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On the K-Jacobsthal Numbers American Review of Mathematics and Statistics March 2014, Vol. 2, No. 1, pp. 67-77 ISSN 2374-2348 (Print) 2374-2356 (Online) Copyright © The Author(s). 2014. All Rights Reserved. Published by American Research Institute for Policy Development On the k-Jacobsthal Numbers Sergio Falcon1 Abstract We introduce a general Jacobsthal sequence that generalizes the classical Jacobsthal sequence. Many properties of these numbers Jk, n , n N are deduced directly from elementary algebra in a similar way that in the case of the k‒Fibonacci numbers. Finally, we will find that the Pascal triangle related with the k‒Jacobsthal numbers coincides with the triangle obtained for the k‒Fibonacci numbers. Keywords: k‒Fibonacci numbers, Formulas of Binnet, Catalan, D'Ocagne and convolution, Pascal triangle 1. Introduction In this section, we introduce the k‒Fibonacci numbers, defined previously by Falcon and Plaza (2007). For any positive real number k, the k‒Fibonacci sequence,say F isdefined k, nn N recurrently by: Fk, n 1 k F k , n F k , n 1 (1.1) for n ≥1, with the initial conditions Fk,0 0 and Fk,1 1. 1 Department of Mathematics, University of Las Palmas de Gran Canaria, 35017 – Las Palmas (Spain). Email: [email protected], Phone +34 928458827, Fax +34 928458811 68 American Review of Mathematics and Statistics, Vol. 2(1), March 2014 Particular cases of the k-Fibonacci sequence are: If k=1, the classical Fibonacci sequence is obtained: F00, F 1 1 and Fn 1 F n F n 1 for n 1. So, Fn 0,1,1,2,3,5,8,... If k=2, the classical Pell sequence appears: P0, P 1 and P 2 P P for n 1. Then P 0,1,2,5,12,70,169,... 0 1n 1 n n 1 2. The k-Jacobsthal Numbers In a similar form to the k-Fibonacci numbers, we define the k-Jacobsthal numbers by mean of the recurrence relation Jk, n 1 J k , n k J k , n 1 for n 1 (2.1) with the initial conditions Jk,0 = 0 and Jk,1 = 1. We will represent the k-Jacobsthal sequence as JJJk 0,1, k,2 , k ,3 ,... For k = 1 and k = 2, the Jacobsthal sequence J1 coincides with the classical Fibonacci sequence and the classical Jacobsthal sequence J 0,1,1,3,5,11,... , respectively. For k = 1, 2…30, all the k-Jacobsthal sequences are listed in Sloane N.J.A. from now on OEIS.In general,we will take k N and the firstk-Jacobsthal numbers are: Table 1 J k,1 1 J k,2 1 Jk,3 1 k Jk,4 1 2 k 2 Jk,5 1 3 k k 2 Jk,6 1 4 k 3 k 2 3 Jk,7 1 5 k 6 k k Sergio Falcon 69 2.1 The Binet Identity for the k-Jacobsthal Numbers The solutions of the characteristic equation r2 r k associated to the 1 1 4k 1 1 4k recurrence relation (2.1) are and , and 1 2 2 2 n n consequently, the solution of the recurrence relation is Jk, n c 1 1 c 2 2 . Then: n0 c1 c 2 0 n1 c1 1 c 2 2 1 From these equations we obtain the general term of the k-Jacobsthal sequence JJ : k k, nn N n n n n 1 1 4k 1 1 4 k 1 J 1 2 (2.2) k, n 4k 1 2 2 1 2 1 5 If k = 1, then is the Golden Ratio Φ. 1 2 2 If 1 or 2 it is k . The characteristic solutions verify the following properties: 1· 2 k 1 2 1 1 2 4 k 1 11 2 0 | 2 | 1 2.2 Two Expressions for the Positive Characteristic roots as Limits Here, two different ways for representing the metallicmeans are given. 70 American Review of Mathematics and Statistics, Vol. 2(1), March 2014 2.2.1 Continued Fractions First, note that from characteristic equation r2 r k itis immediately 2 k obtained 1 1 k 1 1 , from where by repeated substitutions we have: 1 k 1 1 k 1 k 1 k 1 1 ··· Note the last continued fraction represents the positive rootof the characteristic equation, since all the terms are positive. Besides, for different values of the parameter k, we have the continued fraction corresponding of some of the most common k-Jacobsthal sequences. Then, for the classical Fibonacci sequence (k = 1), it 1 is 1 1 1 1 1 1 1 1 1 ··· 2 For the classical Jacobsthal sequence (k = 2), it is 1 1 2 1 2 1 2 1 1 ··· 2.2.2 Nested Radicals From the characteristic equation it is r1 k r ; and applying iteratively this relationwe can write1k 1 k k k k ··· and, as in the case of continued fractions, note thisexpression corresponds to the positive characteristic root. Sergio Falcon 71 2.3 Ratio between two k-Jacobsthal Numbers Jk, n r r If r is a positive integer number, then lim 1 n Jk, n Proof. n n r n r 1 2 r r 2 n r n r 1 2 J k, n r 1 2 1 2 1 r lim lim lim lim 1 n n n n n n n n n J k, n 1 2 1 2 1 2 1 2 1 n 2 because |2 | 1 lim 0 n 1 J F 1 5 Particularly, lim1,n 1 lim n1 , is the golden Ratio, . n n JF1,n n 2 n 2.4 Relation between the Sequences and Jk n For n ≥ 1 and if σ = σ1 or σ= σ2, it is Jk, n J k , n 1 k. Proof. By induction. For n = 1, Jk,1 J k ,0 k . 2 For n = 2, Jk,2 J k ,1 k k . Let us suppose this formula is true for n. Then n1 n 2 Jkn, J kn , 1 k J kn , J kn , 1 k J kn ,() k J kn , 1 k ()Jk, n kJ k , n 1 J k , n k J k , n 1 J k , n k 72 American Review of Mathematics and Statistics, Vol. 2(1), March 2014 Then, the sequence of powers n contains the k-Jacobsthal sequence as coefficients of and also as coefficients of the parameter k. 2.4.1 Proposition 1: Catalan Identity 2n r 2 Forn r : Jk,,,, n r J k n r J k n ( k ) J k r (2.3) Proof. By using Equation (2.1) in the left hand side (LHS) of this relation, and taking into account 1 2 = - k, we obtain: 2 n r n r n r n r n n ()·LHS 1 2 1 2 1 2 1 2 1 2 1 2 2n nrnr nrnr 2 n 2 n nn 2 n 1 1 2 1 2 2 1 2 1 2 2 2 1 2 r r n 2r 2 r 1n2 n 1 n ( k ) 2 1 1 2 1 2 2 1 2 2 4k 1 4 k 1 ( )r 1 2 1 2 ()r r 2 ()() kn r1 2 k n r J 2 4k 1 k, r Note that for r = 1, Equation (2.3) gives the Simson Identity for the k-Jacobsthal sequence: 2n n 1 Jk, n 1 J k , n 1 J k , n ( 1) k (2.4) 2.4.2 Proposition 2: Convolution Identity Jkmn, J km , 1 J kn , k J kmkn , J , 1 Proof. Applying the Binet formula for the k-Jacobsthal numbers to the Second Hand Right (SHR) of this relation, we have: Sergio Falcon 73 1 SHRm1 m 1 n n k m m n 1 n 1 2 1 2 1 2 1 2 1 2 ()1 2 1 mn1 mn 1 n mn 1 nmn 1 mn 1 mn 1 2 [1 1()() k 2 k 2 k 1 k 1 ()1 2 km1 n() k n 1 k m 1 n 2 2 ] 1 m1 n()() 2 k m 1 n 2 k 2 1 1 2 2 ()1 2 1 m1 n()() 2 m 1 n 2 2 1 1 1 2 2 2 1 2 ()1 2 1 1 m1 n ()()() m 1 n m n m n J () 2 11122222 12k , m n 1 2 1 2 Particular cases of the convolution formula: 2 2 2 Even k-Jacobsthal numbers: if m = n, then Jk,2 n J k , n 1 k J k , n 1 2 2 Odd k-Jacobsthal numbers: if m = n+1, then Jk,2 n 1 J k , n 1 k J k , n 3 3 3 3 If m = 2n, then Jk,3 n J k , n 1 k J k , n k J k , n 1 In a similar way that before the following identity is proven. 2.4.3 Proposition 3: D'Ocagne Identity m n 1 n Form > n, Jkmkn, J , 1 J kmkn , 1 J , ( 1) k J kmn , 2.5 Binomial formula for the k-Jacobsthal Numbers ip 1 n j n 1 For n ≥ 1: Jk, n n1 (4 k 1) where ip is the integer part of . 2 j0 2j 1 2 74 American Review of Mathematics and Statistics, Vol. 2(1), March 2014 Proof. If we expand the Binnet Identity, then n n n n 1 1 4k 1 1 4 k 1 J 1 2 k, n 4k 1 2 2 1 2 1 1 n n 3 n 5 n 2 41k (41) k (41)···( k RHS ) 4k 1 2 1 3 5 From this formula it is easy to find any k-Jacobsthal number without having to find before the preceding terms of the k-Jacobsthal sequence.
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