aa r X i v : . [ m a t h . R A ] A ug Tribonacci and Tribonacci-Lucas Sedenions
Y ¨UKSEL SOYKAN
Zonguldak B¨ulent Ecevit University, Department of Mathematics,Art and Science Faculty, 67100, Zonguldak, Turkeye-mail: yuksel [email protected]
Abstract.
The sedenions form a 16-dimensional Cayley-Dickson algebra. In this paper, we introducethe Tribonacci and Tribonacci-Lucas sedenions. Furthermore, we present some properties of these sedenionsand derive relationships between them.
Keywords.
Tribonacci numbers, sedenions, Tribonacci sedenions, Tribonacci-Lucas sedenions.
1. Introduction
Tribonacci sequence { T n } n ≥ and Tribonacci-Lucas sequence { K n } n ≥ are defined by the third-orderrecurrence relations(1.1) T n = T n − + T n − + T n − , T = 0 , T = 1 , T = 1 , and(1.2) K n = K n − + K n − + K n − , K = 3 , K = 1 , K = 3respectively. Tribonacci concept was introduced by 14 year old student M. Feinberg [18] in 1963. Basicproperties of it is given in [4], [35], [36], [39] and Binet formula for the n th number is given in [37]. See also[6],[13],[17], [34], [33], [38], [40].The sequences { T n } n ≥ and { K n } n ≥ can be extended to negative subscripts by defining T − n = − T − ( n − − T − ( n − + T − ( n − and K − n = − K − ( n − − K − ( n − + K − ( n − for n = 1 , , , ... respectively. Therefore, recurrences (1.1) and (1.2) hold for all integer n. It is well known that usual Tribonacci and Tribonacci-Lucas numbers can be expressed using Binet’sformulas(1.3) T n = α n +1 ( α − β )( α − γ ) + β n +1 ( β − α )( β − γ ) + γ n +1 ( γ − α )( γ − β )and K n = α n + β n + γ n respectively, where α, β and γ are the roots of the cubic equation x − x − x − . Moreover, α = 1 + p
19 + 3 √
33 + p − √ ,β = 1 + ω p
19 + 3 √
33 + ω p − √ ,γ = 1 + ω p
19 + 3 √
33 + ω p − √ ω = − i √
32 = exp(2 πi/ , is a primitive cube root of unity. Note that we have the following identities α + β + γ = 1 ,αβ + αγ + βγ = − ,αβγ = 1 . We can give Binet’s formulas of the Tribonacci and Tribonacci-Lucas numbers for the negative subscripts:for n = 1 , , , ... we have T − n = α − n +1 ( α − β )( α − γ ) + β − n +1 ( β − α )( β − γ ) + γ − n +1 ( γ − α )( γ − β )and K − n = α − n + β − n + γ − n . The generating functions for the Tribonacci sequence { T n } n ≥ and Tribonacci-Lucas sequence { K n } n ≥ are ∞ X n =0 T n x n = x − x − x − x and ∞ X n =0 K n x n = 3 − x − x − x − x − x . In this paper, we define Tribonacci and Tribonacci-Lucas sedenions in the next section and give someproperties of them. Before giving their definition, we present some information on Cayley-Dickson algebras.The algebras C (complex numbers), H (quaternions), and O (octonions) are real division algebras ob-tained from the real numbers R by a doubling procedure called the Cayley-Dickson Process (Construction).By doubling R (dim 2 = 1), we obtain the complex numbers C (dim 2 = 2); then C yields the quaternions H (dim 2 = 4); and H produces octonions O (dim 2 = 8). The next doubling process applied to O then RIBONACCI AND TRIBONACCI-LUCAS SEDENIONS 3 produces an algebra S (dim 2 = 16) called the sedenions. This doubling process can be extended beyondthe 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 , A , ... of non-associative R -algebras with involution.The term “conjugation” can be used to refer to the involution because it generalizes the usual conjugationon 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 to be R . Given A n − , the algebra A n is defined additively to be A n − × A n − . Conjugation in A n is defined by( a, b ) = ( a, − b )and multiplication is defined by ( a, b )( c, d ) = ( ac − db, da + bc )and addition is defined by componentwise as( a, b ) + ( c, d ) = ( a + c, b + d ) . Note that A n has dimension 2 n as an R − vector space. If we set, as usual, k x k = p Re( xx ) for x ∈ A n then xx = xx = k x k . Now, suppose that B = { e i ∈ S : i = 0 , , , ..., } is the basis for S , where e is the identity (or unit)and e , e , ..., e are called imaginaries. Then a sedenion S ∈ S can be written as S = X i =0 a i e i = a + X i =1 a i e i where a , a , ..., a are all real numbers. Here a is called the real part of S and P i =1 a i e i is called itsimaginary part.Addition of sedenions is defined as componentwise and multiplication is defined as follows: if S , S ∈ S then we have(1.4) S S = X i =0 a i e i ! X i =0 b i e i ! = X i,j =0 a i b j ( e i e j ) . By setting i ≡ e i where i = 0 , , , ..., , the multiplication rule of the base elements e i ∈ B can besummarized as in the following Table (see [8] and [3]).From the above table, we can see that: e e i = e i e = e i ; e i e i = − e 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 sedenionmultiplication (product) using the naive method requires 256 multiplications and 240 additions, while analgorithm which is given in [5] can compute the same result in only 122 multiplications (or multipliers – inhardware implementation case) and 298 additions, for details see [5].
Y¨UKSEL SOYKAN
Figure 1.
Multiplication table for sedenions imaginary unitsThe problem with Cayley-Dickson Process is that each step of the doubling process results in a progressiveloss of structure. R is an ordered field and it has all the nice properties we are so familiar with in dealing withnumbers 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 multiplicativeidentity element e and multiplicative inverses but it is not a division algebra because it has zero divisors; thismeans that two non-zero sedenions can be multiplied to obtain zero: an example is ( e + e )( e − e ) = 0and the other example is ( e − e )( e + e ) = 0 , see [8].The algebras beyond the complex numbers go by the generic name hypercomplex number. All hyper-complex number systems after sedenions that are based on the Cayley–Dickson construction contain zerodivisors.Note that there is another type of sedenions which is called conic sedenions or sedenions of CharlesMuses, as they are also known, see [26], [27], [30] for more information. The term sedenion is also used forother 16-dimensional algebraic structures, such as a tensor product of two copies of the biquaternions, orthe algebra of 4 by 4 matrices over the reals.In the past, non-associative algebras and related structures with zero divisors have not been givenmuch 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 theoriesof elementary particles. In particular, the octonions O (which is the only non-associative normed divisionalgebra over the reals; see for example [1] and [31]) has been found to be involved in so many unexpected RIBONACCI AND TRIBONACCI-LUCAS SEDENIONS 5 areas (such as topology, quantum theory, Clifford algebras, etc.) and sedenions appear in many areas ofscience 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 and S are sedenions the rules S S = S ( S S ) and S S = ( S S ) S 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 .
2. The Tribonacci and Tribonacci-Lucas Sedenions and Their Generating Functions andBinet Formulas
In this section we define Tribonacci and Tribonacci-Lucas sedenions and give generating functions andBinet formulas for them. First, we give some information about quaternion sequences, octonion sequencesand sedenion sequences from literature.Horadam [22] introduced n th Fibonacci and n th Lucas quaternions as Q n = F n + F n +1 e + F n +2 e + F n +3 e = X s =0 F n + s e s and R n = L n + L n +1 e + L n +2 e + L n +3 e = X s =0 L n + s e s respectively, where F n and L n are the n th Fibonacci and Lucas numbers respectively. He also definedgeneralized Fibonacci quaternion as P n = H n + H n +1 e + H n +2 e + H n +3 e = X s =0 H n + s e s where H n is the n th generalized Fibonacci number (which is now called Horadam number) by the recursiverelation H = p, H = p + q, H n = H n − + H n − ( p and q are arbitrary integers). Many other generalizationof Fibonacci quaternions has been given, see for example Halici and Karata¸s [21], and Polatlı [32].Cerda-Morales [9] defined and studied the generalized Tribonacci quaternion sequence that includes thepreviously introduced Tribonacci, Padovan, Narayana and third order Jacobsthal quaternion sequences. Shedefined generalized Tribonacci quaternion as Q v,n = V n + V n +1 e + V n +2 e + V n +3 e = X s =0 V n + s e s where V n is the n th generalized Tribonacci number defined by the third-order recurrance relations V n = rV n − + sV n − + tV n − , here V = a, V = b, V = c are arbitrary integers and r, s, t are real numbers. Y¨UKSEL SOYKAN
Various families of octonion number sequences (such as Fibonacci octonion, Pell octonion, Jacobsthaloctonion; and third order Jacobsthal octonion) have been defined and studied by a number of authors inmany different ways. For example, Ke¸cilioglu and Akku¸s [24] introduced the Fibonacci and Lucas octonionsas Q n = X s =0 F n + s e s and R n = X s =0 L n + s e s respectively, where F n and L n are the n th Fibonacci and Lucas numbers respectively. In [15], C¸ imen and˙Ipek introduced Jacobsthal octonions and Jacobsthal-Lucas octonions. In [10], Cerda-Morales introducedthird order Jacobsthal octonions and also in [12], she defined and studied tribonacci-type octonions.A number of authors have been defined and studied sedenion number sequences (such as second ordersedenions: Fibonacci sedenion, k-Pell and k-Pell–Lucas sedenions, Jacobsthal and Jacobsthal-Lucas sede-nions). For example, Bilgici, Toke¸ser and ¨Unal [3] introduced the Fibonacci and Lucas sedenions as b F n = X s =0 F n + s e s and b L n = X s =0 L n + s e s respectively, where F n and L n are the n th Fibonacci and Lucas numbers respectively. In [7], Catarinointroduced k-Pell and k-Pell–Lucas sedenions. In [14], C¸ imen and ˙Ipek introduced Jacobsthal and Jacobsthal-Lucas sedenions.G¨ul [20] introduced the k-Fibonacci and k-Lucas trigintaduonions as T F k,n = X s =0 F k,n + s e s and T L k,n = X s =0 L k,n + s e s respectively, where F k,n and L k,n are the n th k-Fibonacci and k-Lucas numbers respectively.We now define Tribonacci and Tribonacci-Lucas sedenions over the sedenion algebra S . The n th Tri-bonacci sedenion is(2.1) b T n = X s =0 T n + s e s = T n + X s =1 T n + s e s and the n th Tribonacci-Lucas sedenion is b K n = X s =0 K n + s e s = K n + X s =1 K n + s e s . RIBONACCI AND TRIBONACCI-LUCAS SEDENIONS 7
For all non-negative integer n, it can be easily shown that(2.2) b T n = b T n − + b T n − + b T n − and(2.3) b K n = b K n − + b K n − + b K n − . The sequences { b T n } n ≥ and { b K n } n ≥ 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 b T − n = − b T − ( n − − b T − ( n − + b T − ( n − and b K − n = − b K − ( n − − b K − ( n − + b K − ( n − , respectively. Thus, the recurrences (2.2) and (2.3) holds for all integer n. The conjugate of b T n and b K n are defined by b T n = T n − X s =1 T n + s e s = T n − T n +1 e − T n +2 e − ... − T n +15 e and b K n = K n − X s =1 K n + s e s = K n − K n +1 e − K n +2 e − ... − K n +15 e respectively. The norms of n th Tribonacci and Tribonacci-Lucas sedenions are (cid:13)(cid:13)(cid:13) b T n (cid:13)(cid:13)(cid:13) = N ( b T n ) = b T n b T n = b T n b T n = T n + T n +1 + ... + T n +15 and (cid:13)(cid:13)(cid:13) b K n (cid:13)(cid:13)(cid:13) = N ( b K n ) = b K n b K n = b K n b K n = K n + K n +1 + ... + K n +15 respectively.Now, we will state Binet’s formula for the Tribonacci and Tribonacci-Lucas sedenions and in the rest ofthe paper we fixed the following notations. b α = X s =0 α s e s , b β = X s =0 β s e s , b γ = X s =0 γ s e s . Y¨UKSEL SOYKAN
Theorem . For any integer n, the n th Tribonacci sedenion is (2.4) b T n = b αα n +1 ( α − β )( α − γ ) + b ββ n +1 ( β − α )( β − γ ) + b γγ n +1 ( γ − α )( γ − β ) and the n th Tribonacci-Lucas sedenion is (2.5) b K n = b αα n + b ββ n + b γγ n . Proof.
Repeated use of (1.3) in (2.1) enable us to write for b α = P s =0 α s e s , b β = P s =0 β s e s and b γ = P s =0 γ s e s : b T n = X s =0 T n + s e s = X s =0 (cid:18) α n +1+ s e s ( α − β )( α − γ ) + β n +1+ s e s ( β − α )( β − γ ) + γ n +1+ s e s ( γ − α )( γ − β ) (cid:19) = b αα n +1 ( α − β )( α − γ ) + b ββ n +1 ( β − α )( β − γ ) + b γγ n +1 ( γ − α )( γ − β ) . Similarly, we can obtain (2.5).We can give Binet’s formula of the Tribonacci and Tribonacci-Lucas sedenions for the negative subscripts:for n = 1 , , , ... we have b T − n = b αα − n +1 ( α − β )( α − γ ) + b ββ − n +1 ( β − α )( β − γ ) + b γγ − n +1 ( γ − α )( γ − β )and b K − n = b αα − n + b ββ − n + b γγ − n , respectively.The next theorem gives us an alternatif proof of the Binet’s formula for the Tribonacci and Tribonacci-Lucas sedenions. For this, we need the quadratik approximation of { T n } n ≥ and { K n } n ≥ : Lemma . For all integer n, we have (a): αα n +2 = T n +2 α + ( T n +1 + T n ) α + T n +1 ,ββ n +2 = T n +2 β + ( T n +1 + T n ) β + T n +1 ,γγ n +2 = T n +2 γ + ( T n +1 + T n ) γ + T n +1 . (b): P α n +2 = K n +2 α + ( K n +1 + K n ) α + K n +1 ,Qβ n +2 = K n +2 β + ( K n +1 + K n ) β + K n +1 ,Rγ n +2 = K n +2 γ + ( K n +1 + K n ) γ + K n +1 , RIBONACCI AND TRIBONACCI-LUCAS SEDENIONS 9 where P = 3 − ( β + γ ) + 3 βγ,Q = 3 − ( α + γ ) + 3 αγ,R = 3 − ( α + βγ ) + 3 αβ. Proof.
See [11] or [12].Alternatif Proof of Theorem 2.1:Note that α b T n +2 + α ( b T n +1 + b T n ) + b T n +1 = α ( T n +2 + T n +3 e + ... + T n +17 e )+ α (( T n +1 + T n ) + ( T n +2 + T n +1 ) e + ... + ( T n +16 + T n +15 ) e )+( T n +1 + T n +2 e + ... + T n +16 e )= α T n +2 + α ( T n +1 + T n ) + T n +1 + ( α T n +3 + ( T n +2 + T n +1 ) + T n +2 ) e +( α T n +4 + ( T n +3 + T n +2 ) + T n +3 ) e ...+( α T n +17 + ( T n +16 + T n +15 ) + T n +16 ) e . From the identity α n +3 = T n +2 α + ( T n +1 + T n ) α + T n +1 for n -th Tribonacci number T n , we have(2.6) α b T n +2 + α ( b T n +1 + b T n ) + b T n +1 = b αα n +3 . Similarly, we obtain(2.7) β b T n +2 + β ( b T n +1 + b T n ) + b T n +1 = b ββ n +3 and(2.8) γ b T n +2 + γ ( b T n +1 + b T n ) + b T n +1 = b γγ n +3 . Substracting (2.7) from (2.6), we have(2.9) ( α + β ) b T n +2 + ( b T n +1 + b T n ) = b αα n +3 − b ββ n +3 α − β . Similarly, substracting (2.8) from (2.6), we obtain(2.10) ( α + γ ) b T n +2 + ( b T n +1 + b T n ) = b αα n +3 − b γγ n +3 α − γ . Finally, substracting (2.10) from (2.9), we get b T n +2 = 1 α − β b αα n +3 − b ββ n +3 α − β − b αα n +3 − b γγ n +3 α − γ ! = b αα n +3 ( α − β )( α − γ ) − b ββ n +3 ( α − β )( β − γ ) + b γγ n +3 ( γ − α )( γ − β )= b αα n +3 ( α − β )( α − γ ) + b ββ n +3 ( β − α )( β − γ ) + b γγ n +3 ( γ − α )( γ − β ) . So this proves (2.4). Similarly we obtain (2.5).Next, we present generating functions.
Theorem . The generating functions for the Tribonacci and Tribonacci-Lucas sedenions are (2.11) g ( x ) = ∞ X n =0 b T n x n = b T + ( b T − b T ) x + b T − x − x − x − x and (2.12) g ( x ) = ∞ X n =0 b K n x n = b K + ( b K − b K ) x + b K − x − x − x − x respectively.Proof. Define g ( x ) = P ∞ n =0 b T n x n . Note that g ( x ) = b T + b T x + b T x + b T x + b T x + b T x + ... + b T n x n + ...xg ( x ) = b T x + b T x + b T x + b T x + b T x + ... + b T n − x n + ...x g ( x ) = b T x + b T x + b T x + b T x + ... + b T n − x n + ...x g ( x ) = b T x + b T x + b T x + ... + b T n − x n + ... Using above table and the recurans b T n = b T n − + b T n − + b T n − we have g ( x ) − xg ( x ) − x g ( x ) − x g ( x )= b T + ( b T − b T ) x + ( b T − b T − b T ) x + ( b T − b T − b T − b T ) x +( b T − b T − b T − b T ) x + ... + ( b T n − b T n − − b T n − − b T n − +) x n + ... = b T + ( b T − b T ) x + ( b T − b T − b T ) x . It follows that g ( x ) = b T + ( b T − b T ) x + ( b T − b T − b T ) x − x − x − x . Since b T − b T − b T = b T − , the generating functions for the Tribonacci sedenion is g ( x ) = b T + ( b T − b T ) x + b T − x − x − x − x . 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.
RIBONACCI AND TRIBONACCI-LUCAS SEDENIONS 11
Theorem . For any integer n, the n th Tribonacci sedenion is b T n = (( α − α ) b T + α b T + b T − ) α n ( α − γ ) ( α − β ) + (( β − β ) b T + β b T + b T − ) β n ( β − γ ) ( β − α )+ (( γ − γ ) b T + γ b T + b T − ) γ n ( γ − β ) ( γ − α ) and the n th Tribonacci-Lucas sedenion is b K n = (( α − α ) b K + α b K + b K − ) α n ( α − γ ) ( α − β ) + (( β − β ) b K + β b K + b K − ) β n ( β − γ ) ( β − α )+ (( γ − γ ) b K + γ b K + b K − ) γ n ( γ − β ) ( γ − α ) . Proof.
We can use generating functions. Since the roots of the equation 1 − x − x − x = 0 are αβ, βγ, αγ and 1 − x − x − x = (1 − αx )(1 − βx )(1 − γx )we can write the generating function of b T n as g ( x )= b T + ( b T − b T ) x + b T − x − x − x − x = b T + ( b T − b T ) x + b T − x (1 − αx )(1 − βx )(1 − γx )= A (1 − αx ) + B (1 − βx ) + C (1 − γx )= A (1 − βx )(1 − γx ) + B (1 − αx )(1 − γx ) + C (1 − αx )(1 − βx )(1 − αx )(1 − βx )(1 − γx )= ( A + B + C ) + ( − Aβ − Aγ − Bα − Bγ − Cα − Cβ ) x + ( Aβγ + Bαγ + Cαβ ) x (1 − αx )(1 − βx )(1 − γx ) . We need to find
A, B and C , so the following system of equations should be solved: A + B + C = b T − Aβ − Aγ − Bα − Bγ − Cα − Cβ = b T − b T Aβγ + Bαγ + Cαβ = b T − We find that A = − α b T + α b T + b T − + α b T α − αβ − αγ + βγ = (( α − α ) b T + α b T + b T − )( α − γ ) ( α − β ) ,B = − β b T + β b T + b T − + β b T β − αβ + αγ − βγ = (( β − β ) b T + β b T + b T − )( β − γ ) ( β − α ) ,C = − γ b T + γ b T + b T − + γ b T γ + αβ − αγ − βγ = (( γ − γ ) b T + γ b T + b T − )( γ − β ) ( γ − α ) . and g ( x ) = (( α − α ) b T + α b T + b T − )( α − γ ) ( α − β ) ∞ X n =0 α n x n + (( β − β ) b T + β b T + b T − )( β − γ ) ( β − α ) ∞ X n =0 β n x n + ( − γ b T + γ b T + b T − + γ b T )( γ − β ) ( γ − α ) ∞ X n =0 γ n x n = ∞ X n =0 (( α − α ) b T + α b T + b T − ) α n ( α − γ )( α − β ) + (( β − β ) b T + β b T + b T − ) β n ( β − γ )( β − α ) + (( γ − γ ) b T + γ b T + b T − ) γ n ( γ − β )( γ − α ) x n . Thus Binet formula of Tribonacci sedenion is b T n = (( α − α ) b T + α b T + b T − ) α n ( α − γ ) ( α − β ) + (( β − β ) b T + β b T + b T − ) β n ( β − γ ) ( β − α )+ (( γ − γ ) b T + γ b T + b T − ) γ n ( γ − β ) ( γ − α ) . 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 b T n , b K n , we have the followingRemark showing relations between b T − , b T , b T ; b K − , b K , b K and b α, b β, b γ. We obtain (b) and (d) after solvingthe system of the equations in (a) and (b) respectively.
Remark . We have the following identities: (a): ( α − α ) b T + α b T + b T − α = b α ( β − β ) b T + β b T + b T − β = b β ( γ − γ ) b T + γ b T + b T − γ = b γ (b): X s =0 T − s e s = b T − = b α ( α − β )( α − γ ) + b β ( β − α )( β − γ ) + b γ ( γ − α )( γ − β ) X s =0 T s e s = b T = b αα ( α − β )( α − γ ) + b ββ ( β − α )( β − γ ) + b γγ ( γ − α )( γ − β ) X s =0 T s e s = b T = b αα ( α − β )( α − γ ) + b ββ ( β − α )( β − γ ) + b γγ ( γ − α )( γ − β ) RIBONACCI AND TRIBONACCI-LUCAS SEDENIONS 13 (c): (( α − α ) b K + α b K + b K − )( α − γ ) ( α − β ) = b α (( β − β ) b K + β b K + b K − )( β − γ ) ( β − α ) = b β (( γ − γ ) b K + γ b K + b K − )( γ − β ) ( γ − α ) = b γ (d): X s =0 K − s e s = b K − = b αα − + b ββ − + b γγ − X s =0 K s e s = b K = b α + b β + b γ X s =0 K s e s = b K = b αα + b ββ + b γγ. Using above Remark we can find b T , b K as follows:(2.13) X s =0 T s e s = b T = b T + b T + b T − = b αα ( α − β )( α − γ ) + b ββ ( β − α )( β − γ ) + b γγ ( γ − α )( γ − β )and(2.14) X s =0 K s e s = b K = b K + b K + b K − = b αα + b ββ + b γγ . 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-Lucasnumbers.
Lemma . For every integer n ≥ , we have (2.15) n X i =0 T i = T + 12 ( T n +2 + T n −
1) = 12 ( T n +2 + T n − and (2.16) n X i =0 K i = K n +2 + K n . Proof. (2.15) and (2.16) can be proved by mathematical induction easily. For a proof of (2.15) with atelescopik 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 = 3 and n P i =1 K i = K n +2 + K n − , it follows that n P i =0 K i = K n +2 + K n . There is also a formula of the summation of the first n negative Tribonacci numbers: n X i =1 T − i = 12 (1 − T − n − − T − n +1 ) . 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-Lucassedenions.
Theorem . The summation formula for Tribonacci and Tribonacci-Lucas sedenions are (2.17) n X i =0 b T i = 12 ( b T n +2 + b T n + c ) and (2.18) n X i =0 b K i = 12 ( b K n +2 + b K n + c ) respectively, where c = − − e − e − e − e − e − e − e − e − e − e − e − e − e − e − e and c = − e − e − e − e − e − e − e − e − e − e − e − e − e − e − e . Proof.
Using (2.1) and (2.15), we obtain n X i =0 b T i = n X i =0 T i + e n X i =0 T i +1 + e n X i =0 T i +2 + ... + e n X i =0 T i +15 = ( T + ... + T n ) + e ( T + ... + T n +1 )+ e ( T + ... + T n +2 ) + ... + e ( T + ... + T n +15 ) . and 2 n X i =0 b T i = ( T n +2 + T n −
1) + e ( T n +3 + T n +1 − − T )+ e ( T n +4 + T n +3 − − T + T ))...+ e ( T n +17 + T n +15 − − T + T + ... + T ))= b T n +2 + b T n + c where c = − e ( − − T ) + e ( − − T + T )) + ... + e ( − − T + ... + T )) . Hence n X i =0 b T i = 12 ( b T n +2 + b T n + c ) . We can compute c as c = − − e − e − e − e − e − e − e − e − e − e − e − e − e − e − e . This proves (2.17). Similarly we can obtain (2.18).
3. Some Identities For The Tribonacci and Tribonacci-Lucas Sedenions
In this section, we give identities about Tribonacci and Tribonacci-Lucas sedenions.
Theorem . For n ≥ , the following identities hold: (a): b K n = 3 b T n +1 − b T n − b T n − , (b): b T n + b T n = 2 T n , b K n + b K n = 2 K n , (c): b T n +1 + b T n = b α ( α + 1) α n +1 ( α − β )( α − γ ) + b β ( β + 1) β n +1 ( β − α )( β − γ ) + b γ ( γ + 1) γ n +1 ( γ − α )( γ − β ) , (d): b K n +1 + b K n = b α ( α + 1) α n + b β ( β + 1) β n + b γ ( γ + 1) γ n , (e): n P i =0 (cid:0) ni (cid:1) b F i = b αα (1 + α ) n ( α − β )( α − γ ) + b ββ (1 + β ) n ( β − α )( β − γ ) + b γγ (1 + γ ) n ( γ − α )( γ − β ) , (f ): n P i =0 (cid:0) ni (cid:1) b K i = b α (1 + α ) n + b β (1 + β ) n + b γ (1 + γ ) n . Proof.
Since K n = 3 T n +1 − T n − T n − (see for example [11]), (a) follows. The others can be establishedeasily. Theorem . For n ≥ , m ≥ , we have (a): b T m + n = T m − b T n +2 + ( T m − + T m − ) b T n +1 + T m − b T n , (b): b T m + n = T m +2 b T n − + ( T m +1 + T m ) b T n − + T m +1 b T n − , (c): b K m + n = K n − b T m +2 + ( b T m +1 + b T m ) K n − + K n − b T m +1 , (d): b K m + n = K m +2 b T n − + ( K m +1 + K m ) b T n − + K m +1 b T n − . Proof. (a) and (d) can be proved by strong induction on m and (c) can be proved by strong inductionon n. For (b), replace n with n − m with m + 3 in (a). References [1] Baez, J., The octonions, Bull. Amer. Math. Soc. 39 (2), 145-205, 2002.[2] Biss, D.K., Dugger, D., and Isaksen, D.C., Large annihilators in Cayley-Dickson algebras, Communication in Algebra,2008.[3] Bilgici, G., Toke¸ser, ¨U,. ¨Unal, Z., Fibonacci and Lucas Sedenions, Journal of Integer Sequences, Article 17.1.8, 20, 1-11.2017. [4] Bruce, I., A modified Tribonacci sequence, The Fibonacci Quarterly, 22 : 3, pp. 244–246, 1984.[5] Cariow, A., Cariowa G., An Algorithm for Fast Multiplication of Sedenios, Information Proccessing Letters, Volume 113,Issue, 9, 324-331, 2013.[6] Catalani, M., Identities for Tribonacci-related sequences - arXiv preprint, https://arxiv.org/pdf/math/0209179.pdfmath/0209179, 2002.[7] Catarino, P., k-Pell, k-Pell–Lucas and modified k-Pell sedenions, Asian-European Journal of Mathematics, 2018[8] Cawagas, E.R., On the Structure and Zero Divisors of the Cayley-Dickson Sedenion Algebra, Discussiones Mathematicae,General Algebra and Applications 24, 251-265, 2004.[9] Cerda-Morales, G., On a Generalization for Tribonacci Quaternions, Mediterranean Journal of Mathematics, 14:239, 1–12,2017.[10] Cerda-Morales, G., The Third Order Jacobsthal Octonions: Some Combinatorial Properties, arXiv:1710.00602v1,[Math.RA], 2 Oct 2017.[11] Cerda-Morales, G., A Three-By-Three Matrix Representation of a Generalized Tribonacci sequence, arXiv:1807.03340v1,[Math.CO], 9 Jul 2018.[12] Cerda-Morales, G., The Unifying Formula for All Tribonacci-Type Octonions Sequences and Their Properties,arXiv:1807.04140v1 [math.RA], [Math.CO], 10 Jul 2018.[13] Choi, E., Modular tribonacci Numbers by Matrix Method, J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. Volume20, Number 3 (August 2013), Pages 207–221, 2013.[14] C¸ imen, C., ˙Ipek, A., On Jacobsthal and Jacobsthal-Lucas Sedenios and Several Identities Involving These Numbers,Mathematica Aeterna, Vol. 7, No.4, 447-454, 2017.[15] C¸ imen, C., ˙Ipek, A, On Jacobsthal and Jacobsthal-Lucas Octonions, Mediterr. J. Math., 14:37, 1-13, 2017.[16] Devbhadra, S. V., Some Tribonacci Identities, Mathematics Today Vol.27(Dec-2011) 1-9, 2011.[17] Elia, M., Derived Sequences, The Tribonacci Recurrence and Cubic Forms, The Fibonacci Quarterly, 39:2, pp. 107-115,2001.[18] Feinberg, M., Fibonacci–Tribonacci, The Fibonacci Quarterly, 1 : 3 (1963) pp. 71–74, 1963.[19] Frontczak, R., Sums of Tribonacci and Tribonacci-Lucas Numbers, International Journal of Mathematical Analysis, Vol.12, No. 1, 19-24, 2018.[20] G¨ul, K., On k-Fibonacci and k-Lucas Trigintaduonions, International Journal of Contemporary Mathematical Sciences,Vol. 13, no. 1, 1 - 10, 2018.[21] Halici, S., Karata¸s, A., On a Generalization for Fibonacci Quaternions. Chaos Solitons and Fractals 98, 178–182, 2017.[22] Horadam, A. F., Complex Fibonacci Numbers and Fibonacci quaternions, Amer. Math. Monthly 70, 289–291, 1963.[23] Imaeda, K., Imaeda, M., Sedenions: algebra and analysis, Applied Mathematics and Computation, 115, 77-88, 2000.[24] Ke¸cilioglu O, Akku¸s, I., The Fibonacci Octonions, Adv. Appl. Clifford Algebr. 25, 151–158, 2015.[25] Kılı¸c, E., Tribonacci Sequences with Certain Indices and Their Sums, Ars. Comb., 86, 13-22, 2008.[26] K¨oplinger, J., Signature of gravity in conic sedenions, Applied Mathematics and Computation, 188, 942-947, 2007.[27] K¨oplinger, J., Gravity and eletromagnetism an conic sedenions, Applied Mathematics and Computation, 188, 948-953,2007.[28] Kuhapatanakul, K., Sukruan, L., The Generalized Tribonacci Numbers With Negative Subscripts, Integer, 14, 1-6, 2014.[29] Moreno, G., The zero divisors of the Cayley-Dickson algebras over the real numbers, Bol. Soc. Mat. Mexicana (3) 4 ,13-28,1998.[30] Muses, C.A., Hypernumber and quantum field theory with a summary of physically applicable hypernumber arithmeticsand their geometrics, Applied Mathematics and Computation, 6, 63-94, 1980.
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