A note on Modified Third-order Jacobsthal numbers

Another important sequence is the Jacobsthal-Lucas sequence. This sequence is defined by the recurrence relation jn+2 = jn+1+2jn, where j0 = 2 and j1 = 1 (see, [7]). In [5] the Jacobsthal recurrence relation is extended to higher order recurrence relations and the basic list of identities provided by A. F. Horadam [7] is expanded and extended to several identities for some of the higher order cases. For example, the third-order Jacobsthal numbers, {J (3) n }n≥0, and third-order Jacobsthal-Lucas numbers, {j n }n≥0, are defined by

Another important sequence is the Jacobsthal-Lucas sequence. This sequence is defined by the recurrence relation j n+2 = j n+1 +2j n , where j 0 = 2 and j 1 = 1 (see, [7]). In [5] the Jacobsthal recurrence relation is extended to higher order recurrence relations and the basic list of identities provided by A. F. Horadam [7] is expanded and extended to several identities for some of the higher order cases. For example, the third-order Jacobsthal numbers, {J (3) n } n≥0 , and third-order Jacobsthal-Lucas numbers, {j (3) n } n≥0 , are defined by 2 = 1, n ≥ 0, and 2 = 5, n ≥ 0, 1 respectively. Some of the following properties given for third-order Jacobsthal numbers and third-order Jacobsthal-Lucas numbers are used in this paper (for more details, see [2,3,4,5]). Note that Eqs. (1.8) and (1.12) have been corrected in this paper, since they have been wrongly described in [5].
n−3 , n ≥ 3. Using standard techniques for solving recurrence relations, the auxiliary equation, and its roots are given by Note that the latter two are the complex conjugate cube roots of unity. Call them ω 1 and ω 2 , respectively. Thus the Binet formulas can be written as respectively. Now, we use the notation , where A = −3 − 2ω 2 and B = −3 − 2ω 1 . Furthermore, note that for all n ≥ 0 we have From the Binet formulas (1.13), (1.14) and Eq. (1.15), we have Motivated essentially by the recent works [5], [2] and [4], in this paper we introduce the Modified third-order Jacobsthal sequences and we give some properties, including the Binet-style formula and the generating functions for these sequences. Some identities involving these sequences are also provided.
2. The Modified Third-order Jacobsthal sequence, Binet's formula and the generating function The principal goals of this section will be to define the Modified third-order Jacobsthal sequence and to present some elementary results involving it.
First of all, we define the Modified third-order Jacobsthal sequence, denoted by {K (3) n } n≥0 , which first terms are {3, 1, 3, 10, 15, 31, 66, ...}. This sequence is defined recursively by n is the n-th third-order Jacobsthal number. In order to find the generating function for the Modified third-order Jacobsthal sequence, we shall write the sequence as a power series where each term of the sequence correspond to coefficients of the series. As a consequence of the definition of generating function of a sequence, the generating function associated to {K Consequently, we obtain the following result: Theorem 2.1. The generating function for the Modified third-order Jacobsthal numbers {K Proof. Using the definition of generating function, we have g K (3) n t n + · · · . Multiplying both sides of this identity by −t, −t 2 and by −2t 3 , and then from ( 0 )t 2 and the result follows.
The following result gives the Binet-style formula for K and ω 1 , ω 2 are the roots of the characteristic equation associated with the respective recurrence relations x 2 + x + 1 = 0.
Proof. Since the characteristic equation has three distinct roots, the sequence K n = a2 n + bω n 1 + cω n 2 is the solution of the Eq. (2.1). Considering n = 0, 1, 2 in this identity and solving this system of linear equations, we obtain a unique value for a, b and c, which are, in this case, a = b = c = 1. So, using these values in the expression of K (3) n stated before, we get the required result.
Using the fact that n are, respectively, the n-th third-order Jacobsthal, third-order Jacobsthal-Lucas and Modified third-order Jacobsthal numbers, then the following identities are true: n+1 + 3M n−1 + 2J n−1 .
(2.7): Using the the Binet formula of K n in Theorem 2.2, we have Then, Then, we obtain the Eq. (2.8) if m = n in Eq. (2.7).

Some identities involving this type of sequence
In this section, we state some identities related with these type of third-order sequence. As a consequence of the Binet formula of Theorem 2.2, we get for this sequence the following interesting identities. n is the n-th Modified third-order Jacobsthal numbers, then the following identity  Using that M s . Then, we obtain Hence the result.
Note that for s = 1 in Catalan's identity obtained, we get the Cassini identity for the Modified third-order Jacobsthal sequence. In fact, for s = 1, the identity stated in Proposition 3.1, yields and using one of the initial conditions of the sequence {U (2) n } in Proposition 3.1 we obtain the following result. n is the n-th Modified third-order Jacobsthal numbers, then the identity The d'Ocagne identity can also be obtained using the Binet formula and in this case we obtain n is the n-th Modified third-order Jacobsthal number, then the following identity Proof. Using the Eq. In addition, some formulae involving sums of terms of the Modified thirdorder Jacobsthal sequence will be provided in the following proposition. are, respectively, the n-th third-order Jacobsthal-Lucas and Modified third-order Jacobsthal numbers, then the following identities are true: Then, the result in Eq. (3.1) is completed. .
Hence we obtain the result.
For negative subscripts terms of the sequence of Modified third-order Jacobsthal we can establish the following result:

Conclusion
Sequences of numbers have been studied over several years, with emphasis on the well known Tribonacci sequence and, consequently, on the Tribonacci-Lucas sequence. In this paper we have also contributed for the study of Modified third-order Jacobsthal sequence, deducing some formulae for the sums of such numbers, presenting the generating functions and their Binet-style formula. It is our intention to continue the study of this type of sequences, exploring some their applications in the science domain. For example, a new type of sequences in the quaternion algebra with the use of this numbers and their combinatorial properties.