Binet Type Formula For The Sequence of Tetranacci Numbers by Alternate Methods

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1 DOI: 545/mjis764 Biet Type Formula For The Sequece of Tetraacci Numbers by Alterate Methods GAUTAMS HATHIWALA AND DEVBHADRA V SHAH CK Pithawala College of Eigeerig & Techology, Surat Departmet of Mathematics, Veer Narmad South Gujarat Uiversity, Surat gautamhathiwala77@gmailcom Received: Jue, 7 Revised: July, 7 Accepted: August 6, 7 Published Olie: September, 7 The Author(s) 7 This article is published with ope access at wwwchitkaraedui/publicatios AbstractThe sequece T{ }of Tetraacci umbers is defied by the recurrece relatio TT= TT 4 T ; 4 with iitial coditio =T T=T = ad T = I this paper, we obtai the eplicit formula Biet type formula for T by two differet methods We use the cocept of eige decompositio as well as of geeratig fuctios to obtai the result Keywords: Biet formula, Fiboacci sequece, Tetraacci sequece Mathematics Subject Classificatio: B9, 5B6 INTRODUCTION Fiboacci sequece is a sequece of umbers defied by the recursive formula F = F F ; for > with iitial coditio F - - =, F = This sequece possesses may iterestig properties which have bee studied i detail ([8], []) Aalogous to Fiboacci sequece, may other sequeces have bee defied, either by chagig the iitial terms or the recursive relatio or both, to obtai the ew sequece which may possess similar properties ([9], []) Oe of the importat result that is associated with Fiboacci umbers ad which has bee φ ( φ) studied for ceturies ow, is the Biet formula give as F = ; 5 where φ = 5 ad F is the th Fiboacci umber Also, by chagig the recursive formula or the iitial terms, we ca have a ew sequece of umbers defied ad correspodigly, a ew Biet type formula Mathematical Joural of Iterdiscipliary Scieces Vol-6, No-, September 7 pp

2 Hathiwala, GS Shah, DV ca be obtaied ([], [6], [7]) I this paper we cosider the sequece of Tetraacci umbers ad obtai the Biet type formula for by two differet methods Defiitio The sequece of Tetraacci umbers is defied by the recurrece relatio T = T T T T 4 ; for 4 with iitial terms T = T = T = ad T = The first few terms of the sequece {T } are,,,,,, 4, 8, 5, 9, 56, 8, 8, 4, 77, 49, 87 May saliet features of this sequece have bee studied i detail ([4], [5], []) PRELIMINARY RESULTS I this sectio, we give some prelimiary results which will be useful to derive the Biet type formula for Propositio T T T T = Proof: Above result ca be proved usig pricipal of mathematical iductio It is clearly observed that result is true for = Let it be true for some positive iteger k This gives, T Tk Tk Tk k = k Now cosider = k The we have RHS = k = k 8

3 Usig the iductio hypothesis, we get RHS = k T = Tk Tk Tk k T = Tk Tk Tk k 4 = LHS Biet Type Formula For The Sequece of Tetraacci Numbers By Alterate Methods Thus result is true for = k also, ad hece for all We et derive the geeratig fuctio for the sequece { T } Propositio : t = 4 Proof: Let t() be the polyomial of ifiite degree with its coefficiets as Tetraacci 4 5 umbers ie let t( )= T = T T T T T4 T5 = Now, multiplyig this polyomial by (-),(- ),(- ) ad ( 4 ) successively ad addig them, we get 4 t = T T T T T T ( T T T T ) 4 ( T T T T T 4 ) 4 This gives, ( ) t( )= t( )= 4, as required We use this result to derive the Biet type formula for T We first use the cocept of eige decompositio of a matri to derive this formula ad the we use the theory of geeratig fuctios to obtai the eplicit formula for T MAIN RESULT Theorem : ( ) ( ) T = ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) where = 9756, = 77484, = i ad = Proof: By propositio, we have t( )= 4 Let f ( )= 4 The for some,, ad, we write f ( )= ( -) ( -) ( - ) ( - ) 9

4 Hathiwala, GS Shah, DV Thus,,, ad roots of are the roots of f() This gives,, ad as the f 4 = = 4 This implies = We ow solve this equatio usig Ferrari s method Cosider the substitutio = y This coverts the above equatio ito depressed quartic form y y y =, 8 which ca be writte as y 5 - y 4 6 = () Net we itroduce a ew variable m o the LHS of () by addig ym m m o both sides Now regroupig the powers of y o the RHS of (), above equatio ca be trasformed as 5 y m my y m m 4 6 = () We ote that () ad () are equivalet for ay value of m We select such a value of m which makes RHS of () a perfect square Thus the discrimiat i y of this quadratic equatio is zero or i other words m is the root of the 5 equatio 4 6 m m m = O simplificatio, we get 5 8m - m m - = () 6 Equatio () is the resolvet cubic equatio of the origial quartic equatio We et apply Carda s method to solve equatio () For that we take the substitutio m= t ad o simplificatio get the depressed cubic equatio as t t = 6 4

5 Now we itroduce two ew variables i ad j, such that i j = t Usig this 7 65 i above equatio, we have i j ij ( i j) = 6 7 Net we impose the coditio ij =,, which implies ij = This gives 65 7 i j = ad i j = Thus, we get a quadratic equatio z z =, whose roots 6 are i ad j We solve this quadratic equatio to get the values of i ad j, ad evetually values of i ad j as Biet Type Formula For The Sequece of Tetraacci Numbers By Alterate Methods i= ad j = 6 Thus, t = i j gives t = This gives, 6 - m = For this value of m, the RHS of equatio () is of the form my 4, which is a perfect square Thus, from (), we have m y m my 4 4 = O solvig, we get the four values m of y as , -4844, i ad i Sice = y 4, we get the values of, ie 4 the roots of equatio = as = 9756, , = i ad = i Now, usig propositio, we write 4

6 Hathiwala, GS Shah, DV 4 T = [ ] (4) Now defie P = We first fid the eigevalues of P For that we cosider the characteristic equatio as det(p - λi) = This gives λ λ λ λ 4 = By above discussio it is clear that the roots of this equatio are,, ad ad they are the eigevalues of P Sice P has four distict eigevalues, it is a diagoalizable matri [] Moreover, for the eige decompositio of P we fid the eigevectors for the correspodig eigevalues Cosider the matri equatio P = a, where is the eigevector correspodig to eigevalue This gives P I = θ Solvig this matri equatio is equivalet to fidig the ull space of (P - I) Thus, we have = θ Solvig this, we get oe of the eigevector as Similarly, for, ad, the correspodig eigevectors are, ad respectively Let M = be the matri formed

7 whose colum vectors are the eigevectors of P correspodig to eigevalues,, ad The by eige decompositio of a matri [], we write P= M M Biet Type Formula For The Sequece of Tetraacci Numbers By Alterate Methods This gives, P = M M (5) By applyig elemetary row operatios o M, we get 4

8 Hathiwala, GS Shah, DV M = ( )( )( ) ( ) ( )( )( ) ( )( )( ) ( )( )( ) ( ) ( )( )( ) ( )( )( ) ( ) ( )( )( ) ( )( )( ) ( )( )( ) ( )( ) ( ) ( )( )( ) ( )( )( ) ( ) ( )( )( ) ( )( )( ) Usig (5) i (4), we get T = [ ] M M 44

9 This gives, ( ) ( ) T = [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) This gives o simplificatio Biet Type Formula For The Sequece of Tetraacci Numbers By Alterate Methods 45

10 Hathiwala, GS Shah, DV ( ) ( ) T = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), as required We et fid the same result by the use of geeratig fuctio for T Theorem : T = ( - )( - )( - ) ( - )( - ) ( - )( - ) ( - )( - ) ; where = 9756, = , = i ad = Proof: By propositio, we have t( )= 4 From theorem, we write ( )( )( ) 4 = This gives t( )= We ow write ( )( )( ) ( -) ( -) ( - ) ( -) A B C D = - ( - ) ( - ) ( - ) (6) The = A( ) ( ) ( ) B( ) ( ) ( ) C( ) ( ) ( ) D( ) ( ) ( ) If we cosider =, we get = A 46

11 This gives A = Similarly, we get ( ) ( ) ( ) B =, ( - )( - )( - ) C= ad D - = ( - )( - ) ( - )( - )( - ) Biet Type Formula For The Sequece of Tetraacci Numbers By Alterate Methods Thus (6) ca be writte as t( )= A( ) B( ) C( ) D i i i i i i i i This gives t( )= A i= B i= o c i= D i= Usig the values A, B, C ad D, we get i i t( )= i ( )( )( ) = ( ) ( ) i i ( ) ( ) ( ) i ( ) ( ) ( ) But, from lemma, t( )= sides, we get = T Thus comparig coefficiets o both ( ) ( ) T = ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) REFERENCES [] Arfke, G: Diagoalizatio of Matrices, Mathematical Methods for Physicists, rd ed Orlado, FL: Academic Press, 985, 7 9 [] Dresde GPB, Du Z: A Simplified Biet Formula for Geeralized Fiboacci Numbers, Joural of Iteger Fuctios, Vol 7, article 447, 4 [] Eige decompositio of a matri: 47

12 Hathiwala, GS Shah, DV [4] Hathiwala G S, Shah D V: Golde proportios for the geeralized Tetraacci umbers, Iteratioal Research Joural of Mathematics, Eigeerig ad IT, Vol, Issue 4, April 6, 9 [5] Hathiwala G S, Shah D V: Periodicity of Tetraacci Numbers Modulo, The Joural of the Idia Academy of Mathematics, Vol 8, Issue, 6, pp 55 65, ISSN: 97 5 [6] Lee G Y, Lee G S, Kim J S, Shi H K: The Biet Formula ad Represetatios of Geeralized Fiboacci Numbers, Fiboacci Quarterly, Vol 9, No, May, [7] Mehta D A: PhD thesis etitled Properties of the Sequeces of Triboacci Numbers ad Geeralized Cut off Numbers, Veer Narmad South Gujarat Uiversity, Surat, Idia, Oct 9 [8] Raab JA: A Geeralizatio of the Coectio betwee the Fiboacci Sequece ad Pascal s Triagle, The Fiboacci Quarterly, Vol, No, Oct 96, [9] Sigh B, Bhatagar S, Sikhwal O: Fiboacci Like Sequece, Iteratioal Joural of Advaced Mathematical Scieces, Vol, No,, 45 5 [] Waddill ME: Some Properties of Tetraacci Numbers modulo, The Fiboacci Quarterly, Vol, No, Aug 99, 8 [] Waddill ME: The Tetraacci Sequece ad Geeralizatios, The Fiboacci Quarterly, Vol, No, Feb 99, 9 [] Wall D D: Fiboacci Series Modulo, America Math Mothly, Vol 67, 96,

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