Tribonacci and Tribonacci-Lucas Sedenions

The sedenions form a 16-dimensional Cayley-Dickson algebra. In this paper, we introduce the Tribonacci and Tribonacci-Lucas sedenions. Furthermore, we present some properties of these sedenions and derive relationships between them.

The generating functions for the Tribonacci sequence {T n } n≥0 and Tribonacci-Lucas sequence {K n } n≥0 are ∞ n=0 T n x n = x 1 − x − x 2 − x 3 and produces an algebra S (dim 2 4 = 16) called the sedenions. This doubling process can be extended beyond the sedenions to form what are known as the 2 n -ions (see for example [23], [29], [2]).
Next, we explain this doubling process.
The Cayley-Dickson algebras are a sequence A 0 , A 1 , ... of non-associative R-algebras with involution.
The term "conjugation" can be used to refer to the involution because it generalizes the usual conjugation on the complex numbers. A full explanation of the basic properties of Cayley-Dickson algebras, see [2].
Cayley-Dickson algebras are defined inductively. We begin by defining A 0 to be R. Given A n−1 , the algebra A n is defined additively to be A n−1 × A n−1 . Conjugation in A n is defined by and multiplication is defined by and addition is defined by componentwise as Note that A n has dimension 2 n as an R−vector space. If we set, as usual, x = Re(xx) for x ∈ A n then Now, suppose that B 16 = {e i ∈ S : i = 0, 1, 2, ..., 15} is the basis for S, where e 0 is the identity (or unit) and e 1 , e 2 , ..., e 15 are called imaginaries. Then a sedenion S ∈ S can be written as where a 0 , a 1 , ..., a 15 are all real numbers. Here a 0 is called the real part of S and 15 i=1 a i e i is called its imaginary part.
Addition of sedenions is defined as componentwise and multiplication is defined as follows: if S 1 , S 2 ∈ S then we have By setting i ≡ e i where i = 0, 1, 2, ..., 15, the multiplication rule of the base elements e i ∈ B 16 can be summarized as in the following Table (see [8] and [3]).
From the above table, we can see that: e 0 e i = e i e 0 = e i ; e i e i = −e 0 for i = 0; e i e j = −e j e i for i = j and i, j = 0.
The operations requiring for the multiplication in (1.4) are quite a lot. The computation of a sedenion multiplication (product) using the naive method requires 256 multiplications and 240 additions, while an algorithm which is given in [5] can compute the same result in only 122 multiplications (or multipliers -in hardware implementation case) and 298 additions, for details see [5]. The problem with Cayley-Dickson Process is that each step of the doubling process results in a progressive loss of structure. R is an ordered field and it has all the nice properties we are so familiar with in dealing with numbers like: the associative property, commutative property, division property, self-conjugate property, etc.
When we double R to have C; C loses the self-conjugate property (and is no longer an ordered field), next H loses the commutative property, and O loses the associative property. When we double O to obtain S; S loses the division property. It means that S is non-commutative, non-associative, and have a multiplicative identity element e 0 and multiplicative inverses but it is not a division algebra because it has zero divisors; this means that two non-zero sedenions can be multiplied to obtain zero: an example is (e 3 + e 10 )(e 6 − e 15 ) = 0 and the other example is (e 2 − e 14 )(e 3 + e 15 ) = 0, see [8].
The algebras beyond the complex numbers go by the generic name hypercomplex number. All hypercomplex number systems after sedenions that are based on the Cayley-Dickson construction contain zero divisors.
Note that there is another type of sedenions which is called conic sedenions or sedenions of Charles Muses, as they are also known, see [26], [27], [30] for more information. The term sedenion is also used for other 16-dimensional algebraic structures, such as a tensor product of two copies of the biquaternions, or the algebra of 4 by 4 matrices over the reals.
In the past, non-associative algebras and related structures with zero divisors have not been given much attention because they did not appear to have any useful applications in most mathematical subjects.
Recently, however, a lot of attention has been centred by theoretical physicists on the Cayley-Dickson algebras O (octonions) and S (sedenions) because of their increasing usefulness in formulating many of the new theories of elementary particles. In particular, the octonions O (which is the only non-associative normed division algebra over the reals; see for example [1] and [31]) has been found to be involved in so many unexpected areas (such as topology, quantum theory, Clifford algebras, etc.) and sedenions appear in many areas of science like linear gravity and electromagnetic theory.
Briefly S, the algebra of sedenions, have the following properties: • S is a 16 dimensional non-associative and non-commutative (Carley-Dickson) algebra over the reals, • S is not a composition algebra or division algebra because of its zero divisors, • S is a non-alternative algebra, i.e., if S 1 and S 2 are sedenions the rules S 2 1 S 2 = S 1 (S 1 S 2 ) and S 1 S 2 2 = (S 1 S 2 )S 2 do not always hold, • S is a power-associative algebra, i.e., if S is an sedenion then S n S m = S n+m .

The Tribonacci and Tribonacci-Lucas Sedenions and Their Generating Functions and Binet Formulas
In this section we define Tribonacci and Tribonacci-Lucas sedenions and give generating functions and Binet formulas for them. First, we give some information about quaternion sequences, octonion sequences and sedenion sequences from literature.
Horadam [22] introduced nth Fibonacci and nth Lucas quaternions as F n+s e s and R n = L n + L n+1 e 1 + L n+2 e 2 + L n+3 e 3 = 3 s=0 L n+s e s respectively, where F n and L n are the nth Fibonacci and Lucas numbers respectively. He also defined generalized Fibonacci quaternion as H n+s e s where H n is the nth generalized Fibonacci number (which is now called Horadam number) by the recursive relation H 1 = p, H 2 = p + q, H n = H n−1 + H n−2 (p and q are arbitrary integers). Many other generalization of Fibonacci quaternions has been given, see for example Halici and Karataş [21], and Polatlı [32].
Cerda-Morales [9] defined and studied the generalized Tribonacci quaternion sequence that includes the previously introduced Tribonacci, Padovan, Narayana and third order Jacobsthal quaternion sequences. She defined generalized Tribonacci quaternion as where V n is the nth generalized Tribonacci number defined by the third-order recurrance relations V n = rV n−1 + sV n−2 + tV n−3 , L n+s e s respectively, where F n and L n are the nth Fibonacci and Lucas numbers respectively. In [15], Ç imen anḋ Ipek introduced Jacobsthal octonions and Jacobsthal-Lucas octonions. In [10], Cerda-Morales introduced third order Jacobsthal octonions and also in [12], she defined and studied tribonacci-type octonions.  L n+s e s respectively, where F n and L n are the nth Fibonacci and Lucas numbers respectively. In [7], Catarino introduced k-Pell and k-Pell-Lucas sedenions. In [14], Ç imen andİpek introduced Jacobsthal and Jacobsthal-Lucas sedenions.
Gül [20] introduced the k-Fibonacci and k-Lucas trigintaduonions as T F k,n = For all non-negative integer n, it can be easily shown that The sequences { T n } n≥0 and { K n } n≥0 can be defined for negative values of n by using the recurrences (2.2) and (2.3) to extend the sequence backwards, or equivalently, by using the recurrences and respectively. Thus, the recurrences (2.2) and (2.3) holds for all integer n.
The conjugate of T n and K n are defined by respectively. Now, we will state Binet's formula for the Tribonacci and Tribonacci-Lucas sedenions and in the rest of the paper we fixed the following notations.
Proof. Repeated use of (1.3) in (2.1) enable us to write for α = .
Next, we present generating functions.
Using above table and the recurans T n = T n−1 + T n−2 + T n−3 we have It follows that Since T 2 − T 1 − T 0 = T −1 , the generating functions for the Tribonacci sedenion is Similarly, we can obtain (2.11).
In the following theorem we present another forms of Binet formulas for the Tribonacci and Tribonacci-Lucas sedenions using generating functions.
Theorem 2.4. For any integer n, the nth Tribonacci sedenion is and the nth Tribonacci-Lucas sedenion is Proof. We can use generating functions. Since the roots of the equation 1−x−x 2 −x 3 = 0 are αβ, βγ, αγ we can write the generating function of T n as g(x) We need to find A, B and C, so the following system of equations should be solved: We find that Thus Binet formula of Tribonacci sedenion is Similarly, we can obtain Binet formula of the Tribonacci-Lucas sedenion.
If we compare Theorem 2.1 and Theorem 2.4 and use the definition of T n , K n , we have the following Remark showing relations between T −1 , T 0 , T 1 ; K −1 , K 0 , K 1 and α, β, γ. We obtain (b) and (d) after solving the system of the equations in (a) and (b) respectively.
Of course, (2.13) and (2.14) can be found directly from (2.4) and (2.5). Now, we present the formulas which give the summation of the first n Tribonacci and Tribonacci-Lucas numbers.
Lemma 2.6. For every integer n ≥ 0, we have and Proof. (2.15) and (2.16) can be proved by mathematical induction easily. For a proof of (2.15) with a telescopik sum method see [16] or with a matrix diagonalisation proof, see [25] or see also [9].
For a proof of (2.16), see [19]. Since K 0 = 3 and For a proof of the above formula, see Kuhapatanakul and Sukruan [28].
Next, we present the formulas which give the summation of the first n Tribonacci and Tribonacci-Lucas sedenions.
We can compute c 1 as This proves (2.17). Similarly we can obtain (2.18).

Some Identities For The Tribonacci and Tribonacci-Lucas Sedenions
In this section, we give identities about Tribonacci and Tribonacci-Lucas sedenions.