k-jacobsthal and k-jacobsthal Lucas Matrix Sequences

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1 International Mathematical Forum, Vol 11, 016, no 3, HIKARI Ltd, wwwm-hikaricom k-jacobsthal and k-jacobsthal Lucas Matrix Sequences S Uygun 1 and H Eldogan Department of Mathematics, Science and Art Faculty Gaziantep University, Campus, 7310, Gaziantep, Turkey Copyright c 015 S Uygun and H Eldogan This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Abstract In this study, we consider sequences named k-jacobsthal, k-jacobsthal Lucas sequences After that, by using these sequences, we define k- Jacobsthal and k-jacobsthal-lucas matrix sequence at the same time Finally we investigate some properties of these sequences, present some important relationship between k-jacobsthal matrix sequence and k- Jacobsthal Lucas matrix sequence Mathematics Subject Classification: 11B83, 11K31, 15A4, 15B99 Keywords: Jacobsthal sequence, Jacobsthal-Lucas sequence, matrix sequences 1 Introduction Integer sequences, such as Fibonacci, Lucas, Jacobsthal, Jacobsthal Lucas, Pell charm us with their abundant applications in science and art, and very interesting properties For instance, it is well known that computers use conditional directives to change the flow of execution of a program In addition to branch instructions, some microcontrollers use skip instructions which conditionally bypass the next instructionthis brings out being useful for one case out of the four possibilities on bits, 3 cases on 3 bits, 5 cases on 4 bits, 11 cases on 5 bits, 1 cases on 6 bits,, which are exactly the Jacobsthal 1 Corresponding author

2 146 S Uygun and H Eldogan numbers Many properties of these sequences were deduced directly from elementary matrix algebra For example F Koken and D Bozkurt in [4] defined a Jacobsthal matrix of the type nxn and using this matrix derived some properties of Jacobsthal numbers Of course the most known integer sequence is made of Fibonacci numbers which are very important because of golden section H Civciv and R Turkmen, in [5,6], is defined s, t-fibonacci and s, t-lucas matrix sequences by using s, t-fibonacci and s, t-lucas sequences Particular cases of Jacobsthal and Jacobsthal Lucas numbers were investigated earlier by Horadam [1-] Ş Uygun, in [3] is defined s, t-jacobsthal and s, t-jacobsthal Lucas sequences Also K Uslu and Ş Uygun, in [7] are defined s, t-jacobsthal and s, t-jacobsthal Lucas matrix sequences by using s, t-jacobsthal and s, t-jacobsthal Lucas sequences In this study, firstly we define k-jacobsthal and k-jacobsthal Lucas sequences, then by using these sequences, we also define k-jacobsthal and k- Jacobsthal Lucas matrix sequences We derive numerous interesting properties of these sequences Then we investigate the relationship between k-jacobsthal and k-jacobsthal Lucas matrix sequences Additionally, in [1], the Jacobsthal and Jacobsthal Lucas sequences are defined recurrently by j n j n 1 + j n, j 0 0, j 1 1 c n c n 1 + c n, c 0, c 1 1 where n 1 any integer These sequences can be generalized by preserving the relation of sequence, altering the initial conditions or by altering the relation of sequence preserving the initial conditions Main Results Firstly, let us first consider the following definition of k-jacobsthal sequence which will be needed for the definition of k-jacobsthal matrix sequence Definition 1 Let be n N, k > 0 any real number Then k-jacobsthal sequence {ĵ k,n } n N is defined by the following equation: with initial conditions ĵ k,0 0, ĵ k,1 1 ĵ k,n kĵ k,n 1 + ĵ k,n, 1 First few terms of the k Jacobsthal number sequences are ĵ k,0 0, ĵ k,1 1, ĵ k,, ĵ k,3 k +, ĵ k,4 k 3 + 4k, ĵ k,5 k 4 + 6k + 4, ĵ k,6 k 5 + 8k 3 + 1k

3 k-jacobsthal and k-jacobsthal Lucas matrix sequences 147 If k 1 we have the classic Jacobsthal sequence {0, 1, 1, 3, 5, 11, 1, }A [8] If k we have the sequence {0, 1,, 6, 16, 44, 10, }A00605 [8] Definition For n N, k > 0 any real number, then k-jacobsthal Lucas sequence {ĉ k,n } n N is defined by the following equation: ĉ k,n kĉ k,n 1 + ĉ k,n, with initial conditions ĉ 0, ĉ 1 k If Jacobsthal Lucas sequence k 1, then we have the classic Some of the interesting properties that the k-jacobsthal sequence satisfies are summarized as below: Lemma 3 For n 0 any integer, the Binet formulas for nth k Jacobsthal number and nth k Jacobsthal Lucas number are given by and ĵ k,n rn 1 r n, ĉ k,n r n 1 + r n respectively where r 1 k+ k +8 and r k k +8, are the roots of the characteristic equation x kx+ associated to the recurrence relation defined in 1 We can see easily r 1 r, r 1 + r k, k + 8 Proof For the proof of the first equality we use the principle of induction n For n 0, we have ĵ k,0 r0 1 r0 r 1 r 0 Also for n 1, we have ĵ k,n r1 1 r1 r 1 r 1 We assume that the statement is true for n m, ĵ k,m rm 1 rm r 1 r For n m+1, ĵ k,m+1 kĵ k,m + ĵ k,m 1 k rm 1 r m + rm 1 1 r m 1 r 1 r r1 k m + r1 r k m + r r 1 r r1 m kr 1 + r 1 + r m kr + r rm+1 1 r m+1

4 148 S Uygun and H Eldogan Now, let us prove the second equality It follows from by using r 1 r, we have ĉ k,n kĵ k,n + 4ĵ k,n 1 r n k 1 r n r n 1 1 r n r 1 r [ ] 1 r1 k n + 4r1 r k n + 4r [ ] 1 r r1 n 1 + r r n + r 1 r r1 n + r n Definition 4 For n N, k > 0 any real number, then k-jacobsthal matrix sequence Ĵk,n is defined by the following equation: n N Ĵ k,n+ kĵk,n+1 + Ĵk,n 3 with the initial conditions Ĵ k, and Ĵk,1 k 1 0 Definition 5 For n N, k > 0 any real number, then k-jacobsthal Lucas matrix sequence Ĉk,n is defined by the following equation: n N with initial conditions k 4 Ĉ k,0 and k Ĉk,1 k-jacobsthal {Ĵk,n } n N Ĉ k,n+ kĉn+1 + Ĉn 4 k + 4 k k 4 and k-jacobsthal Lucas {Ĉk,n } n N matrix sequences are defined by carrying to matrix theory k-jacobsthal and k-jacobsthal Lucas sequences The following theorem shows us the nth general term of the Jacobsthal matrix sequence given in 3 Theorem 6 For n is any positive integer, k > 0 any real number, we have ĵk,n+1 ĵ Ĵ k,n k,n 5 ĵ k,n ĵ k,n 1

5 k-jacobsthal and k-jacobsthal Lucas matrix sequences 149 Proof Let us consider n 1 in 5 We clearly know that ĵ 0 0, ĵ 1 1, ĵ k, so ĵ ĵ Ĵ k,1 1 k ĵ 1 ĵ As a next step of that, for n, we also get ĵ3 ĵ Ĵ k, k + k ĵ ĵ 1 k By iterating this procedure and considering induction steps, let us assume that the equality in 5 holds for all m n Z + To end up the proof, we have to show that the case also holds for n + 1 Therefore, we get Ĵ k,n+1 kĵk,n + Ĵk,n 1 ĵk,n+1 ĵ k k,n ĵk,n ĵ + k,n 1 ĵ k,n ĵ k,n 1 ĵ k,n 1 ĵ k,n kĵk,n+1 + ĵ k,n kĵ k,n + 4ĵ k,n 1 kĵ k, + ĵ k,n 1 kĵ k,n 1 + 4ĵ k,n ĵk,n+ ĵ k,n+1 Hence the result ĵ k,n+1 ĵ k,n Theorem 7 For n N, k > 0 any real number, we have Ĵ k,m+n Ĵk,mĴk,n 6 Proof It s proven by induction We can easily see the truth of the hypothesis for n 0 Let us suppose that the equality in 6 holds for all p n Z + After that,we want to show that the equality is true for p n + 1 Ĵ k,m+n+1 Ĵk,m+n + Ĵk,m+n 1 Ĵk,mĴk,n + Ĵk,mĴk,n 1 Ĵk,mĴk,n + Ĵk,n 1 Ĵk,mĴk,n+1 Theorem 8 For any integer n 1, we get Ĵ k,n Ĵ n k,1

6 150 S Uygun and H Eldogan Proof It s proven by induction We can easily see the truth of the hypothesis for n 1 Let us suppose that the equality in 6 holds for all m n Z + After that,we want to show that the equality is true for m n + 1 Ĵ k,n+1 Ĵk,1Ĵk,n Ĵk,1Ĵ n k,1 Ĵ n+1 k,1 Theorem 9 For n is any positive integer, k > 0 any real number, we have ĉk,n+1 ĉ Ĉ k,n k,n ĉ k,n ĉ k,n 1 Proof We use the method of induction For n 1, we have k Ĉ k,1 + 4 k k 4 And for n, we also have k Ĉ k, 3 + 6k k + 8 k + 4 k Let us suppose that the equality in 6 holds for all m n Z + To end up the proof, we have to show that the case also holds for n + 1 We get Ĉ k,n+1 kĉk,n + Ĉk,n 1 ĉk,n+1 ĉ k k,n ĉ k,n ĉ k,n 1 ĉk,n ĉ + k,n 1 ĉ k,n 1 ĉ k,n kĉ k,n+1 + ĉ k,n kĉ k,n + 4ĉ k,n 1 kĉ k,n + ĉ k,n 1 kĉ k,n 1 + 4ĉ k,n ĉk,n+ ĉ k,n+1 ĉ k,n+1 ĉ k,n Theorem 10 For n 0 any integer, k > 0 any real number, we have Ĉ k,n+1 Ĉk,1Ĵk,n Proof For n 0 it can be easily seen the truth of the hypothesis due to product of identity matrix For n 1, it is obvious from Ĉk,1 + 4 k k k 4

7 k-jacobsthal and k-jacobsthal Lucas matrix sequences 151 and Ĵk,1 k 1 0 Ĉ k, Ĉk,1Ĵk,1 k + 4 k k k k 3 + 6k k + 8 k + 4 k ĉk,3 ĉ k, ĉ k, ĉ k,1 We assume that it is true for all integers m n Now we show that it is true for m n + 1 : Ĉ k,1 Ĵk,n+1 Ĉk,1Ĵk,nĴk,1 Ĉk,n+1Ĵk,1 ĉk,n+ ĉ k,n+1 ĉ k,n+1 ĉ k,n ĉk,n+3 ĉ k,n+ Ĉk,n+ ĉ k,n+ ĉ k,n+1 k 1 0 Theorem 11 For n > 0 any integer, k > 0 any real number we have Proof For n 1, it is obvious: Ĉ k,n kĵk,n + 4Ĵk,n 1 Ĉ k,1 kĵk,1 + 4Ĵk,0 k + 4 k k 4 k k For n we get Ĉ k,n Ĉk,1Ĵk,n 1 [ kĵk,1 + 4Ĵk,0] Ĵk,n 1 kĵk,n + 4Ĵk,n 1

8 15 S Uygun and H Eldogan Theorem 1 For n, m 0 any integers, k > 0 any real number, we have the commutative property Ĵ k,m Ĉ k,n+1 Ĉk,n+1Ĵk,m Ĵ k,m Ĉ k,n+1 Ĵk,mĈk,1Ĵk,n [ Ĵk,m kĵk,1 + 4Ĵk,0] Ĵk,n kĵk,n+m+1 + 4Ĵk,n+m [ kĵk,1 + 4Ĵk,0] Ĵk,n+m Ĉk,1Ĵk,nĴk,m Ĉk,n+1Ĵk,m Theorem 13 For n 0 any integer, we have a Ĉ k,n+1 Ĉ k,1 Ĵk,n b Ĉ k,n+1 Ĉk,n+1 c Ĉk,n+1 Ĵk,nĈk,n+1 Proof For the proof of a For the proof of b Ĉ k,n+1 Ĉk,n+1Ĉk,n+1 Ĉk,1Ĵk,nĈk,1Ĵk,n Ĉ k,1ĵk,n For the proof of c Ĉ k,n+1 Ĉ k,1ĵk,n Ĉk,1Ĉk,1Ĵk,n Ĉk,1Ĉk,n+1 Ĉ k,n+1 Ĉk,1Ĵk,n Ĵk,nĈk,n+1 Corollary 14 For n 0 any integer, we have i ĉ k,n+ + ĉ k,n+1 k + 8k ĵ k,n+3 ii ĉ k,n+ + ĉ k,n+1 ĉ k,n+4 + ĉ k,n+ iii ĉ k,n ĵ k,n ĉ k,n+1 + ĉ k,n ĵ k,n 1

9 k-jacobsthal and k-jacobsthal Lucas matrix sequences 153 Proof k Ĉk,n+1 Ĉ k,1ĵk,n + 4 k ĵk,n+1 ĵ k,n k 4 ĵ k,n ĵ k,n 1 ĉk,n+ ĉ k,n+1 k k + 16 k k ĵk,n+1 ĵ k,n k 3 + 8k k + 16 ĉ k,n+1 ĉ k,n ĵ k,n ĵ k,n 1 From the equality of the entries of 1,1 matrices, we have k k + 16 ĵ k,n+1 + k k ĵ k,n k 3 kĵ k,n+1 + kĵ k,n + 8k kĵ k,n+1 + kĵ k,n + 16ĵ k,n+1 k 3 ĵ k,n+ + 10kĵ k,n+ 4kĵ k,n + 16ĵ k,n+1 + k ĵ k,n+1 +8kĵ k,n+ + 16ĵ k,n+1 k ĵ k,n+3 + 8kĵ k,n+3 k + 8k ĵ k,n+3 Ĉ k,n+1 Ĉk,1Ĉk,n+1 k + 4 k k 4 ĉk,n+ ĉ k,n+1 ĉ k,n+1 ĉ k,n ĉ k,n+ + ĉ k,n+1 ĉ k,n+4 + ĉ k,n+ Theorem 15 For n 0, we get Ĵ k,n Ĵk,1 r Ĵ k,0 r n 1 Ĵk,1 r 1 Ĵ k,0 r n Proof Ĵk,1 r Ĵ k,0 Ĵk,1 r 1 Ĵ k,0 Ĵ k,n r1 n r n r1 n k r k r1 1 r 1 r 1 1 k r n 1 r n r 1 r r n 1 1 r n 1 r1 n r n ĵk,n+1 ĵ k,n ĵ k,n ĵ k,n 1 r n r1 n r n r 1 r r n 1 1 r n 1

10 154 S Uygun and H Eldogan References [1] A F Horadam, Jacobsthal representation numbers, The Fibonacci Quarterly, , no, 40-5 [] A F Horadam, Jacobsthal representation polynomials, The Fibonacci Quarterly, , no, [3] Ş Uygun, The s,t-jacobsthal and s,t-jacobsthal Lucas sequences, Applied Mathematical Sciences, , no 9, [4] F Köken, D Bozkurt, On the Jacobsthal Numbers by Matrix Methods, Int Jour Contemp Math Sciences, 3 008, no 13, [5] H Civciv, R Turkmen, On the s,t Fibonacci and Fibonacci matrix sequences, ARS Combinatoria, , [6] H Civciv, R Turkmen, Notes on the s,t Lucas and Lucas matrix sequences, ARS Combinatoria, , [7] K Uslu, Ş Uygun, The s,t Jacobsthal and s,t Jacobsthal-Lucas Matrix sequences, ARS Combinatoria, , 13- [8] N J A Sloane, The On-Line Encyclopedia of Integer Sequences, 006 [9] T Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley and Sons Inc, New York, [10] Eric W Weisstein, Jacobsthal Number, Wolfram Mathworld, Retrieved 007, [online] Received: December 15, 015; Published: January 6, 016