On Second Order Additive Coupled Fibonacci Sequences

Transcription

1 MAYFEB Joural of Mathematics O Secod Order Additive Coupled Fiboacci Sequeces Shikha Bhatagar School of Studies i Mathematics Vikram Uiversity Ujjai (M P) Idia suhai_bhatagar@rediffmailcom Omprakash Sikhwal Devashi Tutorial Keshw Kuj Madsaur (MP) Idia opbhsikhwal@rediffmailcom` Abstract - The coupled differece equatios or recurrece relatios ivolve two sequeces of itegers i which the elemets of oe sequece are part of the geeralizatio of the other ad vice versa K T Ataassov was first itroduced cocept of additive coupled Fiboacci sequeces i 1985 The coupled Fiboacci sequeces are ew directio of geeralizatio of Fiboacci sequece He defied four differet schemes with properties of specific schemes for additive coupled Fiboacci sequeces of secod order I this paper we preset some properties of additive coupled Fiboacci sequeces of secod order for real umbers Keywords- Fiboacci Sequece Additive Coupled Fiboacci Sequeces I INTRODUCTION Fiboacci sequece stads as a kid of super sequece with fabulous properties Fiboacci umbers are a sequece of umbers i which each successive umber is the sum of two previous umbers: Fiboacci sequece is defied by the recurrece relatio: (1) F F F with F 0 F 1 where The Fiboacci sequece has bee geeralized i a umber of ways [1-3] The coupled Fiboacci sequeces are ew directio i geeralizatio of Fiboacci sequece 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 We ca say that these are geeralizatio of ordiary recursive sequeces ad may results ca be developed for cosiderig the two sequeces are idetical They ca be cosidered as complemetary cocept of itersectios of liear sequeces The coupled sequeces provides visual patter of relatioship The cocept of additive coupled Fiboacci sequeces of secod order was first itroduced by K T Ataassov et al [4] He defied for differet schemes for additive coupled Fiboacci sequeces of secod order ad called them 2-Fiboacci sequeces (or 2-F sequeces) Let ad 0 0 a b c ad be two ifiite sequeces with iitial values 0 a 0 b 1 c ad 1 d where d are four arbitrary real umbers The four differet schemes of additive coupled Fiboacci sequeces of secod order are defied as follows: 30

2 MAYFEB Joural of Mathematics First Scheme (2) Secod Scheme (3) Third Scheme (4) Fourth Scheme (5) Further K T Ataassov et al [4-6] have bee studied the properties of first ad secod schemes M Sigh et al [7] preseted coupled Fiboacci sequeces of fifth order with some properties for positive ad egative itegers O Sikhwal [8] preseted properties of additive ad multiplicative coupled Fiboacci sequeces of secod order ad higher order I this paper we state ad prove some properties of third ad fourth schemes for additive coupled Fiboacci sequeces of secod order II RESULTS OF THIRD SCHEME I this sectio some properties of third scheme discussed All results derived by mathematical iductio method First few terms of scheme (4) are as follows: TABLE I FEW TERMS OF THIRD SCHEME OF ADDITVE COUPLED FIBONACCI SEQUENCE 0 a b 1 c d 2 a d b c 3 b 2c a 2d 4 2a 3d 2b 3c 5 3b 5c 3a 5d Theorem (21) If 0 is ay iteger the ( a) Proof ( a ) If 0 the 31

3 MAYFEB Joural of Mathematics Also Thus result is true for 0 Now assume that result is true for some iteger 1 the Also

4 MAYFEB Joural of Mathematics Hece the result is true for all itegers 0 Theorem (22) If 0 is ay iteger the F F Proof If 0 the F F Thus result is true for 0 Now suppose that result is true for some iteger 1 the F F F F F F F F F F Hece the result is true for all 0 Theorem (23) If 0 is ay iteger the k k k 0 Proof If 0 the F F 1 F F Thus result is true for 0 Now suppose that result is true for some iteger 1 the 1 k k 1 1 k k k0 k F F 33

5 MAYFEB Joural of Mathematics 1 F F F F theorem 22) F F F F F F Hece the result is true for all 0 III RESULTS OF FOURTH SCHEME I this sectio some properties of fourth scheme discussed All results derived by mathematical iductio method First few terms of scheme (5) are as follows: TABLE I FEW TERMS OF FOURTH SCHEME OF ADDITVE COUPLED FIBONACCI SEQUENCE 0 a b 1 c d 2 a c b d 3 a 2c b 2d 4 2a 3c 2b 3d 5 3a 5c 3b 5d Theorem (31) If 0 is ay iteger the ( a) ( b) Proof ( a) If 0 the Thus result is true for 0 Now assume that result is true for some iteger 1 the

6 MAYFEB Joural of Mathematics Hece the result is true for all itegers 0 Proof ( b ) Proof ca be give same as part ( a ) Theorem (32) If 0 is ay iteger the F F Proof If 0 the F F Thus result is true for 0 Now suppose that result is true for some iteger 1 the F F F F F F F F F F Hece the result is true for all 0 Theorem (33) If 0 is ay iteger the ( a) 2F 2 F ( b) 2F 2 F Proof ( a ) If 0 the 2F 2F Thus result is true for 0 Now suppose that result is true for some iteger 1 the 35

7 MAYFEB Joural of Mathematics F F F F F F 2 F F F 2F Hece the result is true for all 0 Proof ( b ) Proof ca be give same as part ( a) Theorem (34) If 0 is ay iteger the (a) 2F 2 F (b) 2F 2 F Proof ( a ) If 0 the 2F 2F Thus result is true for 0 Now suppose that result is true for some iteger 1 the F F F F F F 2 F F F 2F Hece the result is true for all 0 Proof () b Proof ca be give same as part ( a) Theorem (35) If 0 is a iteger the 36

8 MAYFEB Joural of Mathematics (a) 2F 2 F (b) 2F 2 F (c) 2F 2 F (d) 2F 2 F (e) 2F 2 F (f) 2F 2 F Proof ( a ) If 0 the 2F 2F Thus result is true for 0 Now suppose that result is true for some iteger 1 the F F 4 F F By simple computatio we have 2 F F 2F 2F Hece the result is true for all 0 Proof () b to ( f ) Proof ca be give same as part ( a ) IV CONCLUSION I preset paper we have preseted properties of specific schemes for additive coupled Fiboacci sequeces of secod order May ew results ca be established by varyig the patter i higher order additive coupled Fiboacci sequeces 37

9 MAYFEB Joural of Mathematics V ACKNOWLEDGMENT The authors are thakful to the reviewers for their costructive suggestios ad commets for improvig the expositio of the origial versio REFERENCES [1] A F Horadam A geeralized Fiboacci sequece Amer Math Mothly Vol 68 No 5 pp [2] A F Horadam Basic properties of certai geeralized sequece of umbers The Fiboacci Quarterly Vol 3 No 3 pp [3] J Z Lee ad J S Lee Some properties of the geeralizatio of the Fiboacci sequece The Fiboacci Quarterly Vol 25 No 2 pp [4] K T Ataassov L C Ataassov D D Sasselov A ew perspective to the geeralizatio of the Fiboacci sequece The Fiboacci Quarterly Vol 23 No 1 pp [5] K T Ataassov O a secod ew geeralizatio of Fiboacci sequece The Fiboacci Quarterly Vol 24 No 4 pp [6] K T Ataassov V Ataassov A Shao ad J Turer New visual perspective o Fiboacci umbers Sigapore World Scietific Publishig Compay 2002 [7] M Sigh M O Sikhwal S Jai Coupled Fiboacci sequeces of fifth order ad some properties Iteratioal Joural of Mathematical Aalysis Vol 4 No 25 pp [8] O Sikhwal Geeralizatio of Fiboacci sequece: A itriguig sequece Germay Lap Lambert Academic Publishig GmbH & Co KG