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On r-Jacobsthal and r-Jacobsthal-Lucas Numbers

On r-Jacobsthal and r-Jacobsthal-Lucas Numbers Annales Mathematicae Silesianae 37 (2023), no. 1, 16–31 DOI: 10.2478/amsil-2023-0001 ON r-JACOBSTHAL AND r-JACOBSTHAL–LUCAS NUMBERS Göksal Bilgici , Dorota Bród Abstract. Recently, Bród introduced a new Jacobsthal-type sequence which is called r-Jacobsthal sequence in current study. After defining the appro- priate r-Jacobsthal–Lucas sequence for the r-Jacobsthal sequence, we obtain some properties of these two sequences. For simpler results, we define two new sequences and examine their properties, too. Finally, we generalize some well-known identities. 1. Introduction The Lucas sequences generalize many famous integer sequences defined by a second order linear recurrence relation such as the Fibonacci num- bers, the Lucas numbers, the Pell numbers, the Pell–Lucas numbers, the Jacobsthal numbers and the Jacobsthal–Lucas numbers. Let A and B be 2 2 integers. The roots of x Ax + B = 0 are x = (A + A 4B) and x = (A A 4B). The Lucas sequences (see [3] for details) are defined by the following Binet-like formula n n n n x x x + x 1 2 1 2 P = and Q = : n n x x x + x 1 2 1 2 Received: 07.03.2022. Accepted: 05.01.2023. Published online: 07.02.2023. (2020) Mathematics Subject Classification: 11B83, 11B37. Key words and phrases: r-Jacobsthal numbers, r-Jacobsthal–Lucas numbers, Binet formula. c 2023 The Author(s). This is an Open Access article distributed under the terms of the Creative Commons Attribution License CC BY (http://creativecommons.org/licenses/by/4.0/). On r-Jacobsthal and r-Jacobsthal–Lucas numbers 17 For (A; B) = (1;1), (2;1) and (1;2), the sequence fP g gives the Fi- bonacci, the Pell and the Jacobsthal numbers, respectively. Similarly, for (A; B) = (1;1), (2;1) and (1;2), the sequence fQ g gives the Lucas, the Pell–Lucas and the Jacobsthal–Lucas numbers, respectively. The Jacobsthal sequence fJ g and the Jacobsthal–Lucas sequence fj g n n are defined by the same second order recurrence relation, namely T = T + 2T n n1 n2 except the initial conditions. While the initial conditions of the Jacobsthal sequence are J = 0 and J = 1, the initial conditions of the Jacobsthal– 0 1 Lucas sequence are j = 2 and j = 1. Binet formulas for the Jacobsthal and 0 1 the Jacobsthal–Lucas numbers are n n n n J = and j = ; n n respectively, where = 2 and  = 1 are roots of the characteristic equation x x 2 = 0. Generating functions for the sequences fJ g and fj g are n n 1 1 X X x 2 x n n J x = and j x = ; n n 2 2 1 x 2x 1 x 2x n=0 n=0 respectively. A comprehensive study about the Jacobsthal and the Jacobsthal– Lucas numbers was made by Horadam ([7]). He gave a lot of interesting prop- erties and beautiful identities of these numbers. We recall some of them. J + j = 2J ; n n n+1 2 2 (1.1) j + 9J = 2j ; 2n n n 2 2 n n+2 (1.2) j 9J = (1) 2 ; n n (1.3) J j + J j = 2J ; m n n m n+m n n+1 (1.4) J j J j = (1) 2 J : m n n m mn There are some generalizations of the Jacobsthal and the Jacobsthal–Lucas numbers defined in different ways. Falcon ([5]) defined the k-Jacobsthal num- bers, Jhala, Sisodiya and Rathore ([8]) gave another definition for the k- Jacobsthal numbers, Dasdemir ([2]) defined the Jacobsthal p-numbers and Uygun ([10]) introduced the (s; t)-Jacobsthal numbers. Similarly, Uygun and Owusu ([11]) defined the bi-periodic Jacobsthal numbers. All of these authors 18 Göksal Bilgici, Dorota Bród changed the recurrence relation of the Jaconsthal sequence, slightly, while preserving the initial conditions. Some general sequences also generalize the Jacobsthal and the Jacobsthal- -Lucas numbers. We can refer to the Horadam sequence ([6]) and the bi- periodic generalized Fibonacci sequence ([4]) as examples of this approach. In [1], Bród defined another one parameter Jacobsthal sequence fJ g r;n (for integers n  0 and r  0) which is called r-Jacobsthal sequence, by the recurrence relation r r r (1.5) J = 2 J + (2 + 4 )J ; n  2 r;n r;n1 r;n2 r+1 with the initial conditions J = 1 and J = 1 + 2 . It is clear that r;0 r;n J = J . n 0;n2 The r-Jacobsthal sequence has an application in the theory of graphs (see [1]). Recall this graph interpretation of the r-Jacobsthal numbers. Let G be a finite, undirected, simple graph with vertex set V (G) and edge set E(G). A set S  V (G) is an independent set of G if for any two distinct vertices x; y 2 S holds xy 62 E(G). A subset of V (G) containing only one vertex and the empty set are independent sets of G, too. The number of inde- pendent sets of a graph G is denoted by NI(G). The parameter NI(G) was studied not only in mathematical literature. In 1989, Merrifield and Simmons ([9]) introduced the number of independent sets into the chemical literature as the index . They showed a correlation between this index in a molecular graph and some chemical properties. The parameter NI(G) is often repre- sented by the Fibonacci numbers and the Lucas numbers. This fact may be a motivation to ask the following question: Are there any generalizations of the Fibonacci numbers that have graph interpretations due to the number of independent sets in the graph? Consider a graph G (Figure 1), where n  1, r  0. n;r y y y y y G : 1 2 n2 n1 n n;r r r : : : rr r r @ @ @ @ @ @ v r rz 1 1 @ @ @ . .. @ r @r rr @ r @ r . .. : : : . .. x x@ x@ x@ x@ 1 2 n2 n1 n r r : : :@r r : : :@r r : : :@r @r v z r r u u w w h h 1 r 1 r 1 r Figure 1. A graph G n;r In [1], the following result was proved. Theorem 1.1 ([1]). Let n; r be integers such that n  1, r  0. Then NI(G ) = J : n;r r;n On r-Jacobsthal and r-Jacobsthal–Lucas numbers 19 The graph interpretation of the r-Jacobsthal numbers can be used for proving some identities. Theorem 1.2 ([1] convolution identity). Let n; m; r be integers such that m  3; n  2, r  0. Then r r r J = 2 J J + (4 + 8 )J J : r;n r;m1 r;n r;m2 r;n1 Corollary 1.3 ([1]). J = J J + 2J J : m+n m n+1 m1 n Bród gave the following Binet formula for the r-Jacobsthal numbers p p r n r n r r r r ( 4 2 + 5 4 + 3 2 + 2) + ( 4 2 + 5 4 3 3 2) 1 2 J = ; r;n r r 2 4 2 + 5 4 2 r r r where  and  are roots of the characteristic equation  2 (2 +4 ) = 0, 1 2 namely p p r r r r r r 2 + 4 2 + 5 4 2 4 2 + 5 4 = and  = : 1 2 2 2 We can change this Binet formula easily with n  n 1 1 2 2 (1.6) J = ; r;n 1 2 r  r where  = 2 + 1 +  and  = 2 + 1 +  . 1 2 1 2 Now we define the r-Jacobsthal–Lucas sequence fK g with the same r;n n=0 recurrence relation r r r K = 2 K + (2 + 4 )K ; n  2; r;n r;n1 r;n2 1r r with the initial conditions K = 3+2 and K = 3+42 . It is easily seen r;0 r;1 that j = K . Some initials terms of the r-Jacobsthal–Lucas sequence are n 0;n2 1r K = 3 + 2 ; r;0 K = 3 + 4 2 ; r;1 r r K = 2 + 8 2 + 7 4 ; r;2 r r r K = 5 2 + 15 4 + 11 8 ; r;3 r r r r K = 2 2 + 15 4 + 30 8 + 18 16 : r;4 20 Göksal Bilgici, Dorota Bród Binet formula for the r-Jacobsthal–Lucas numbers is given in the following theorem. Theorem 1.4. For any nonnegative integer n, the nth r-Jacobsthal–Lucas number is n  n 1 1 2 2 K = : r;n 1 2 Proof. For the r-Jacobsthal–Lucas sequence fK g , we have r;n n=0 n n (1.7) K = A  + A  : r;n 1 2 1 2 By using the initial conditions of r-Jacobsthal–Lucas sequence, we get the following system of equations 1r A + A = 3 + 2 ; 1 2 A  + A  = 3 + 4 2 : 1 1 2 2 Solutions of the system are r r 1 + 2 +  1 + 2 + 1 2 A = and A = : 1 2 r r 2 2 After substitution A and A into (1.7), we obtain the theorem. 1 2 Now we present generating function for the r-Jacobsthal–Lucas sequence fK g . r;n n=0 Theorem 1.5. The generating function for the r-Jacobsthal–Lucas se- quence is 1r r 3 + 2 + (2 + 1)x K x = : r;n r r r 2 1 2 x (2 + 4 )x n=0 Proof. Let us define K(x) = K x . Then we obtain r;n n=0 1r r n (1.8) K(x) = 3 + 2 + (3 + 4 2 )x + K x : r;n n=2 On r-Jacobsthal and r-Jacobsthal–Lucas numbers 21 r r r 2 By multiplying both sides of (1.8) by 2 x and (2 + 4 )x , we have r r r n (1.9) 2 xK(x) = (2 + 3 2 )x 2 K x r;n1 n=2 and r r 2 r r n (1.10) (2 + 4 )x K(x) = (2 + 4 )K x ; r;n n=2 respectively. Adding side by side equalities (1.8), (1.9) and (1.10) gives r r r 2 1 2 x (2 + 4 )x K(x) 1r r r r r n = 3 + 2 + (2 + 1)x + K 2 K (2 + 4 )K x : r;n r;n1 r;n2 n=2 The last equation and the recurrence relation for the r-Jacobsthal–Lucas num- bers complete the proof. It should be noted that we need the following equation for later use r 2 = (2 + 1) : 1 2 2. Second type of r-Jacobsthal and r-Jacobsthal–Lucas numbers For simpler results, we need a new type of the r-Jacobsthal and the r- Jacobsthal–Lucas numbers. We preserve the recurrence relation but change the initial conditions. Namely, the second type of the r-Jacobsthal numbers satisfies the recurrence relation 0 r 0 r r 0 J = 2 J + (2 + 4 )J ; n  2 r;n r;n1 r;n2 0 0 with the initial conditions J = 0 and J = 1, and the second type of the r;0 r;1 r-Jacobsthal–Lucas numbers satisfies the recurrence relation 0 r 0 r r 0 (2.1) K = 2 K + (2 + 4 )K ; n  2 r;n r;n1 r;n2 0 1r 0 with the initial conditions K = 2 and K = 1. r;0 r;1 22 Göksal Bilgici, Dorota Bród Binet formula for the r-Jacobsthal and the r-Jacobsthal–Lucas numbers can be found in the following theorem. Theorem 2.1. For any nonnegative integer n, the nth second type of the r-Jacobsthal and the r-Jacobsthal–Lucas numbers are n n 0 1 2 (2.2) J = r;n 1 2 and n n 0 1 2 K = ; r;n 1 2 respectively. Proof. Proofs can be done with the similar way to the proof of Theo- rem 1.4. 0 1 Theorem 2.2. The generating functions for the sequences fJ g and r;n n=0 0 1 fK g are r;n n=0 0 n J x = r;n r r r 2 1 2 x (2 + 4 )x n=0 and 1r 2 x 0 n K x = ; r;n r r r 2 1 2 x (2 + 4 )x n=0 respectively. Proof. Proofs can be done by using the similar way to the proof of The- orem 1.5. The following lemma presents some connections between the r-Jacobsthal and the r-Jacobsthal–Lucas numbers. Lemma 2.3. For any positive integers r and n, we have 1r K = 2 J J r;n r;n+1 r;n and 0 1r 0 0 K = 2 J J : r;n r;n+1 r;n On r-Jacobsthal and r-Jacobsthal–Lucas numbers 23 Proof. From the Binet formula for the r-Jacobsthal numbers (1:6), we have 1r  n+1  n+1  n  n 2 (    )   + 1r 1 1 2 2 1 1 2 2 2 J J = p r;n+1 r;n r r 4 2 + 5 4 n 1r  n 1r (2  1)   (2  1) 1 2 1 1 2 2 = : r r 4 2 + 5 4 1r r 1r r r If we substitute the identities 2  1 = 2 4 2 + 5 4 and 2  1 = 1 2 r r 2 4 2 + 5 4 into the last expression, we obtain the first identity in the theorem. The second identity can be proved similarly. The connections between both types of the r-Jacobsthal numbers and be- tween both types of the r-Jacobsthal–Lucas numbers are given in the following lemma. Lemma 2.4. For any positive integers r and n, we have 0 r 0 J = J + (2 + 1)J r;n r;n+1 r;n and 0 r 0 K = K + (2 + 1)K : r;n r;n+1 r;n Proof. The proofs can be done, easily, by using the Binet formulas for the r-Jacobsthal and the r-Jacobsthal–Lucas numbers. The next lemma gives the second type of the r-Jacobsthal–Lucas and the r-Jacobsthal–Lucas numbers with negative indices. Lemma 2.5. For any positive integers r and n, we have n+1 (1) 0 0 J = J r;n r;n r r n (2 + 4 ) and (1) 0 0 K = K : r;n r;n r r n (2 + 4 ) 24 Göksal Bilgici, Dorota Bród Proof. From the Binet formula (2:2) for the second type of the r-Jacobsthal numbers, we have n n 0 1 2 J = p r;n r r 4 2 + 5 4 1 1 1 = p n n r r 4 2 + 5 4 1 2 n n 1 2 = : r r n (   ) 4 2 + 5 4 1 2 r r The equality   = (2 + 4 ) gives the first identity in lemma. The second 1 2 identity can be obtained in a similar way. Similarly, the first type of the r-Jacobsthal and the r-Jacobsthal–Lucas numbers with negative indices can be obtained as in the following by the help of Lemma 2.4 and Lemma 2.5. Lemma 2.6. For any positive integers r and n, we have n+1 (1) r 0 0 J = 2 J J r;n r;n r;n1 r r n1 (2 + 4 ) and (1) r 0 0 K = 2 K K : r;n r;n r;n1 r r n1 (2 + 4 ) Lemma 2.6 provides us to expand all the results about the r-Jacobsthal- -Lucas and the r-Jacobsthal–Lucas numbers to integers. Although Bród re- stricted r to nonnegative integers, we should emphasize that there is no need such a restriction. Namely, r can be an arbitrary integer. 3. Some properties of r-Jacobsthal and r-Jacobsthal–Lucas numbers In this section, we give some results for the r-Jacobsthal and the r-Jacobsthal–Lucas numbers starting with Vajda’s identities. On r-Jacobsthal and r-Jacobsthal–Lucas numbers 25 Theorem 3.1. For any integers, r; n; m and k, we have (3.1) J J J J r;n+m r;n+k r;n r;n+m+k r 2 0 0 0 0 = (2 + 1) J J J J r;n+m r;n+k r;n r;n+m+k r 2 n r r n 0 0 = (2 + 1) (1) (2 + 4 ) J J r;m r;k and K K K K r;n+m r;n+k r;n r;n+m+k r 2 0 0 0 0 = (2 + 1) K K K K r;n+m r;n+k r;n r;n+m+k r 2 n+1 r r n r r (2 + 1) (1) (2 + 4 ) (4 2 + 5 4 ) 0 0 = J J : r;m r;k Proof. Using the Binet formula (1:6) for the r-Jacobsthal numbers, we have J J J J r;n+m r;n+k r;n r;n+m+k n+m  n+m  n+k  n+k = (    )(    ) 1 1 2 2 1 1 2 2 r r 2 (4 2 + 5 4 ) n  n  n+m+k  n+m+k (    )(    ) 1 1 2 2 1 1 2 2 r 2 n (2 + 1) (  ) 1 2 m k k m m+k m+k =     +  + 1 2 1 2 1 2 r r 2 (4 2 + 5 4 ) r 2 n r r n (2 + 1) (1) (2 + 4 ) m m k k = (  )(  ) : 1 2 1 2 r r 2 (4 2 + 5 4 ) The last equality gives the first identity in (3.1). The others can be obtained similarly. If we take k ! m, Theorem 3.1 gives Catalan’s identities (3.2) J J J r;n+m r;nm r;n h i r 2 0 0 0 = (2 + 1) J J J r;n+m r;nm r;n r 2 n+m+1 r r nm 0 = (2 + 1) (1) (2 + 4 ) J r;m 26 Göksal Bilgici, Dorota Bród and (3.3) K K K r;n+m r;nm r;n h i r 2 0 0 0 = (2 + 1) K K K r;n+m r;nm r;n r 2 n+m r r nm r r (2 + 1) (1) (2 + 4 ) (4 2 + 5 4 ) = J : r;m If we take m ! 1, Catalan’s identities (3.2) and (3.3) give Cassini’s iden- tities (3.4) J J J r;n+1 r;n1 r;n h i r 2 0 0 0 = (2 + 1) J J J r;n+1 r;n1 r;n r 2 n r r n1 = (2 + 1) (1) (2 + 4 ) and K K K r;n+1 r;n1 r;n h i r 2 0 0 0 = (2 + 1) K K K r;n+1 r;n1 r;n r 2 n+1 r r n1 r r (2 + 1) (1) (2 + 4 ) (4 2 + 5 4 ) = : In the next theorem d’Ocagne’s identities for the r-Jacobsthal and the r-Jacobsthal–Lucas numbers are given. Theorem 3.2. For any integers, r; m and n, we have (3.5) J J J J r;m r;n+1 r;m+1 r;n r 2 0 0 0 0 = (2 + 1) J J J J r;m r;n+1 r;m+1 r;n r 2 m+1 r r m 0 = (2 + 1) (1) (2 + 4 ) J r;nm and K K K K r;m r;n+1 r;m+1 r;n r 2 0 0 0 0 = (2 + 1) K K K K r;m r;n+1 r;m+1 r;n r 2 2r r r m r r m 0 = (2 + 1) 2 (4 2 + 5 4 )(1) (2 + 4 ) J : r;nm On r-Jacobsthal and r-Jacobsthal–Lucas numbers 27 Proof. From the Binet formula (1:6) for the r-Jacobsthal numbers, we have J J J J r;m r;n+1 r;m+1 r;n n+1 n+1 m+1 m+1 m m n n (    +   +   ) 1 2 1 2 2 1 2 1 1 2 r r 4 2 + 5 4 r 2 m n n m (2 + 1)   (  )   (  ) 1 2 1 2 1 2 1 2 r r 4 2 + 5 4 nm nm r 2 m (2 + 1) (  ) (  ) 1 2 1 2 = p : r r 4 2 + 5 4 From the last equality, we obtain the first identity in (3.5). The other identities can be obtained similarly. Some connections between the r-Jacobsthal and the r-Jacobsthal–Lucas numbers are presented in the next theorem. Theorem 3.3. For any integers, r; m and n, we have r 2 r r 2 (3.6) 4 K + (4 2 + 5 4 )J = 2K ; r;2n+2 r;n r;n r 0 2 r 0 2 0 2 (K ) + (5 2 + 4)(J ) = 2K ; r;n r;n r;2n r 2 r r 2 r 2 n r r n 4 K (4 2 + 5 4 )J = 4(2 + 1) (1) (2 + 4 ) ; r;n r;n r 0 2 r 0 2 n r r n 2 (K ) (5 2 + 4)(J ) = 4(1) (2 + 4 ) ; r;n r;n 12r (3.7) J K +J K = 2 J ; r;m r;n r;n r;m r;m+n+2 0 0 0 0 1r 0 J K +J K = 2 J ; r;m r;n r;n r;m r;m+n 1r r 2 n r r 2 0 J K J K = 2 (2 + 1) (1) (2 + 4 ) J ; r;m r;n r;n r;m r;mn 0 0 0 0 1r n r r 2 0 J K J K = 2 (1) (2 + 4 ) J : r;m r;n r;n r;m r;mn Proof. All the proofs are based on the Binet formula and we prove two of them. We have r 2 r r 2 4 K + (4 2 + 5 4 )J r;n r;n n  n 2  n  n 2 = (  +   ) + (    ) 1 1 2 2 1 1 2 2 2 2n  2 2n = 2( )  + 2( ) 1 1 2 2 28 Göksal Bilgici, Dorota Bród r  2n r  2n = 2 (2 + 1 +  )  + (2 + 1 +  ) 1 2 1 1 2 2 r  2n  2n  2n+1  2n+1 = 2 (2 + 1)(  +   ) + (  +   ) 1 1 2 2 1 1 2 2 r+1 r = 2 (2 + 1)K +K : r;2n r;2n+1 The recurrence relation (2.1) gives (3.6). Now we prove (3.7). From Binet formula for the r-Jacobsthal and the r-Jacobsthal–Lucas numbers, we have J K +J K r;m r;n r;n r;m m  m  n  n = (    )(  +   ) 1 1 2 2 1 1 2 2 r r r 2 4 2 + 5 4 n  n  m  m + (    )(  +   ) 1 1 2 2 1 1 2 2 2 m+n  2 m+n = 2( )  2( ) 1 2 1 2 r r r 2 4 2 + 5 4 r  m+n = (2 + 1 +  ) r r r 2 4 2 + 5 4 r  m+n (2 + 1 +  ) r  m+n  m+n = p (2 + 1)(  +   ) 1 1 2 2 r r r 2 4 2 + 5 4 m+n+1  m+n+1 + (  +   ) 1 1 2 2 = J + (2 + 1)J : r;m+n+1 r;m+n From the recurrence relation (1.5), we obtain (3.7). The other identities can be proved similarly. By Theorem 3.3, for r = 0, we obtain known identities (1.1), (1.2), (1.3), (1.4) for the classical Jacobsthal and Jacobsthal–Lucas numbers. In the next theorem we give summation formulas for the r-Jacobsthal- -Lucas numbers and the second types of the r-Jacobsthal and the r-Jacobsthal- -Lucas numbers. Theorem 3.4. Let n; r be integers. Then n1 r r r 1r K + (2 + 4 )K 2 2 4 r;n r;n1 (3.8) K = ; r;i r+1 r 2 + 4 1 i=0 n1 0 r r 0 J + (2 + 4 )J 1 r;n r;n1 (3.9) J = ; r;i r+1 r 2 + 4 1 i=0 On r-Jacobsthal and r-Jacobsthal–Lucas numbers 29 n1 0 r r 0 1r K + (2 + 4 )K + 1 2 r;n r;n1 (3.10) K = : r;i r+1 r 2 + 4 1 i=0 Proof. For (3.8), on account of (1.7) we get n1 n1 n n X X 1  1 i i 1 2 K = (A  + A  ) = A + A r;i 1 2 1 2 1 2 1  1 1 2 i=0 i=0 n n n1 n1 A + A (A  + A  ) (A  + A  ) +   (A  + A  ) 1 2 1 2 2 1 1 2 1 2 1 2 1 2 1 2 1 ( +  ) + 1 2 1 2 r r A + A (    )K (2 + 4 )K 1 2 1 2 2 1 r;n r;n1 = : r r r 1 2 (2 + 4 ) 1r By simple calculations we have A + A = 3 + 2 , A  + A  = 1 2 1 2 2 1 (1 + 2 ). Hence n1 1r r r r 4 + 2 + 2 K (2 + 4 )K r;n r;n1 K = r;i r+1 r (2 + 4 1) i=0 r r r 1r K + (2 + 4 )K 2 2 4 r;n r;n1 = : r+1 r 2 + 4 1 In the same way one can easily prove (3.9) and (3.10). 4. Matrix generators Now we give the matrix generators of the numbers J and K . r;n r;n Theorem 4.1. Let n; r be integers. Then n1 J J J J 2 1 r;n+1 r;n r;2 r;1 (4.1) =  : r r J J J J 2 + 4 0 r;n r;n1 r;1 r;0 Proof. (by induction on n) It is easily seen that for n = 1 the result is obvious. Assuming that the formula (4.1) holds for n  1, we will prove it for n + 1. 30 Göksal Bilgici, Dorota Bród Using induction’s hypothesis and the recurrence formula for the r-Jacobsthal numbers, we have n1 r r J J 2 1 2 1 r;2 r;1 r r r r J J 2 + 4 0 2 + 4 0 r;1 r;0 J J 2 1 r;n+1 r;n r r J J 2 + 4 0 r;n r;n1 r r r 2 J + (2 + 4 )J J r;n+1 r;n r;n+1 r r r 2 J + (2 + 4 )J J r;n r;n1 r;n J J r;n+2 r;n+1 = ; J J r;n+1 r;n which ends the proof. As a consequence of Theorem 4.1 we get Cassini’s identity (3.4) for the r-Jacobsthal numbers. Corollary 4.2. Let n; r be integers. Then 2 n r r n1 r 2 J J J = (1) (2 + 4 ) (2 + 1) : r;n+1 r;n1 r;n Proof. Calculating determinants in formula (4.1), we obtain J J r;n+1 r;n 2 = J J J ; r;n+1 r;n1 r;n J J r;n r;n1 r r r J J 3 4 + 2 2 2 2 + 1 r;2 r;1 r 2 = = (2 + 1) ; J J 2 2 + 1 1 r;1 r;0 2 1 r r = (2 + 4 ): r r 2 + 4 0 Hence we get 2 r 2 n1 r r n1 J J J = (2 + 1) (1) (2 + 4 ) r;n+1 r;n1 r;n n r r n1 r 2 = (1) (2 + 4 ) (2 + 1) ; which completes the proof. Similarly to Theorem 4.1 and Corollary 4.2, we can prove the following results. On r-Jacobsthal and r-Jacobsthal–Lucas numbers 31 Theorem 4.3. Let n; r be integers. Then n1 K K K K 2 1 r;n+1 r;n r;2 r;1 =  : r r K K K K 2 + 4 0 r;n r;n1 r;1 r;0 Corollary 4.4. Let n; r be integers. Then 2 n1 r r n1 2r r r K K K = (1) (2 + 4 ) (13 + 2 + 14 2 + 5 4 ): r;n+1 r;n1 r;n References [1] D. Bród, On a new Jacobsthal-type sequence, Ars Combin. 150 (2020), 21–29. [2] A. Daşdemir, The representation, generalized Binet formula and sums of the general- ized Jacobsthal p-sequence, Hittite J. Sci. Eng. 3 (2016), no. 2, 99–104. [3] L.E. Dickson, History of the Theory of Numbers. Vol. I: Divisibility and Primality, Chelsea Publishing Co., New York, 1952. [4] M. Edson and O. Yayenie, A new generalization of Fibonacci sequence & extended Binet’s formula, Integers 9 (2009), no. 6, 639–654. [5] S. Falcon, On the k-Jacobsthal numbers, American Review of Mathematics and Sta- tistics 2 (2014), no. 1, 67–77. [6] A.F. Horadam, Basic properties of a certain generalized sequence of numbers, Fi- bonacci Quart. 3 (1965), no. 3, 161–176. [7] A.F. Horadam, Jacobsthal representation numbers, Fibonacci Quart. 34 (1996), no. 1, 40–54. [8] D. Jhala, K. Sisodiya, and G.P.S. Rathore, On some identities for k-Jacobsthal num- bers, Int. J. Math. Anal. (Ruse) 7 (2013), no. 12, 551–556. [9] R.E. Merrifield and H.E. Simmons, Topological Methods in Chemistry, John Wiley & Sons, New York, 1989. [10] S. Uygun, The (s; t)-Jacobsthal and (s; t)-Jacobsthal Lucas sequences, Appl. Math. Sci. (Ruse) 9 (2015), no. 70, 3467–3476. [11] S. Uygun and E. Owusu, A new generalization of Jacobsthal numbers (bi-periodic Jacobsthal sequences), J. Math. Anal. 7 (2016), no. 5, 28–39. Göksal Bilgici Elementary Mathematics Education Kastamonu University 37200 Kastamonu Turkey e-mail: gbilgici@kastamonu.edu.tr Dorota Bród Department of Discrete Mathematics Faculty of Mathematics and Applied Physics Rzeszow University of Technology Poland e-mail: dorotab@prz.edu.pl http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annales Mathematicae Silesianae de Gruyter

On r-Jacobsthal and r-Jacobsthal-Lucas Numbers

Bilgici, Göksal; Bród, Dorota
Annales Mathematicae Silesianae , Volume 37 (1): 16 – Mar 1, 2023
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Annales Mathematicae Silesianae 37 (2023), no. 1, 16–31 DOI: 10.2478/amsil-2023-0001 ON r-JACOBSTHAL AND r-JACOBSTHAL–LUCAS NUMBERS Göksal Bilgici , Dorota Bród Abstract. Recently, Bród introduced a new Jacobsthal-type sequence which is called r-Jacobsthal sequence in current study. After defining the appro- priate r-Jacobsthal–Lucas sequence for the r-Jacobsthal sequence, we obtain some properties of these two sequences. For simpler results, we define two new sequences and examine their properties, too. Finally, we generalize some well-known identities. 1. Introduction The Lucas sequences generalize many famous integer sequences defined by a second order linear recurrence relation such as the Fibonacci num- bers, the Lucas numbers, the Pell numbers, the Pell–Lucas numbers, the Jacobsthal numbers and the Jacobsthal–Lucas numbers. Let A and B be 2 2 integers. The roots of x Ax + B = 0 are x = (A + A 4B) and x = (A A 4B). The Lucas sequences (see [3] for details) are defined by the following Binet-like formula n n n n x x x + x 1 2 1 2 P = and Q = : n n x x x + x 1 2 1 2 Received: 07.03.2022. Accepted: 05.01.2023. Published online: 07.02.2023. (2020) Mathematics Subject Classification: 11B83, 11B37. Key words and phrases: r-Jacobsthal numbers, r-Jacobsthal–Lucas numbers, Binet formula. c 2023 The Author(s). This is an Open Access article distributed under the terms of the Creative Commons Attribution License CC BY (http://creativecommons.org/licenses/by/4.0/). On r-Jacobsthal and r-Jacobsthal–Lucas numbers 17 For (A; B) = (1;1), (2;1) and (1;2), the sequence fP g gives the Fi- bonacci, the Pell and the Jacobsthal numbers, respectively. Similarly, for (A; B) = (1;1), (2;1) and (1;2), the sequence fQ g gives the Lucas, the Pell–Lucas and the Jacobsthal–Lucas numbers, respectively. The Jacobsthal sequence fJ g and the Jacobsthal–Lucas sequence fj g n n are defined by the same second order recurrence relation, namely T = T + 2T n n1 n2 except the initial conditions. While the initial conditions of the Jacobsthal sequence are J = 0 and J = 1, the initial conditions of the Jacobsthal– 0 1 Lucas sequence are j = 2 and j = 1. Binet formulas for the Jacobsthal and 0 1 the Jacobsthal–Lucas numbers are n n n n J = and j = ; n n respectively, where = 2 and  = 1 are roots of the characteristic equation x x 2 = 0. Generating functions for the sequences fJ g and fj g are n n 1 1 X X x 2 x n n J x = and j x = ; n n 2 2 1 x 2x 1 x 2x n=0 n=0 respectively. A comprehensive study about the Jacobsthal and the Jacobsthal– Lucas numbers was made by Horadam ([7]). He gave a lot of interesting prop- erties and beautiful identities of these numbers. We recall some of them. J + j = 2J ; n n n+1 2 2 (1.1) j + 9J = 2j ; 2n n n 2 2 n n+2 (1.2) j 9J = (1) 2 ; n n (1.3) J j + J j = 2J ; m n n m n+m n n+1 (1.4) J j J j = (1) 2 J : m n n m mn There are some generalizations of the Jacobsthal and the Jacobsthal–Lucas numbers defined in different ways. Falcon ([5]) defined the k-Jacobsthal num- bers, Jhala, Sisodiya and Rathore ([8]) gave another definition for the k- Jacobsthal numbers, Dasdemir ([2]) defined the Jacobsthal p-numbers and Uygun ([10]) introduced the (s; t)-Jacobsthal numbers. Similarly, Uygun and Owusu ([11]) defined the bi-periodic Jacobsthal numbers. All of these authors 18 Göksal Bilgici, Dorota Bród changed the recurrence relation of the Jaconsthal sequence, slightly, while preserving the initial conditions. Some general sequences also generalize the Jacobsthal and the Jacobsthal- -Lucas numbers. We can refer to the Horadam sequence ([6]) and the bi- periodic generalized Fibonacci sequence ([4]) as examples of this approach. In [1], Bród defined another one parameter Jacobsthal sequence fJ g r;n (for integers n  0 and r  0) which is called r-Jacobsthal sequence, by the recurrence relation r r r (1.5) J = 2 J + (2 + 4 )J ; n  2 r;n r;n1 r;n2 r+1 with the initial conditions J = 1 and J = 1 + 2 . It is clear that r;0 r;n J = J . n 0;n2 The r-Jacobsthal sequence has an application in the theory of graphs (see [1]). Recall this graph interpretation of the r-Jacobsthal numbers. Let G be a finite, undirected, simple graph with vertex set V (G) and edge set E(G). A set S  V (G) is an independent set of G if for any two distinct vertices x; y 2 S holds xy 62 E(G). A subset of V (G) containing only one vertex and the empty set are independent sets of G, too. The number of inde- pendent sets of a graph G is denoted by NI(G). The parameter NI(G) was studied not only in mathematical literature. In 1989, Merrifield and Simmons ([9]) introduced the number of independent sets into the chemical literature as the index . They showed a correlation between this index in a molecular graph and some chemical properties. The parameter NI(G) is often repre- sented by the Fibonacci numbers and the Lucas numbers. This fact may be a motivation to ask the following question: Are there any generalizations of the Fibonacci numbers that have graph interpretations due to the number of independent sets in the graph? Consider a graph G (Figure 1), where n  1, r  0. n;r y y y y y G : 1 2 n2 n1 n n;r r r : : : rr r r @ @ @ @ @ @ v r rz 1 1 @ @ @ . .. @ r @r rr @ r @ r . .. : : : . .. x x@ x@ x@ x@ 1 2 n2 n1 n r r : : :@r r : : :@r r : : :@r @r v z r r u u w w h h 1 r 1 r 1 r Figure 1. A graph G n;r In [1], the following result was proved. Theorem 1.1 ([1]). Let n; r be integers such that n  1, r  0. Then NI(G ) = J : n;r r;n On r-Jacobsthal and r-Jacobsthal–Lucas numbers 19 The graph interpretation of the r-Jacobsthal numbers can be used for proving some identities. Theorem 1.2 ([1] convolution identity). Let n; m; r be integers such that m  3; n  2, r  0. Then r r r J = 2 J J + (4 + 8 )J J : r;n r;m1 r;n r;m2 r;n1 Corollary 1.3 ([1]). J = J J + 2J J : m+n m n+1 m1 n Bród gave the following Binet formula for the r-Jacobsthal numbers p p r n r n r r r r ( 4 2 + 5 4 + 3 2 + 2) + ( 4 2 + 5 4 3 3 2) 1 2 J = ; r;n r r 2 4 2 + 5 4 2 r r r where  and  are roots of the characteristic equation  2 (2 +4 ) = 0, 1 2 namely p p r r r r r r 2 + 4 2 + 5 4 2 4 2 + 5 4 = and  = : 1 2 2 2 We can change this Binet formula easily with n  n 1 1 2 2 (1.6) J = ; r;n 1 2 r  r where  = 2 + 1 +  and  = 2 + 1 +  . 1 2 1 2 Now we define the r-Jacobsthal–Lucas sequence fK g with the same r;n n=0 recurrence relation r r r K = 2 K + (2 + 4 )K ; n  2; r;n r;n1 r;n2 1r r with the initial conditions K = 3+2 and K = 3+42 . It is easily seen r;0 r;1 that j = K . Some initials terms of the r-Jacobsthal–Lucas sequence are n 0;n2 1r K = 3 + 2 ; r;0 K = 3 + 4 2 ; r;1 r r K = 2 + 8 2 + 7 4 ; r;2 r r r K = 5 2 + 15 4 + 11 8 ; r;3 r r r r K = 2 2 + 15 4 + 30 8 + 18 16 : r;4 20 Göksal Bilgici, Dorota Bród Binet formula for the r-Jacobsthal–Lucas numbers is given in the following theorem. Theorem 1.4. For any nonnegative integer n, the nth r-Jacobsthal–Lucas number is n  n 1 1 2 2 K = : r;n 1 2 Proof. For the r-Jacobsthal–Lucas sequence fK g , we have r;n n=0 n n (1.7) K = A  + A  : r;n 1 2 1 2 By using the initial conditions of r-Jacobsthal–Lucas sequence, we get the following system of equations 1r A + A = 3 + 2 ; 1 2 A  + A  = 3 + 4 2 : 1 1 2 2 Solutions of the system are r r 1 + 2 +  1 + 2 + 1 2 A = and A = : 1 2 r r 2 2 After substitution A and A into (1.7), we obtain the theorem. 1 2 Now we present generating function for the r-Jacobsthal–Lucas sequence fK g . r;n n=0 Theorem 1.5. The generating function for the r-Jacobsthal–Lucas se- quence is 1r r 3 + 2 + (2 + 1)x K x = : r;n r r r 2 1 2 x (2 + 4 )x n=0 Proof. Let us define K(x) = K x . Then we obtain r;n n=0 1r r n (1.8) K(x) = 3 + 2 + (3 + 4 2 )x + K x : r;n n=2 On r-Jacobsthal and r-Jacobsthal–Lucas numbers 21 r r r 2 By multiplying both sides of (1.8) by 2 x and (2 + 4 )x , we have r r r n (1.9) 2 xK(x) = (2 + 3 2 )x 2 K x r;n1 n=2 and r r 2 r r n (1.10) (2 + 4 )x K(x) = (2 + 4 )K x ; r;n n=2 respectively. Adding side by side equalities (1.8), (1.9) and (1.10) gives r r r 2 1 2 x (2 + 4 )x K(x) 1r r r r r n = 3 + 2 + (2 + 1)x + K 2 K (2 + 4 )K x : r;n r;n1 r;n2 n=2 The last equation and the recurrence relation for the r-Jacobsthal–Lucas num- bers complete the proof. It should be noted that we need the following equation for later use r 2 = (2 + 1) : 1 2 2. Second type of r-Jacobsthal and r-Jacobsthal–Lucas numbers For simpler results, we need a new type of the r-Jacobsthal and the r- Jacobsthal–Lucas numbers. We preserve the recurrence relation but change the initial conditions. Namely, the second type of the r-Jacobsthal numbers satisfies the recurrence relation 0 r 0 r r 0 J = 2 J + (2 + 4 )J ; n  2 r;n r;n1 r;n2 0 0 with the initial conditions J = 0 and J = 1, and the second type of the r;0 r;1 r-Jacobsthal–Lucas numbers satisfies the recurrence relation 0 r 0 r r 0 (2.1) K = 2 K + (2 + 4 )K ; n  2 r;n r;n1 r;n2 0 1r 0 with the initial conditions K = 2 and K = 1. r;0 r;1 22 Göksal Bilgici, Dorota Bród Binet formula for the r-Jacobsthal and the r-Jacobsthal–Lucas numbers can be found in the following theorem. Theorem 2.1. For any nonnegative integer n, the nth second type of the r-Jacobsthal and the r-Jacobsthal–Lucas numbers are n n 0 1 2 (2.2) J = r;n 1 2 and n n 0 1 2 K = ; r;n 1 2 respectively. Proof. Proofs can be done with the similar way to the proof of Theo- rem 1.4. 0 1 Theorem 2.2. The generating functions for the sequences fJ g and r;n n=0 0 1 fK g are r;n n=0 0 n J x = r;n r r r 2 1 2 x (2 + 4 )x n=0 and 1r 2 x 0 n K x = ; r;n r r r 2 1 2 x (2 + 4 )x n=0 respectively. Proof. Proofs can be done by using the similar way to the proof of The- orem 1.5. The following lemma presents some connections between the r-Jacobsthal and the r-Jacobsthal–Lucas numbers. Lemma 2.3. For any positive integers r and n, we have 1r K = 2 J J r;n r;n+1 r;n and 0 1r 0 0 K = 2 J J : r;n r;n+1 r;n On r-Jacobsthal and r-Jacobsthal–Lucas numbers 23 Proof. From the Binet formula for the r-Jacobsthal numbers (1:6), we have 1r  n+1  n+1  n  n 2 (    )   + 1r 1 1 2 2 1 1 2 2 2 J J = p r;n+1 r;n r r 4 2 + 5 4 n 1r  n 1r (2  1)   (2  1) 1 2 1 1 2 2 = : r r 4 2 + 5 4 1r r 1r r r If we substitute the identities 2  1 = 2 4 2 + 5 4 and 2  1 = 1 2 r r 2 4 2 + 5 4 into the last expression, we obtain the first identity in the theorem. The second identity can be proved similarly. The connections between both types of the r-Jacobsthal numbers and be- tween both types of the r-Jacobsthal–Lucas numbers are given in the following lemma. Lemma 2.4. For any positive integers r and n, we have 0 r 0 J = J + (2 + 1)J r;n r;n+1 r;n and 0 r 0 K = K + (2 + 1)K : r;n r;n+1 r;n Proof. The proofs can be done, easily, by using the Binet formulas for the r-Jacobsthal and the r-Jacobsthal–Lucas numbers. The next lemma gives the second type of the r-Jacobsthal–Lucas and the r-Jacobsthal–Lucas numbers with negative indices. Lemma 2.5. For any positive integers r and n, we have n+1 (1) 0 0 J = J r;n r;n r r n (2 + 4 ) and (1) 0 0 K = K : r;n r;n r r n (2 + 4 ) 24 Göksal Bilgici, Dorota Bród Proof. From the Binet formula (2:2) for the second type of the r-Jacobsthal numbers, we have n n 0 1 2 J = p r;n r r 4 2 + 5 4 1 1 1 = p n n r r 4 2 + 5 4 1 2 n n 1 2 = : r r n (   ) 4 2 + 5 4 1 2 r r The equality   = (2 + 4 ) gives the first identity in lemma. The second 1 2 identity can be obtained in a similar way. Similarly, the first type of the r-Jacobsthal and the r-Jacobsthal–Lucas numbers with negative indices can be obtained as in the following by the help of Lemma 2.4 and Lemma 2.5. Lemma 2.6. For any positive integers r and n, we have n+1 (1) r 0 0 J = 2 J J r;n r;n r;n1 r r n1 (2 + 4 ) and (1) r 0 0 K = 2 K K : r;n r;n r;n1 r r n1 (2 + 4 ) Lemma 2.6 provides us to expand all the results about the r-Jacobsthal- -Lucas and the r-Jacobsthal–Lucas numbers to integers. Although Bród re- stricted r to nonnegative integers, we should emphasize that there is no need such a restriction. Namely, r can be an arbitrary integer. 3. Some properties of r-Jacobsthal and r-Jacobsthal–Lucas numbers In this section, we give some results for the r-Jacobsthal and the r-Jacobsthal–Lucas numbers starting with Vajda’s identities. On r-Jacobsthal and r-Jacobsthal–Lucas numbers 25 Theorem 3.1. For any integers, r; n; m and k, we have (3.1) J J J J r;n+m r;n+k r;n r;n+m+k r 2 0 0 0 0 = (2 + 1) J J J J r;n+m r;n+k r;n r;n+m+k r 2 n r r n 0 0 = (2 + 1) (1) (2 + 4 ) J J r;m r;k and K K K K r;n+m r;n+k r;n r;n+m+k r 2 0 0 0 0 = (2 + 1) K K K K r;n+m r;n+k r;n r;n+m+k r 2 n+1 r r n r r (2 + 1) (1) (2 + 4 ) (4 2 + 5 4 ) 0 0 = J J : r;m r;k Proof. Using the Binet formula (1:6) for the r-Jacobsthal numbers, we have J J J J r;n+m r;n+k r;n r;n+m+k n+m  n+m  n+k  n+k = (    )(    ) 1 1 2 2 1 1 2 2 r r 2 (4 2 + 5 4 ) n  n  n+m+k  n+m+k (    )(    ) 1 1 2 2 1 1 2 2 r 2 n (2 + 1) (  ) 1 2 m k k m m+k m+k =     +  + 1 2 1 2 1 2 r r 2 (4 2 + 5 4 ) r 2 n r r n (2 + 1) (1) (2 + 4 ) m m k k = (  )(  ) : 1 2 1 2 r r 2 (4 2 + 5 4 ) The last equality gives the first identity in (3.1). The others can be obtained similarly. If we take k ! m, Theorem 3.1 gives Catalan’s identities (3.2) J J J r;n+m r;nm r;n h i r 2 0 0 0 = (2 + 1) J J J r;n+m r;nm r;n r 2 n+m+1 r r nm 0 = (2 + 1) (1) (2 + 4 ) J r;m 26 Göksal Bilgici, Dorota Bród and (3.3) K K K r;n+m r;nm r;n h i r 2 0 0 0 = (2 + 1) K K K r;n+m r;nm r;n r 2 n+m r r nm r r (2 + 1) (1) (2 + 4 ) (4 2 + 5 4 ) = J : r;m If we take m ! 1, Catalan’s identities (3.2) and (3.3) give Cassini’s iden- tities (3.4) J J J r;n+1 r;n1 r;n h i r 2 0 0 0 = (2 + 1) J J J r;n+1 r;n1 r;n r 2 n r r n1 = (2 + 1) (1) (2 + 4 ) and K K K r;n+1 r;n1 r;n h i r 2 0 0 0 = (2 + 1) K K K r;n+1 r;n1 r;n r 2 n+1 r r n1 r r (2 + 1) (1) (2 + 4 ) (4 2 + 5 4 ) = : In the next theorem d’Ocagne’s identities for the r-Jacobsthal and the r-Jacobsthal–Lucas numbers are given. Theorem 3.2. For any integers, r; m and n, we have (3.5) J J J J r;m r;n+1 r;m+1 r;n r 2 0 0 0 0 = (2 + 1) J J J J r;m r;n+1 r;m+1 r;n r 2 m+1 r r m 0 = (2 + 1) (1) (2 + 4 ) J r;nm and K K K K r;m r;n+1 r;m+1 r;n r 2 0 0 0 0 = (2 + 1) K K K K r;m r;n+1 r;m+1 r;n r 2 2r r r m r r m 0 = (2 + 1) 2 (4 2 + 5 4 )(1) (2 + 4 ) J : r;nm On r-Jacobsthal and r-Jacobsthal–Lucas numbers 27 Proof. From the Binet formula (1:6) for the r-Jacobsthal numbers, we have J J J J r;m r;n+1 r;m+1 r;n n+1 n+1 m+1 m+1 m m n n (    +   +   ) 1 2 1 2 2 1 2 1 1 2 r r 4 2 + 5 4 r 2 m n n m (2 + 1)   (  )   (  ) 1 2 1 2 1 2 1 2 r r 4 2 + 5 4 nm nm r 2 m (2 + 1) (  ) (  ) 1 2 1 2 = p : r r 4 2 + 5 4 From the last equality, we obtain the first identity in (3.5). The other identities can be obtained similarly. Some connections between the r-Jacobsthal and the r-Jacobsthal–Lucas numbers are presented in the next theorem. Theorem 3.3. For any integers, r; m and n, we have r 2 r r 2 (3.6) 4 K + (4 2 + 5 4 )J = 2K ; r;2n+2 r;n r;n r 0 2 r 0 2 0 2 (K ) + (5 2 + 4)(J ) = 2K ; r;n r;n r;2n r 2 r r 2 r 2 n r r n 4 K (4 2 + 5 4 )J = 4(2 + 1) (1) (2 + 4 ) ; r;n r;n r 0 2 r 0 2 n r r n 2 (K ) (5 2 + 4)(J ) = 4(1) (2 + 4 ) ; r;n r;n 12r (3.7) J K +J K = 2 J ; r;m r;n r;n r;m r;m+n+2 0 0 0 0 1r 0 J K +J K = 2 J ; r;m r;n r;n r;m r;m+n 1r r 2 n r r 2 0 J K J K = 2 (2 + 1) (1) (2 + 4 ) J ; r;m r;n r;n r;m r;mn 0 0 0 0 1r n r r 2 0 J K J K = 2 (1) (2 + 4 ) J : r;m r;n r;n r;m r;mn Proof. All the proofs are based on the Binet formula and we prove two of them. We have r 2 r r 2 4 K + (4 2 + 5 4 )J r;n r;n n  n 2  n  n 2 = (  +   ) + (    ) 1 1 2 2 1 1 2 2 2 2n  2 2n = 2( )  + 2( ) 1 1 2 2 28 Göksal Bilgici, Dorota Bród r  2n r  2n = 2 (2 + 1 +  )  + (2 + 1 +  ) 1 2 1 1 2 2 r  2n  2n  2n+1  2n+1 = 2 (2 + 1)(  +   ) + (  +   ) 1 1 2 2 1 1 2 2 r+1 r = 2 (2 + 1)K +K : r;2n r;2n+1 The recurrence relation (2.1) gives (3.6). Now we prove (3.7). From Binet formula for the r-Jacobsthal and the r-Jacobsthal–Lucas numbers, we have J K +J K r;m r;n r;n r;m m  m  n  n = (    )(  +   ) 1 1 2 2 1 1 2 2 r r r 2 4 2 + 5 4 n  n  m  m + (    )(  +   ) 1 1 2 2 1 1 2 2 2 m+n  2 m+n = 2( )  2( ) 1 2 1 2 r r r 2 4 2 + 5 4 r  m+n = (2 + 1 +  ) r r r 2 4 2 + 5 4 r  m+n (2 + 1 +  ) r  m+n  m+n = p (2 + 1)(  +   ) 1 1 2 2 r r r 2 4 2 + 5 4 m+n+1  m+n+1 + (  +   ) 1 1 2 2 = J + (2 + 1)J : r;m+n+1 r;m+n From the recurrence relation (1.5), we obtain (3.7). The other identities can be proved similarly. By Theorem 3.3, for r = 0, we obtain known identities (1.1), (1.2), (1.3), (1.4) for the classical Jacobsthal and Jacobsthal–Lucas numbers. In the next theorem we give summation formulas for the r-Jacobsthal- -Lucas numbers and the second types of the r-Jacobsthal and the r-Jacobsthal- -Lucas numbers. Theorem 3.4. Let n; r be integers. Then n1 r r r 1r K + (2 + 4 )K 2 2 4 r;n r;n1 (3.8) K = ; r;i r+1 r 2 + 4 1 i=0 n1 0 r r 0 J + (2 + 4 )J 1 r;n r;n1 (3.9) J = ; r;i r+1 r 2 + 4 1 i=0 On r-Jacobsthal and r-Jacobsthal–Lucas numbers 29 n1 0 r r 0 1r K + (2 + 4 )K + 1 2 r;n r;n1 (3.10) K = : r;i r+1 r 2 + 4 1 i=0 Proof. For (3.8), on account of (1.7) we get n1 n1 n n X X 1  1 i i 1 2 K = (A  + A  ) = A + A r;i 1 2 1 2 1 2 1  1 1 2 i=0 i=0 n n n1 n1 A + A (A  + A  ) (A  + A  ) +   (A  + A  ) 1 2 1 2 2 1 1 2 1 2 1 2 1 2 1 2 1 ( +  ) + 1 2 1 2 r r A + A (    )K (2 + 4 )K 1 2 1 2 2 1 r;n r;n1 = : r r r 1 2 (2 + 4 ) 1r By simple calculations we have A + A = 3 + 2 , A  + A  = 1 2 1 2 2 1 (1 + 2 ). Hence n1 1r r r r 4 + 2 + 2 K (2 + 4 )K r;n r;n1 K = r;i r+1 r (2 + 4 1) i=0 r r r 1r K + (2 + 4 )K 2 2 4 r;n r;n1 = : r+1 r 2 + 4 1 In the same way one can easily prove (3.9) and (3.10). 4. Matrix generators Now we give the matrix generators of the numbers J and K . r;n r;n Theorem 4.1. Let n; r be integers. Then n1 J J J J 2 1 r;n+1 r;n r;2 r;1 (4.1) =  : r r J J J J 2 + 4 0 r;n r;n1 r;1 r;0 Proof. (by induction on n) It is easily seen that for n = 1 the result is obvious. Assuming that the formula (4.1) holds for n  1, we will prove it for n + 1. 30 Göksal Bilgici, Dorota Bród Using induction’s hypothesis and the recurrence formula for the r-Jacobsthal numbers, we have n1 r r J J 2 1 2 1 r;2 r;1 r r r r J J 2 + 4 0 2 + 4 0 r;1 r;0 J J 2 1 r;n+1 r;n r r J J 2 + 4 0 r;n r;n1 r r r 2 J + (2 + 4 )J J r;n+1 r;n r;n+1 r r r 2 J + (2 + 4 )J J r;n r;n1 r;n J J r;n+2 r;n+1 = ; J J r;n+1 r;n which ends the proof. As a consequence of Theorem 4.1 we get Cassini’s identity (3.4) for the r-Jacobsthal numbers. Corollary 4.2. Let n; r be integers. Then 2 n r r n1 r 2 J J J = (1) (2 + 4 ) (2 + 1) : r;n+1 r;n1 r;n Proof. Calculating determinants in formula (4.1), we obtain J J r;n+1 r;n 2 = J J J ; r;n+1 r;n1 r;n J J r;n r;n1 r r r J J 3 4 + 2 2 2 2 + 1 r;2 r;1 r 2 = = (2 + 1) ; J J 2 2 + 1 1 r;1 r;0 2 1 r r = (2 + 4 ): r r 2 + 4 0 Hence we get 2 r 2 n1 r r n1 J J J = (2 + 1) (1) (2 + 4 ) r;n+1 r;n1 r;n n r r n1 r 2 = (1) (2 + 4 ) (2 + 1) ; which completes the proof. Similarly to Theorem 4.1 and Corollary 4.2, we can prove the following results. On r-Jacobsthal and r-Jacobsthal–Lucas numbers 31 Theorem 4.3. Let n; r be integers. Then n1 K K K K 2 1 r;n+1 r;n r;2 r;1 =  : r r K K K K 2 + 4 0 r;n r;n1 r;1 r;0 Corollary 4.4. Let n; r be integers. Then 2 n1 r r n1 2r r r K K K = (1) (2 + 4 ) (13 + 2 + 14 2 + 5 4 ): r;n+1 r;n1 r;n References [1] D. Bród, On a new Jacobsthal-type sequence, Ars Combin. 150 (2020), 21–29. [2] A. Daşdemir, The representation, generalized Binet formula and sums of the general- ized Jacobsthal p-sequence, Hittite J. Sci. Eng. 3 (2016), no. 2, 99–104. [3] L.E. Dickson, History of the Theory of Numbers. Vol. I: Divisibility and Primality, Chelsea Publishing Co., New York, 1952. [4] M. Edson and O. Yayenie, A new generalization of Fibonacci sequence & extended Binet’s formula, Integers 9 (2009), no. 6, 639–654. [5] S. Falcon, On the k-Jacobsthal numbers, American Review of Mathematics and Sta- tistics 2 (2014), no. 1, 67–77. [6] A.F. Horadam, Basic properties of a certain generalized sequence of numbers, Fi- bonacci Quart. 3 (1965), no. 3, 161–176. [7] A.F. Horadam, Jacobsthal representation numbers, Fibonacci Quart. 34 (1996), no. 1, 40–54. [8] D. Jhala, K. Sisodiya, and G.P.S. Rathore, On some identities for k-Jacobsthal num- bers, Int. J. Math. Anal. (Ruse) 7 (2013), no. 12, 551–556. [9] R.E. Merrifield and H.E. Simmons, Topological Methods in Chemistry, John Wiley & Sons, New York, 1989. [10] S. Uygun, The (s; t)-Jacobsthal and (s; t)-Jacobsthal Lucas sequences, Appl. Math. Sci. (Ruse) 9 (2015), no. 70, 3467–3476. [11] S. Uygun and E. Owusu, A new generalization of Jacobsthal numbers (bi-periodic Jacobsthal sequences), J. Math. Anal. 7 (2016), no. 5, 28–39. Göksal Bilgici Elementary Mathematics Education Kastamonu University 37200 Kastamonu Turkey e-mail: gbilgici@kastamonu.edu.tr Dorota Bród Department of Discrete Mathematics Faculty of Mathematics and Applied Physics Rzeszow University of Technology Poland e-mail: dorotab@prz.edu.pl

Journal

Annales Mathematicae Silesianae – de Gruyter

Published: Mar 1, 2023

Keywords: r -Jacobsthal numbers; r -Jacobsthal–Lucas numbers; Binet formula; 11B83; 11B37

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