i the case of two sequeces []. I itroduce their recurret formulas below. Mathematics Subject Classificatio: B39 B37 Keywords: Fiboacci sequece multiplicative riboacci sequece multiplicative etraacci sequece.. INRODUCION: I the recet years much work has bee doe i this field but its multiplicative form is less kow. he coupled Fiboacci sequece was first itroduced by K.. Ataassov ad also discussed may curious properties ad ew directio of geeralizatio of Fiboacci sequece i []. He defied ad studied about four differet ways to geerate coupled sequeces ad called them coupled Fiboacci sequeces (or -F sequeces). K.. Ataassov [] otifies four differet schemes i multiplicative form for coupled Fiboacci sequeces.. RECURREN FORMULAS OF HE GENERALIZED MULIPLICAIVE FIBONACCI SEQUENCE OF HIRD ORDER: Orgaised by Departmet of Mathematics M.S.P.Madal'sV.P.College VaijapurAuragabad
2 Natioal Coferece o 'Advaces i Computatioal Mathematics' 7-8 Sept.03 :- 50 We ca costruct 8 differet schemes of geeralized multiplicative Fiboacci sequece i the case of two sequeces. We itroduce their recurret formulas below. Everywhere let X0 C0Y 0 C X CY C3 X C4Y C5 ad assume that 0 is a atural umber where C0 C C C3 C4 C 5 are give costats ad Z is oe of the symbols X or Y. he differet schemes are as follows: :{ X 3 X X X 3 Y Y Y :{ X 3 X X Y 3 Y Y X :{ X 3 3 X X 3 Y X Y :{ X 3 4 X Y 3 Y X X :{ X 3 5 X X 3 X Y Y :{ X 3 6 X Y 3 X Y X :{ X 3 7 X 3 X X Y :{ X 3 8 Y 3 X X X he first scheme is trivial. All of the others are otrivial; they have the followig recurret formulas for 0: ----For : Z 6 Z 5 Z 4 Z 3 5 4Z ----For : Z For 3 : Z 6 Z 4 5 Z Z ----For 4 : Z 6 Z 4 Orgaised by Departmet of Mathematics M.S.P.Madal'sV.P.College VaijapurAuragabad
3 Natioal Coferece o 'Advaces i Computatioal Mathematics' 7-8 Sept.03 : For 5 : Z 6 Z For : Z Z Z For 7 : Z 6 Z For : Z Z RECURREN FORMULAS OF HE GENERALIZED FIBONACCI SEQUENCE OF HIRD ORDER: We ca costruct 8 differet schemes of geeralized Fiboacci sequece of third order i the case of two sequeces. We itroduce their recurret formulas below. Everywhere let X0 C0Y 0 C X CY C3 X C4Y C5 ad assume that 0 is a atural umber where C0 C C C3 C4 C 5 are give costats ad Z is oe of the symbols X or Y. he differet schemes are as follows: :{ X 3 X X X 3 Y Y Y :{ X 3 X X Y 3 Y Y X :{ X 3 3 X X 3 Y X Y :{ X 3 5 X X 3 X Y Y :{ X 3 7 X 3 X X Y :{ X 3 4 X Y 3 Y X X :{ X 3 6 X Y 3 X Y X :{ X 3 8 Y 3 X X X he first scheme is trivial. All of the others are otrivial; they have the followig recurret formulas for 0: ----For : Z 6 Z 5 Z 4 Z For : Z 6 Z 5 Z 4 Z 3 Z Z ----For 3 : Z 6 Z 5 Z 4 Z 3 Z Z Orgaised by Departmet of Mathematics M.S.P.Madal'sV.P.College VaijapurAuragabad
4 Natioal Coferece o 'Advaces i Computatioal Mathematics' 7-8 Sept.03 : For 4 : Z 6 Z 5 Z 4 Z Z Z ----For 5 : Z 6 3Z 4 Z 3 Z Z Z ----For 6 : Z 6 3Z 4 Z Z ----For 7 : Z 6 Z 4 4Z 3 Z Z ----For 8 : Z 6 Z 4 Z 3 3Z Z Z 4. RECURREN FORMULAS OF HE GENERALIZED MULIPLICAIVE FIBONACCI SEQUENCE OF FOURH ORDER: We ca costruct 6 differet schemes of geeralized multiplicative Fiboacci sequece i the case of two sequeces. We itroduce their recurret formulas below. Everywhere let X0 C0Y 0 C X CY C3 X C4Y C5 X 3 C6Y3 C7 ad assume that 0 is a atural umber where C0 C C C3 C4 C5 C6 C 7 are give costats ad Z is oe of the symbols X or Y. he differet schemes are as follows: X X X X 4 Y 3 Y Y X X X Y 4 Y 3 Y X 4 3 Y 4 Y 3 X Y X X X Y X 3 :{ 4 X X Y 4 Y 3 X X 5 X X X 4 Y 3X Y Y 6 X X Y 4 Y 3X Y X 4 3 :{ X 8 X X Y 4 X 3Y Y X 7 X X X 4 X 3Y Y Y 9 X X 4 Y 3X X Y 0 X Y 4 Y 3X X X X X 4 X 3X Y Y X Y 4 X 3X Y X Orgaised by Departmet of Mathematics M.S.P.Madal'sV.P.College VaijapurAuragabad
5 Natioal Coferece o 'Advaces i Computatioal Mathematics' 7-8 Sept.03 : X X 4 X 3Y X Y 4 X Y 4 X 3Y X X 5 X 4 X 3X X Y 6 Y 4 X 3X X X he first scheme is trivial. All of the others are otrivial; they have the followig recurret formulas for 0: ----For : Z 8 Z 7 Z 6 Z 5 Z 4 7 6Z ----For : Z 8 3 Z Z For 3 : Z Z ----For 4 : Z Z For5 : Z 8 Z 6 3 Z 7 5 Z ----For6 : Z For 7 : Z Z For 8 : Z Z For 9 : Z Z Z ----For 0 : Z 8 Z Z For : Z 8 Z For : Z 8 Z Z For3 : Z Orgaised by Departmet of Mathematics M.S.P.Madal'sV.P.College VaijapurAuragabad
6 Natioal Coferece o 'Advaces i Computatioal Mathematics' 7-8 Sept.03 : For : Z Z For5 : Z 8 Z For : RECURREN FORMULAS OF HE GENERALIZED FIBONACCI SEQUENCE OF FOURH ORDER: We ca costruct 6 differet schemes of geeralized Fiboacci sequece of fourth order i the case of two sequeces. We itroduce their recurret formulas below. Everywhere let X0 C0Y 0 C X CY C3 X C4Y C5 X 3 C6Y3 C7 ad assume that 0 is a atural umber where C0 C C C3 C4 C5 C6 C 7 are give costats ad Z is oe of the symbols X or Y. he differet schemes are as follows: X X X X 4 Y 3 Y Y Y X X X Y 4 Y 3 Y Y X 3 X X X 4 Y 3 Y X Y 4 X X Y 4 Y 3 Y X X 5 X X X 4 Y 3 X Y Y 6 X X Y 4 Y 3 X Y X 7 X X X 4 X 3 Y Y Y 8 X X Y 4 X 3 Y Y X 9 X X 4 Y 3 X X Y 0 X Y 4 Y 3 X X X X X 4 X 3 X Y Y X Y 4 X 3 X Y X 3 X X 4 X 3 Y X Y 4 X Y 4 X 3 Y X X 5 X 4 X 3 X X Y 6 Y 4 X 3 X X X Orgaised by Departmet of Mathematics M.S.P.Madal'sV.P.College VaijapurAuragabad
7 Natioal Coferece o 'Advaces i Computatioal Mathematics' 7-8 Sept.03 :- 55 he first scheme is trivial. All of the others are otrivial; they have the followig recurret formulas for 0: ----For : Z 8 Z 7 Z 6 Z 5 Z For : Z 8 Z 7 Z 6 3Z 4 Z 3 Z ----For 3 : Z 8 Z 7 Z 6 Z 5 Z 4 Z 3 Z Z ----For 4 : Z 8 Z 7 Z 6 Z 5 Z 4 Z 3 Z Z ----For 5 : Z 8 Z 7 Z 6 Z 5 Z 4 Z 3 Z Z Z ----For 6 : Z 8 Z 7 Z 6 Z 5 Z 4 Z Z : Z Z Z Z 3Z 3Z Z Z ----For For 8 : Z 8 3Z 6 Z 5 Z 4 Z Z : Z Z Z Z Z Z Z ----For For 0 : Z 8 Z 7 Z 6 Z 4 Z 3 3Z Z Z : Z Z 4Z 3Z Z Z Z ----For For : Z 8 3Z 6 Z 4 Z 3 Z Z Z ----For 3 : Z 8 3Z 6 3Z 4 Z Z : Z Z 4Z Z Z Z Z ----For : Z Z Z 5Z Z Z Z ----For : Z Z Z 3Z 4Z 3Z Z Z ----For Coclusio: I this paper I itroduced recurret formulas for coupled Fiboacci sequeces of third & fourth order uder differet schemes. he proofs for these facts ca be show by iductio. A ope problem is the costructio of a explicit formula for each of the schemes give above. Ackowledgemet: I am thakful to aoymous referees for valuable suggestios. Orgaised by Departmet of Mathematics M.S.P.Madal'sV.P.College VaijapurAuragabad
8 Refereces: []. K. Ataassov L. Ataassov & D. Sasselov. "A New Perspective to the Geeralizatio of he Fiboacci Sequece." Fiboacci Quarterly 3. (985):-8. []. K. Ataassov. "O a Geeralizatio of the Fiboacci Sequece i the Case of hree Sequeces." Fiboacci Quarterly 7. (989):7-0. [3]. J.-Z. Lee & J. -S. Lee. "Some Properties of the Geeralizatio of the Fiboacci Sequece. Fiboacci Quarterly 5. (987):-7. Orgaised by Departmet of Mathematics M.S.P.Madal'sV.P.College VaijapurAuragabad
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