International J.Math. Combin. Vol.1(2017), 36-52 Some New Generalizations of the Lucas Sequence F¨ugen TORUNBALCI AYDIN Yildiz Technical University, Faculty of Chemical and Metallurgical Engineering, Department of Mathematical Engineering, 34220, Istanbul, Turkey Salim YUCE¨ Yildiz Technical University, Faculty of Arts and Sciences Department of Mathematics, 34220, Istanbul, Turkey E-mail: [email protected], [email protected] Abstract: In this paper, we investigate the generalized Lucas, the generalized complex Lucas and the generalized dual Lucas sequence using the Lucas number. Also, we investigate special cases of these sequences. Furthermore, we give recurrence relations, vectors, the golden ratio and Binet’s formula for the generalized Lucas and the generalized dual Lucas sequence. Key Words: Smarandache-Fibonacci triple, Fibonacci number, Lucas number, Lucas sequence, generalized Fibonacci sequence, generalized complex Lucas sequence, generalized dual Lucas sequence. AMS(2010): 11B37, 11B39. §1. Introduction Let S(n),n 0 with S(n) = S(n 1) + S(n 2) be a Smarandache-Fibonacci triple, where ≥ − − S(n) is the Smarandache function for integers n 0. Particularly, let S(n) be F (n) or L(n), ≥ we get the Fibonacci or Lucas sequence as follows: A Fibonacci sequence 1 , 1 , 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, , F , · · · n · · · is defined by the recurrence relation Fn = Fn 1 + Fn 2 , (n > 3), − − with F1 = F2 = 1, where Fn is the n-th term of the Fibonacci sequence (Fn) (Leonardo Fibonacci, 1202). The Fibonacci sequence is named after Italian mathematician Leonardo of Pisa, known as Fibonacci. The name “Fibonacci Sequnce” was first used by the 19th- century number theorist Edouard Lucas. Some recent generalizations for the Fibonacci sequence have 1Received May 17, 2016, Accepted February 8, 2017. Some New Generalizations of the Lucas Sequence 37 produced a variety of new and extended results, [1],[5],[6],[9],[13]. A Lucas sequence 2 , 1 , 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, ,L , · · · n · · · is defined by the recurrence relation Ln = Ln 1 + Ln 2 , (n > 3), − − with L1 =2 ,L2 = 1, where Ln is the n-th term of the Lucas sequence (Ln) (Fran¸cois Edouard Anatole Lucas, 1876). There are a lot of generalizations of the Lucas sequences,[15],[16],[17]. The generalized Fibonacci sequence defined by Hn = Hn 1 + Hn 2 , (n > 3) (1.1) − − with H1 = p , H2 = p+q where p, q are arbitrary integers [3]. That is, the generalized Fibonacci sequence is p, p + q , 2p + q, 3p +2q, 5p +3q, 8p +5q, , (p q)F + qF , (1.2) · · · − n n+1 · · · Using the equations (1.1) and (1.2) , it was obtained Hn+1 = q Fn + p Fn+1 (1.3) Hn+2 = p Fn + (p + q) Fn+1. For the generalized Fibonacci sequence, it was obtained the following properties: 2 2 Hn 1 + Hn = (2p q)H2n 1 e F2n 1 , (1.4) − − − − − 2 2 Hn+1 Hn 1 = (2p q)H2n e F2n , (1.5) − − − − 2 n Hn 1 Hn+1 Hn = ( 1) e , (1.6) − − − Hn+r = Hn 1Fr + HnFr+1 (n > 3) (1.7) − 2 n r 2 Hn+1 r Hn+1+r Hn+1 = ( 1) − e Fr , (1.8) − − − 2 2 Hn+1 + e Fn = p H2n+1 , (1.9) n+s Hn Hn+1+r Hn s Hn+r+s+1 = ( 1) e FsFr+s+1 , (1.10) − − − 2 2 2 2 [2Hn+1Hn+2] + [HnHn+3] = 2Hn+1Hn+2 + Hn (1.11) r Hn+r + ( 1) Hn r r − − = Fr+1 + ( 1) Fr 1 (1.12) Hn − − where e = p2 p q q2. − − 38 F¨ugen TORUNBALCI AYDIN and Salim YUCE¨ Also, for p =1, q = 0 , we get the following well-known results: 2 2 Fn 1 + Fn = F2n 1 , (Catalan), (1.13) − − 2 n Fn 1 Fn+1 Fn = ( 1) , (Simpson or Cassini), (1.14) − − − 2 2 Fn+1 + Fn = F2n+1 (Lucas). (1.15) In this paper, we will define the generalized Lucas, the generalized complex Lucas and the generalized dual Lucas sequences respectively, denoted by Gn , Cn , Dn . §2. Generalized Lucas Sequence and Lucas Vectors In this section, we will define the generalized Lucas sequence denoted by Ln. The generalized Lucas sequence defined by Ln = Ln 1 + Ln 2 , (n > 3), (2.1) − − with L1 = 2 p q , L2 = p +2 q where p, q are arbitrary integers,[3]. That is, the generalized − Lucas sequence is 2 p q,p +2 q , 3 p + q , 4 p +3 q , 7 p +4 q , 11 p +7 q , , (p q)L + qL , (2.2) − · · · − n n+1 · · · Using the equations (2.1) and (2.2) , we get Ln+1 = qLn + pLn+1 , (2.3) Ln+2 = pLn + (p + q) Ln+1. Putting n = r in (2.3) and using (2.1), we find in turn Lr+3 = (2 p + q )Lr+1 + (p + q ) Lr = H3Lr+1 + H2Lr (2.4) Lr+4 = (3 p +2 q) Lr+1 + (2 p + q) Lr = H4Lr+1 + H3Lr So, in general, we have obtain relations between generalized Lucas sequence and generalized Fibonacci sequence as follows: Ln+r = Hn 1Lr + HnLr+1 (2.5) − Also, certain results follow almost immediately from (2.1) Ln+2 2 Ln Ln 1 =0, (2.6) − − − Ln+1 2 Ln + Ln 2 =0, (2.7) − − Some New Generalizations of the Lucas Sequence 39 n 1 − L2i+1 = L2n (2 p q), (2.8) − − i=0 X n L2i = L2n+1 (p +2 q), (2.9) − i=1 X n (L2i 1 L2i)= L2n 1 p +3 q. (2.10) − − − − − i=1 X For the generalized Lucas sequence, we have the following properties: 2 2 Ln 1 + Ln = (2p q)(L2n 2 + L2n) eL (L2n 2 + L2n) , (2.11) − − − − − 2 2 Ln+1 Ln 1 = (2p q)(L2n+2 L2n 2) eL (L2n+2 L2n 2) , (2.12) − − − − − − − − 2 n+1 Ln 1 Ln+1 Ln = 5 ( 1) eL , (2.13) − − − 2 2 Ln+1 + eL Ln = p (L2n+2 + L2n) , (2.14) r Ln+r + ( 1) Ln r − − = Lr (2.15) Ln where e = p2 p q q2. L − − Theorem 2.1 If Ln is the generalized Lucas number, then Ln+1 pα + q lim = , n Ln qα + (p q) →∞ − where α = (1+ √5) 2=1.618033 is the golden ratio. · · · Proof We have for the Lucas number Ln, L lim n+1 = α, n →∞ Ln where α = (1+ √5) 2=1.618033 · · · . is the golden ratio [12]. Then for the generalized Lucas number Ln, we obtain Ln+1 pLn+1 + qLn pα + q lim = lim = . (2.16) n Ln n qLn+1 + (p q)Ln qα + (p q) →∞ →∞ − − ¾ 40 F¨ugen TORUNBALCI AYDIN and Salim YUCE¨ Theorem 2.2 The Binet’s formula2 for the generalized Lucas sequence is as follows; n n Ln =(¯α α + β¯ β ) (2.17) where α¯ = α(2 p q) (p +2 q), β¯ = (p +2 q) β (2 p q). − − − − Proof The characteristic equation of recurrence relation Ln+2 = Ln+1 + Ln is t2 t 1=0. (2.18) − − The roots of this equation are 1+ √5 1 √5 α = and β = − , (2.19) 2 2 where α + β =1 , α β = √5 , αβ = 1. − − Using recurrence relation and initial values L0 = (2 p q), L1 = (p +2 q) the Binet’s − formula for Ln, we get n n n n Ln = A α + Bβ = α¯ α + β¯ β , (2.20) where L1 β L0 α L0 L1 A = − , B = − α β α β − − ¯ andα ¯ = α(2 p q) (p +2 q), β = (p +2 q) β (2 p q). ¾ − − − − A generalized Lucas vector is defined by L−→n = (Ln , Ln+1, Ln+2) Also, from equation (2.2) it can be expressed as L−→n = (p q)−→L n + q−→L n+1 (2.21) − where −→L n = ( Ln , Ln+1, Ln+2) and −→L n+1 = ( Ln+1 , Ln+2, Ln+3) are the Lucas vectors. The product of L−→n and λ R is given by ∈ λ L−→n = (λ L−→n, λ L−−−→n+1, λ L−−−→n+2) 2Binet’s formula is the explicit formula to obtain the n-th Fibonacci and Lucas numbers. It is well known that for the Fibonacci and Lucas numbers, Binet’s formulas are αn βn Fn = − α β − and n n Ln = α + β respectively, where α + β = 1 , α β = √5 , αβ = 1 and α = (1+ √5) 2 , − − β = (1 √5) 2 ,[7], [8]. − . Some New Generalizations of the Lucas Sequence 41 and L−→n and L−→m are equal if and only if Ln = Lm Ln+1 = Lm+1 Ln+2 = Lm+2. Theorem 2.3 Let L−→n and L−→m be two generalized Lucas vectors. The dot product of L−→n and L−→m is given by 2 L−→n, L−→m = p (5 Fn+m+3 + Ln Lm) D E +p q [2 Ln+m 1 + 10 Fn+m+2 ] − 2 +q (5 Fn+m+1 + Ln 1Lm 1) . (2.22) − − Proof The dot product of L−→n = (Ln, Ln+1, Ln+2) and L−→m = (Lm, Lm+1, Lm+2) defined by L−→n, L−→m = LnLm + Ln+1Lm+1 + Ln+2Lm+2. D E Also, using the equations (2.1), (2.2) and (2.3), we obtain 2 2 Ln Lm = p (Ln Lm)+ p q [LnLm 1 + Ln 1 Lm]+ q (Ln 1 Lm 1), (2.23) − − − − 2 2 Ln+1 Lm+1 = p (Ln+1 Lm+1)+ p q [Ln+1Lm + Ln Lm+1]+ q (Ln Lm), (2.24) 2 Ln+2 Lm+2 = p (Ln+2Lm+2)+ p q [Ln+2Lm+1 + Ln+1Lm+2] (2.25) 2 +q (Ln+1Lm+1).