Ln≥0 (Or Shortly )

SARAJEVO JOURNAL OF MATHEMATICS DOI: 10.5644/SJM.16.01.08 Vol.16 (29), No.1, (2020), 103 – 122

ON GENERALIZED TRIBONACCI SEDENIONS

YUKSEL¨ SOYKAN, INC˙ I˙ OKUMUS¸, AND ERKAN TAS¸DEMIR*˙

ABSTRACT. The sedenions form a 16-dimensional Cayley-Dickson algebra. In this work, we introduce the generalized Tribonacci sedenion and present some properties of this sedenion.

1. INTRODUCTION

The generalized Tribonacci sequence Vn(V0,V1,V2;r,s,t) n 0 (or shortly { } ≥ Vn n 0) is deﬁned as follows: { } ≥ Vn = rVn 1 + sVn 2 +tVn 3, V0 = a,V1 = b,V2 = c, n 3 (1.1) − − − ≥ where V0,V1,V2 are arbitrary integers and r,s,t are real numbers. This sequence has been studied by many authors, see for example [6], [10], [19], [22], [32], [37], [39], [41], [44], [50], [51]. The sequence Vn n 0 can be extended to negative subscripts by deﬁning { } ≥ s r 1 V n = V (n 1) V (n 2) + V (n 3) − −t − − − t − − t − − for n = 1,2,3,... when t = 0. Therefore, recurrence (1.1) holds for all integer n. 6 If we set r = s = t = 1 and V0 = 0,V1 = 1,V2 = 1 then Vn is the well-known { } Tribonacci sequence and if we set r = s = t = 1 and V0 = 3,V1 = 1,V2 = 3 then Vn is the well-known Tribonacci-Lucas sequence. { In} fact, the generalized Tribonacci sequence is the generalization of the well- known sequences like Tribonacci, Tribonacci-Lucas, Padovan (Cordonnier), Per- rin, Padovan-Perrin, Narayana, third order Jacobsthal and third order Jacobsthal- Lucas. In literature, for example, the following names and notations (see Table 1) are used for the special case of r,s,t and initial values.

2010 Mathematics Subject Classiﬁcation. 11B39, 11B83, 17A45, 05A15. Key words and phrases. Tribonacci numbers, sedenions, Tribonacci sedenions. 104 YUKSEL¨ SOYKAN, INC˙ I˙ OKUMUS¸, AND ERKAN TAS¸DEMIR*˙

Sequences (Numbers) Notation Tribonacci Tn = Vn(0,1,1;1,1,1) { } { } Tribonacci-Lucas Kn = Vn(3,1,3;1,1,1) { } { } Padovan (Cordonnier) Pn = Vn(1,1,1;0,1,1) { } { } Pell-Padovan Rn = Vn(1,1,1;0,2,1) { } { } Jacobsthal-Padovan JPn = Vn(1,1,1;0,1,2) { } { } Perrin Qn = Vn(3,0,2;0,1,1) { } { } Pell-Perrin pQn = Vn(3,0,2;0,2,1) { } { } Jacobsthal-Perrin JQn = Vn(3,0,2;0,1,2) { } { } Padovan-Perrin Sn = Vn(0,0,1;0,1,1) { } { } Narayana Nn = Vn(0,1,1;1,0,1) { } { } third order Jacobsthal Jn = Vn(0,1,1;1,1,2) { } { } third order Jacobsthal-Lucas jn = Vn(2,1,5;1,1,2) { } { } Table 1. A few values of generalized Tribonacci sequences.

As Vn is a third order recurrence sequence (difference equation), it’s charac- teristic{ equation} is x3 rx2 sx t = 0, whose roots are − − − r α = α(r,s,t)= + A + B 3 r β = β(r,s,t)= + ωA + ω2B 3 r γ = γ(r,s,t)= + ω2A + ωB 3 where r3 rs t 1/3 r3 rs t 1/3 A = + + + √∆ , B = + + √∆ 27 6 2 27 6 2 − r3t r2s2 rst s3 t2 1 + i√3 ∆ = ∆(r,s,t)= + + , ω = − = exp(2πi/3) 27 − 108 6 − 27 4 2 Note that we have the following identities α + β + γ =r, αβ + αγ + βγ = s, − αβγ =t. From now on, we assume that ∆(r,s,t) > 0, so that the Eq.(1.1) has one real (α) and two non-real solutions with the latter being conjugate complex. So, in this case, it is well known that generalized Tribonacci numbers can be expressed, for all integers n, using Binet’s formula Pαn Qβn Rγn V = + + (1.2) n (α β)(α γ) (β α)(β γ) (γ α)(γ β) − − − − − − ON GENERALIZED TRIBONACCI SEDENIONS 105 where

P = V2 (β+γ)V1 +βγV0, Q = V2 (α+γ)V1 +αγV0, R = V2 (α+β)V1 +αβV0. − − − Note that the Binet form of a sequence satisfying (1.1) for non-negative integers is valid for all integers n, for a proof of this result see [27]. This result of Howard and Saidak [27] is even true in the case of higher-order recurrence relations. We can give Binet’s formula of the generalized Tribonacci numbers for the negative subscripts: for n = 1,2,3,... we have 2 1 n 2 1 n α rα s Pα − β rβ s Qβ − V n = − − + − − − t (α β)(α γ) t (β α)(β γ) − − − − γ2 rγ s Rγ1 n + − − − . t (γ α)(γ β) − − In this paper, we intruduce and study the generalized Tribonacci sedenions in the next section and give some properties of them. Before giving its definition, we present some information on Cayley-Dickson algebras. The process of extending the field R of reals to C is known as Cayley–Dickson process (construction). It can be carried further to higher dimensions using a doubling procedure, yielding the quaternions H, octonions O and sedenions S which (as a real vector space) are of dimension 4, 8, and 16, respectively. This doubling process can be extended beyond the sedenions to form what are known as the 2n- ions (see for example [28], [35], [4]). However, just as applying the construction to reals loses the property of order- ing, more properties familiar from real and complex numbers vanish with increas- ing dimension. The quaternions are only a skew field, i.e. for some x,y we have x y = y x for two quaternions, the multiplication of octonions fails (in addition to not· 6 being· commutative) to be associative: for some x,y,z we have (x.y).z = x.(y.z). Real numbers, complex numbers, quaternions and octonions are all normed6 division algebras over R. However, by Hurwitz’s theorem, they are the only ones. The next step in the Cayley–Dickson precess, the sedenions, in fact fails to have this structure. S 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 (e3 + e10)(e6 e15)= 0 and the other example is (e2 e14)(e3 + e15)= 0, for details see [13]. − − In the following, we explain this doubling process. The Cayley-Dickson algebras are a sequence A0,A1,... of non-associative R- algebras with involution. The term “conjugation” is used to refer to the involution because it generalizes the usual conjugation on the complex numbers. See [4] for a full explanation of the basic properties of Cayley-Dickson algebras. We define Cayley-Dickson algebras inductively. We begin by defining A0 to be R. Given An 1, the algebra An is defined additively to be An 1 An 1. Conjugation, multi- − − × − plication and addition in An are defined by 106 YUKSEL¨ SOYKAN, INC˙ I˙ OKUMUS¸, AND ERKAN TAS¸DEMIR*˙

(a,b) = (a, b), − (a,b)(c,d) = (ac db,da + bc), − (a,b) + (c,d) = (a + c,b + d), n respectively. Note that An has dimension 2 as an R vector space. If we set, as −2 usual, x = Re(xx) for x An then xx = xx = x . k k ∈ k k Now, assumep that B16 = ei S : i = 0,1,2,...,15 is the basis for S, where e0 { ∈ } is the identity (or unit) and e1,e2,...,e15 are called imaginaries. Then a sedenion S S can be written as ∈ 15 15 S = ∑ aiei = a0 + ∑ aiei i=0 i=1 ∑15 where a0,a1,...,a15 are all real numbers. a0 is called the real part of S and i=1 aiei is called its imaginary part. Addition of sedenions is defined as componentwise and multiplication is defined as follows: if S1,S2 S, then we have ∈ 15 15 15 S1S2 = ∑ aiei ∑ biei = ∑ aib j(eie j). (1.3) i=0 i=0 i, j=0 The operations requiring for the multiplication in (1.3) are quite a lot. The compu- tation of a sedenion multiplication using the naive method requires 240 additions and 256 multiplications, while an algorithm which is given in [8] can compute the same result in only 298 additions and 122 multiplications, see [8] for details. Moreover, efficient algorithms for the multiplication of quaternions, octonions and trigintaduonions with reduced number of real multiplications is already exist in literature, see [33], [7] and [9], respectively. Recently, a lot of attention has been centred by theoretical physicists on the Cayley-Dickson algebras O (octonions) and S (sedenions) because of their increas- ing usefulness in formulating many of the new theories of elementary particles. In particular, the octonions O (see for example [2] and [36]) has been found to be involved in so many unexpected areas (such as Clifford algebras, quantum theory, topology, etc.) and sedenions appear in many areas of science like electromagnetic theory and linear gravity.

2. GENERALIZED TRIBONACCI SEDENIONSAND THEIR GENERATING FUNCTIONAND BINET FORMULA In this section, we define generalized Tribonacci 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 [25] introduced nth Fibonacci and nth Lucas quaternions as 3 Qn = Fn + Fn+1e1 + Fn+2e2 + Fn+3e3 = ∑ Fn+ses s=0 ON GENERALIZED TRIBONACCI SEDENIONS 107 and 3 Rn = Ln + Ln+1e1 + Ln+2e2 + Ln+3e3 = ∑ Ln+ses s=0 respectively, where Fn and Ln are the nth Fibonacci and Lucas numbers respectively. He also defined generalized Fibonacci quaternion as 3 Pn = Hn + Hn+1e1 + Hn+2e2 + Hn+3e3 = ∑ Hn+ses s=0 where Hn is the nth generalized Fibonacci number (which is now called Horadam number) by the recursive relation H1 = p, H2 = p + q, Hn = Hn 1 + Hn 2 (p and q are arbitrary integers). Many other generalization of Fibonacci− quaternions− has been given, see for example Halici and Karatas¸[24], and Polatlı [38]. See also Tasci [47] for k-Jacobsthal and k-Jacobsthal-Lucas Quaternions. Cerda-Morales [14] defined and studied the generalized Tribonacci quaternion sequence that includes the previously introduced Tribonacci, Padovan, Narayana and third order Jacobsthal quaternion sequences. He defined generalized Tribonacci quaternion as 3 Qv,n = Vn +Vn+1e1 +Vn+2e2 +Vn+3e3 = ∑ Vn+ses s=0 where Vn is the nth generalized Tribonacci number defined by the third-order recurrence relations Vn = rVn 1 + sVn 2 +tVn 3, − − − here V0 = a,V1 = b,V2 = c are arbitrary integers and r,s,t are real numbers. See also [1], [46] for Tribonacci quaternions, [16] for third order Jacobsthal quaternions, [17] for dual third order Jacobsthal quaternions and [48] for Padovan and Pell- Padovan quaternions. Various families of octonion number sequences (such as Fibonacci octonion, Pell octonion, Jacobsthal octonion; and third order Jacobsthal octonion) have been defined and studied by a number of authors in many different ways. For example, Keçilioglu and Akkus¸[31] introduced the Fibonacci and Lucas octonions as 7 Qn = ∑ Fn+ses s=0 and 7 Rn = ∑ Ln+ses s=0 respectively, where Fn and Ln are the nth Fibonacci and Lucas numbers respectively. In [21], Çimen and Ipek˙ introduced Jacobsthal octonions and Jacobsthal- Lucas octonions. In [15], Cerda-Morales introduced third order Jacobsthal octonions and also in [18], he defined and studied tribonacci-type octonions. See also Ipek˙ and Çimen [29] and Karatas¸and Halici [30] for Horadam type octonions. Fur- thermore, we refer to Szynal-Liana and Włoch [45] for Pell quaternions and Pell 108 YUKSEL¨ SOYKAN, INC˙ I˙ OKUMUS¸, AND ERKAN TAS¸DEMIR*˙ octonions and Catarino [11] for Modified Pell and Modified k-Pell Quaternions and Octonions. A number of authors have been defined and studied sedenion number sequences (such as second order sedenions: Fibonacci sedenion, k-Pell and k-Pell–Lucas sedenions, Jacobsthal and Jacobsthal-Lucas sedenions). For example, Bilgici, Tokes¸er and Unal¨ [5] introduced the Fibonacci and Lucas sedenions as 15 Fn = ∑ Fn+ses s=0 and b 15 Ln = ∑ Ln+ses s=0 respectively, where Fn and Ln areb the nth Fibonacci and Lucas numbers respectively. In [12], Catarino introduced k-Pell and k-Pell–Lucas sedenions. In [20], Çimen and Ipek˙ introduced Jacobsthal and Jacobsthal-Lucas sedenions. In [43], Soykan introduced the nth Tribonacci sedenion as 15 15 Tn = ∑ Tn+ses = Tn + ∑ Tn+ses. s=0 s=1 (see also preprint versionb [42] of it). Gül [23] introduced the k-Fibonacci and k-Lucas trigintaduonions as 31 TFk,n = ∑ Fk,n+ses s=0 and 31 TLk,n = ∑ Lk,n+ses s=0 respectively, where Fk,n and Lk,n are the nth k-Fibonacci and k-Lucas numbers respectively. We now define the generalized Tribonacci sedenion over the sedenion algebra S. The nth generalized Tribonacci sedenion is 15 15 Vn = ∑ Vn+ses = Vn + ∑ Vn+ses (2.1) s=0 s=1 b where Vn is the nth generalized Tribonacci number defined by Eq.(1.1). For all non-negative integer n, it can be easily shown that the following iterative relation holds: Vn = rVn 1 + sVn 2 +tVn 3. (2.2) − − − The sequence Vn n 0 canb be definedb forb negativeb values of n by using the recurrence (2.2) to{ extend} ≥ the sequence backwards, or equivalently, by using the recurrence b s r 1 V n = V (n 1) V (n 2) + V (n 3). − −t − − − t − − t − − b b b b ON GENERALIZED TRIBONACCI SEDENIONS 109

Thus, the recurrence (2.2) holds for all integer n. The conjugate of Vn is deﬁned by 15 b Vn = Vn ∑ Vn+ses = Vn Vn+1e1 Vn+2e2 ... Vn+15e15. − s=1 − − − − The norm ofb nth generalized Tribonacci sedenion is 2 2 2 2 2 Vn = N (Vn)= VnVn = VnVn = Vn +Vn+1 + ... +Vn+15.

Now, we willb state Binet’sb formulab b forb b the nth generalized Tribonacci sedenion. From now on, we ﬁxed the following notations. 15 15 15 s s s α = ∑ α es, β = ∑ β es, γ = ∑ γ es. s=0 s=0 s=0 b Theorem 2.1. For anyb non-negative integer n, theb nth generalized Tribonacci sedenion is Pααn Qββn Rγγn V = + + (2.3) n (α β)(α γ) (β α)(β γ) (γ α)(γ β) − b − − b − − b − where P,Q andb R are as in Eq.(1.2). Proof. Repeated use of (1.2) in (2.1) enable us to write: 15 Pαn Qβn Rγn V = ∑ V e = + + n n+s s (α β)(α γ) (β α)(β γ) (γ α)(γ β) s=0 − − − − − − b Pαn+1 Qβn+1 Rγn+1 + + + e1 (α β)(α γ) (β α)(β γ) (γ α)(γ β) − − − − − − Pαn+2 Qβn+2 Rγn+2 + + + e2 (α β)(α γ) (β α)(β γ) (γ α)(γ β) − − − − − − . . Pαn+15 Qβn+15 Rγn+15 + + + e15 (α β)(α γ) (β α)(β γ) (γ α)(γ β) − − − − − − Pααn Qββn Rγγn = + + . (α β)(α γ) (β α)(β γ) (γ α)(γ β) − b − − b − − b − We can give Binet’s formula of the nth generalized Tribonacci sedenion for the negative subscripts: for n = 1,2,3,... we have 2 1 n 2 1 n α rα s Pαα − β rβ s Qββ − V n = − − + − − − t (α β)(α γ) t (β α)(β γ) −b − − b − b γ2 rγ s Rγγ1 n + − − − . t (γ α)(γ β) − b − 110 YUKSEL¨ SOYKAN, INC˙ I˙ OKUMUS¸, AND ERKAN TAS¸DEMIR*˙

The next theorem gives us an alternative proof of the Binet’s formula for the nth generalized Tribonacci sedenion. For this, we need the quadratic approximation of Vn n 0: { } ≥ Lemma 2.2. [18]. For all integer n, we have n+2 2 Pα = α Vn+2 + α(sVn+1 +tVn)+tVn+1, n+2 2 Qβ = β Vn+2 + β(sVn+1 +tVn)+tVn+1, n+2 2 Rγ = γ Vn+2 + γ(sVn+1 +tVn)+tVn+1, where P,Q and R are as in Eq.(1.2). Alternative Proof of Theorem 2.1: Note that 2 α Vn+2 + α(sVn+1 +tVn)+tVn+1 α2 = (bVn+2 +Vn+b3e1 + ...b+Vn+b17e15) + α((sVn+1 +tVn) + (sVn+2 +tVn+1)e1 + ... + (sVn+16 +tVn+15)e15)

+t(Vn+1 +Vn+2e1 + ... +Vn+16e15) 2 =α Vn+2 + α(sVn+1 +tVn)+tVn+1 2 + (α Vn+3 + α(sVn+2 +tVn+1)+tVn+2)e1 2 + (α Vn+4 + α(sVn+3 +tVn+2)+tVn+3)e2 . . 2 + (α Vn+17 + α(sVn+16 +tVn+15)+tVn+16)e15. n+2 2 From the identity Pα = α Vn+2 + α(sVn+1 + tVn)+ tVn+1 for nth generalized Tribonacci number Vn, we have 2 n+2 α Vn+2 + α(sVn+1 +tVn)+tVn+1 = Pαα . (2.4) Similarly, we obtain b b b b b 2 n+2 β Vn+2 + β(sVn+1 +tVn)+tVn+1 = Qββ (2.5) and b b b b b 2 n+2 γ Vn+2 + γ(sVn+1 +tVn)+tVn+1 = Rγγ . (2.6) Substracting (2.5) from (2.4), we have b b b b b ααn+2 ββn+2 α β P Q ( + )Vn+2 + (sVn+1 +tVn)= α − β . (2.7) b − b Similarly, substracting (2.6)b fromb (2.4), web obtain ααn+2 γγn+2 α γ P R ( + )Vn+2 + (sVn+1 +tVn)= α −γ . (2.8) b b b b b − ON GENERALIZED TRIBONACCI SEDENIONS 111

Finally, substracting (2.8) from (2.7), we get 1 Pααn+2 Qββn+2 Pααn+2 Rγγn+2 Vn 2 = − − + β γ α β − α γ − b − b b − b b Pααn+2 Qββn+2 Rγγn+2 = + (α β)(α γ) − (α β)(β γ) (γ α)(γ β) −b − −b − − b − Pααn+2 Qββn+2 Rγγn+2 = + + . (α β)(α γ) (β α)(β γ) (γ α)(γ β) −b − − b − − b − So this proves (2.3). Next, we present generating function of Vn. Theorem 2.3. The generating function for theb nth generalized Tribonacci sedenion is ∞ 2 n V0 + (V1 rV0)x +tV 1x g(x)= ∑ V x = − − . (2.9) n 1 rx sx2 tx3 n=0 b −b −b − b ∞ bn Proof. Deﬁne g(x)= ∑n=0 Vnx . Note that 2 3 4 n g(x)= V0 + V1xb+ V2x + V3x + V4x + ... + Vnx + ... 2 3 4 n rxg(x)= rV0x + rV1x + rV2x + rV3x + ... + rVn 1x + ... 2 b b b 2 b 3 b 4 b − n sx g(x)= sV0x + sV1x + sV2x + ... + sVn 2x + ... 3 b b b 3 b 4 b − n tx g(x)= tV0x + tV1x + ... + tVn 3x + ... b b b b −

Using above table and the recurence Vbn = rVn b1 + sVn 2 +tVbn 3, we have − − − (1 rx sx2 tx3)g(x)=g(x) rxg(x) sx2g(x) tx3g(x) − − − − b −b b− b 2 =V0 + (V1 rV0)x + (V2 rV1 sV0)x − − − 3 b+ (V3 b rV2 b sV1 btV0)x b b − − − 4 + (Vb4 rVb3 sVb2 tVb1)x − − − n + ...b+ (Vnb rVnb 1 bsVn 2 tVn 3+)x + ... − − − − − − 2 =V0 + (V1 b rV0)bx + (V2 b rV1 bsV0)x . − − − It follows that b b b b b b 2 V0 + (V1 rV0)x + (V2 rV1 sV0)x g(x)= − − − . 1 rx sx2 tx3 b b −b − b − b b Since V2 rV1 sV0 = tV 1, the generating function for the Tribonacci sedenion is − − − 2 b b b b V0 + (V1 rV0)x +tV 1x g(x)= − − . 1 rx sx2 tx3 b −b −b − b In the following theorem, we present another form of Binet formula for the nth generalized Tribonacci sedenions by using generating function. 112 YUKSEL¨ SOYKAN, INC˙ I˙ OKUMUS¸, AND ERKAN TAS¸DEMIR*˙

Theorem 2.4. For any integer n, the nth generalized Tribonacci sedenion is

2 n 2 n (α rα)V0 + αV1 +tV 1 α (β rβ)V0 + βV1 +tV 1 β V = − − + − − n α γ α β β γ β α −b b− b −b b− b b 2 n ((γ rγ)V0 + γV1 +tV 1)γ + − γ α γ β − . −b b− b Proof. We can use generating function. Since

1 rx sx2 tx3 = (1 αx)(1 βx)(1 γx), − − − − − − we can write the generating function of Vn as

2 2 V0 + (V1 rV0)x +tV 1x bV0 + (V1 rV0)x +tV 1x g(x)= − − = − − 1 rx sx2 tx3 (1 αx)(1 βx)(1 γx) b −b −b − b b − b −b −b D E F = + + (1 αx) (1 βx) (1 γx) − − − D(1 βx)(1 γx)+ E(1 αx)(1 γx)+ F(1 αx)(1 βx) = − − − − − − (1 αx)(1 βx)(1 γx) − − − D + E + F ( Dβ Dγ Eα Eγ Fα Fβ)x = + − − − − − − (1 αx)(1 βx)(1 γx) (1 αx)(1 βx)(1 γx) − − − − − − (Dβγ + Eαγ + Fαβ)x2 + (1 αx)(1 βx)(1 γx) − − − We need to ﬁnd D,E and F, so the following system of equations should be solved:

D + E + F =V0

Dβ Dγ Eα Eγ Fα Fβ =Vb1 rV0 − − − − − − − Dβγ + Eαγ + Fαβ =tbV 1 b − We ﬁnd that b 2 2 tV 1 + αV1 + α V0 rαV0 (α rα)V0 + αV1 +tV 1 D = − − = − − α2 αβ αγ + βγ α γ α β b −b − b b −b −b b 2 2 tV 1 + βV1 + β V0 rβV0 (β rβ)V0 + βV1 +tV 1 E = − − = − − β2 αβ + αγ βγ β γ β α b − b b− −b −b b 2 2 tV 1 + γV1 + γ V0 rγV0 (γ rγ)V0 + γV1 +tV 1 F = − − = − − γ2 + αβ αγ βγ γ α γ β b b − b− b −b −b b and ON GENERALIZED TRIBONACCI SEDENIONS 113

α2 α α ∞ β2 β β ∞ ( r )V0 + V1 +tV 1 n n ( r )V0 + V1 +tV 1 n n g x − ∑ α x − ∑ β x ( )= − α γ α β + − β γ β α −b −b b n=0 −b −b b n=0 γ2 γ γ ∞ ( r )V0 + V1 +tV 1 n n − ∑ γ x + − γ α γ β −b −b b n=0 2 n 2 n (α rα)V0 + αV1 +tV 1 α (β rβ)V0 + βV1 +tV 1 β − − + − − ∞  α γ α β β γ β α  ∑ b b b b b b xn = − − γ2 γ γ −γn − . n 0  ( r )V0 + V1 +tV 1  =  + − −   γ α γ β   −b b− b  Thus, Binet formula of Tribonacci sedenion is 2 n 2 n (α rα)V0 + αV1 +tV 1 α (β rβ)V0 + βV1 +tV 1 β V = − − + − − n α γ α β β γ β α −b b− b −b b− b b 2 n (γ rγ)V0 + γV1 +tV 1 γ + − − . γ α γ β −b b− b If we compare Theorem 2.1 and Theorem 2.4 and use the deﬁnition of Vn, we have the following Remark showing relations between T ,T ,T and α,β,γ. We 1 0 1 b obtain (b) after solving the system of the equations in (a).− b b b b b b Remark 2.1. We have the following identities:

(a): 2 ((α rα)V0 + αV1 +tV 1)=Pα − − β2 β β β (( r )Vb0 + Vb1 +tVb 1)=Qb − − 2 ((γ rγ)bV0 + γVb1 +tVb 1)=Rγb − − (b): b b b b 15 Pβγ(γ β)α + Qαγ(α γ)β + Rαβ(β α)γ ∑ V 1+ses =V 1 = − − − − − t γ β α γ α β s=0 b − − b − b 15 b P(β γ)α +Q(γ α)β+ R(α β)γ ∑ V e V s s = 0 = − β γ α − γ α β − s=0 b− − b − b b 15 αE + βE + γE ∑ V e V 1 2 3 1+s s = 1 = γ β α γ α β s=0 −b b− b − b where E1 = P β γ r + β + γ , E2 = Q γ α r + α + γ , − − − − E3 = R α β r+ α + β . − − Using above Remark, we can find V2 as follows: 15 b αc1 + βc2 + γc3 ∑ V2+ses = V2 = rV1 + sV0 +tV 1 = − β γ α γ α β s=0 −b b− b − b b b b 114 YUKSEL¨ SOYKAN, INC˙ I˙ OKUMUS¸, AND ERKAN TAS¸DEMIR*˙ where 2 c1 =P(β γ)(s rβ rγ + βγ + r ), − − − 2 c2 =Q(γ α)(s rα rγ + αγ + r ), − − − 2 c3 =R(α β)(s rα rβ + αβ + r ). − − − Now, we present the formula which gives the summation of the first n generalized Tribonacci numbers. Lemma 2.5 ([14]). For every integer n 0, we have n ≥ Vn+2 + (1 r)Vn+1 +tVn + (r + s 1)V0 + (r 1)V1 V2 ∑Vi = − ε − − − (2.10) i=0 r,s,t where ε = εr s t = r + s +t 1. , , − Next, we present the formula which gives the summation of the first n generalized Tribonacci sedenion. Theorem 2.6. The summation formula for the generalized Tribonacci sedenion is n Vn+2 + (1 r)Vn+1 +tVn + ϕr,s,t ∑Vi = − ε (2.11) i=0 b br,s,t b where ε = εr s t = r + s +bt 1, µ = µr s t = (r + s 1)V0 + (r 1)V1 V2 and ϕ = , , − , , − − − ϕr s t = µ + e1(µ εV0)+ e2(µ ε(V0 +V1)+ .... + e15(µ ε(V0 +V1 + ... +V14)). , , − − − Proof. Using (2.1), we obtain n n n n n ∑Vi = ∑Vi + e1 ∑Vi+1 + e2 ∑Vi+2 + ... + e15 ∑Vi+15 i=0 i=0 i=0 i=0 i=0 b =(V0 + ... +Vn)+ e1(V1 + ... +Vn+1) + e2(V2 + ... +Vn+2)+ ... + e15(V15 + ... +Vn+15).

Using (2.10) and the notation µr,s,t = (r+s 1)V0 +(r 1)V1 V2, it can be written that − − − n εr s t ∑ Vi =(Vn 2 + (1 r)Vn 1 +tVn + µr s t ) , , i=0 + − + , , + e1(Vn 3 + (1 r)Vn 2 +tVn 1 + µ εV0) b + − + + − + e2(Vn+4 + (1 r)Vn+3 +tVn+2 + µ ε(V0 +V1)) . − − . + e15(Vn 17 + (1 r)Vn 16 +tVn 15 + µ ε(V0 + ... +V14)) + − + + − =Vn 2 + (1 r)Vn 1 +tVn + ϕr s t + − + , , where ϕ = ϕr s t = µ + e1(µ εV0)+ e2(µ ε(V0 +V1)+ .... + e15(µ ε(V0 +V1 + , , b − b b− − ... +V14)). It now follows that n Vn+2 + (1 r)Vn+1 +tVn + ϕr,s,t ∑Vi = − ε i=0 b br,s,t b and this completes the proof.b The formula (2.12) of next Theorem gives the norm of the nth generalized Tri- bonacci sedenion. ON GENERALIZED TRIBONACCI SEDENIONS 115

Theorem 2.7. The norm of generalized Tribonacci sedenion is given by 2 (β α)2P2αα2n + (α γ)2Q2ββ2n + (α β)2R2γγ2n 2M V = (2.12) n − − ψ2 − −

e e e where b ψ =ψ(α,β,γ) = (α β)(α γ)(β γ), − − − α =1 + α2 + α4 + ... + α30, β 1 β2 β4 β30 e = + + + ... + , γ γ2 γ4 γ30 e =1 + + + ... + , (2.13) αβ αβ αβ 2 αβ 3 αβ 15 e =1 + + ( ) + ( ) + ... + ( ) , αγ αγ αγ 2 αγ 3 αγ 15 f =1 + + ( ) + ( ) + ... + ( ) , βγ 1 βγ βγ 2 βγ 3 βγ 15 f = + + ( ) + ( ) + ... + ( ) , n n Me =(α γ)(β γ)PQαβ(αβ) + (α β)(β γ)PQαγ(αγ) − − − − + (α β)(α γ)PQβγ(βγ)n. (2.14) − − f f 2 e ∑15 2 Proof. We know that Vn = s=0 Vn+s. Using Binet formula of Vn, we have

n n n ψVn =b (β γ)Pα + (γ α)Qβ + (α β)Rγ − − − where ψ is as in (2.13). It follows that ψ2V 2 =(β γ)2P2α2n + (γ α)2Q2β2n n − − + (α β)2R2γ2n + 2(β γ)(γ α)PQ(αβ)n − − − + 2(β γ)(α β)PR(αγ)n + 2(γ α)(α β)QR(βγ)n − − − − and 2 ψ2 ψ2 2 2 2 2 Vn = (Vn +Vn+1 +Vn+2 + ... +Vn+15)

b =(β γ)2P2αα2n + (γ α)2Q2ββ2n − − 2 2 2n n + (α β) R γγ + 2(β γ)(eγ α)PQαβ(αβ) − e − − + 2(β γ)(α β)PRαγ(αγ)n + 2(γ α)(α β)QRβγ(βγ)n − e− − f − where α,β,γ,αβ,αγ and βγ are as in (2.13).f e Now, we present the quadratic approximation of Vn. e e e f f e Theorem 2.8. For all integer n 0, we have b ≥ n+2 2 Pαα =α Vn+2 + α(sVn+1 +tVn)+tVn+1, Qββn+2 β2V β sV tV tV b = bn+2 + ( bn+1 + bn)+ bn+1, γγn+2 γ2 γ Rb = Vbn+2 + (sVbn+1 +tVbn)+tVbn+1, where P,Q and R areb as in (1.2).b b b b 116 YUKSEL¨ SOYKAN, INC˙ I˙ OKUMUS¸, AND ERKAN TAS¸DEMIR*˙

Proof. Using Binet’s formula of Vn, we obtain Pααn(α3 + (s + βγ)α +t) αV + (s + βγ)V +b tV = n+2 n+1 n (α β)(α γ) b − − b b b Qββn(αβ2 + (s + βγ)β +t) + (β α)(β γ) b − − Rγγn(αγ2 + (s + βγ)γ +t) + . (γ α)(γ β) b − − Since αβ2 + (s + βγ)β +t = 0, αγ2 + (s + βγ)γ +t = 0 and

α α β α γ = α3 α2β α2γ+αβγ = α3 (αβ+αγ)α+t = α3 +(s+βγ)α+t − − − − − it follows that n+1 αVn+2 + (s + βγ)Vn+1 +tVn = Pαα . (2.15) Multiplying (2.15) by α and using again αβγ = t, we obtain b b b b n+2 2 2 Pαα = α Vn+2 + α(s + βγ)Vn+1 + αtVn = α Vn+2 + α(sVn+1 +tVn)+tVn+1. α β γ If web change theb role of with band , web obtainb the desiredb result. b b

3. SOME IDENTITIES FOR SPECIAL CASES OF THE GENERALIZED TRIBONACCI SEDENION In this section, we give identities about Tribonacci and Tribonacci-Lucas sedenions. Theorem 3.1. For n 1, the following identities hold: ≥ (a): Kn = 3Tn+1 2Tn Tn 1, here Tn =V (0,1,1;1,1,1) and Kn =V (3,1,3;1,1,1). − − − (b): Vbn +Vnb= 2Vn b b n n n b b Pα α + 1 α Qβ β + 1 β Rγ γ + 1 γ (c): Vn 1 +Vn = + + , + (αb β)(α γ) (βb α)(β γ) (γb α)(γ β) − − − − − − bn b Pαα Qββ (d): ∑ n V = (1 + α)n + (1 + β)n i i (α β)(α γ) (β α)(β γ) i=0 b b Rγγb − − − − : + (1 + γ)n, (γ α)(γ β) − b − Proof. Since Kn = 3Tn+1 2Tn Tn 1, (a) follows. The others can be established easily. − − − Theorem 3.2. For n 0, m 3, we have ≥ ≥ (a): Km+n = Kn 1Tm+2 + (Tm+1 + Tm)Kn 2 + Kn 3Tm+1, − − − (b): Km+n = Km+2Tn 1 + (Km+1 + Km)Tn 2 + Km+1Tn 3. b b − b b − b − b b b b ON GENERALIZED TRIBONACCI SEDENIONS 117

Proof. (a) can be proved by strong induction on n and (b) can be proved by strong induction on m. Similar formula can be given for the other Fibonacci sequences (for example, for Pm+n and Rm+n).

Theoremb 3.3 b([26]). Consider the sequences Jn which is called generalized Tribo- nacci-Lucas sequence, and Vn given by 2 Jn = Vn(3,r,r + 2s;r,s,t) . { } and Vn = Vn(V0,V1,V2;r,s,t) . Then, for all integers n and m we have { } m m Vn+2m = JmVn+m t J mVn +t Vn m. − − − Using above theorem, we have the following corollary. Corollary 3.1. For all integers n and m, the followings are true:

(a): Tn+2m = KmTn+m K mTn + Tn m − − − (b): Kn+2m = KmKn+m K mKn + Kn m. − − − Proof.

(a): Take Vn = Vn(0,1,1;1,1,1) = Tn and Jn = Vn(3,1,3;1,1,1) = { } { } { } Kn . { } (b): Take Vn = Vn(3,1,3;1,1,1) = Kn and Jn = Vn(3,1,3;1,1,1) = { } { } { } Kn . { } 4. MATRICES AND DETERMINANTS RELATED WITH TRIBONACCIAND TRIBONACCI-LUCAS SEDENIONS

Let Xn and Yn be any sequences generated by (1.1). Deﬁne the 4 4 deter- { } { } × minant Dn, for all integers n, by

Xn Yn Yn+1 Yn+2 X2 Y2 Y3 Y4 Dn = . X1 Y1 Y2 Y3

X0 Y0 Y1 Y2

Theorem 4.1. Dn = 0 for all integers n. Proof. This is a special case of a result in [34]. Theorem 4.2. The followings are true.

(a): 44Tn = 10Kn 2 6Kn 1 8Kn. + − + − (b): Kn = Tn 2 + 4Tn 1 Tn. b − +b +b− b Proof. Takingb Xn b= Tn b, Yn =b Kn and applying Theorem 4.1 and expanding { } { } { } { } Dn along the top row gives 44Tn = 10Kn 2 6Kn 1 8Kn and now (a) follows. + − + − Taking Xn = Kn , Yn = Tn and applying Theorem 4.1 and expanding Dn { } { } { } { } along the top row gives Kn = Tn 2 + 4Tn 1 Tn and now (b) follows. − + + − 118 YUKSEL¨ SOYKAN, INC˙ I˙ OKUMUS¸, AND ERKAN TAS¸DEMIR*˙

Note that similar formulas can be given for the other Tribonacci sequences (for example, taking Xn = Pn , Yn = Rn ). { } { } { } { } Consider the sequence Un which is deﬁned by the third-order recurrence relation { } Un = Un 1 +Un 2 +Un 3, U0 = U1 = 0,U2 = 1. − − − Note that some authors call Un as a Tribonacci sequence instead of Tn . The { } { } numbers Un can be expressed using Binet’s formula αn βn γn U = + + n (α β)(α γ) (β α)(β γ) (γ α)(γ β) − − − − − − and the negative numbers U n (n = 1,2,3,...) satisﬁes the recurrence relation − Un+1 Un+2 2 U n = = Un+1 Un+2Un. − Un Un+1 −

The matrix method is very useful method in order to obtain some identities for special sequences. We deﬁne the square matrix M of order 3 as: r s t M = 1 0 0 0 1 0 such that detM = t. Note that Un+2 sUn+1 +tUn tUn+1 n M = Un+1 sUn +tUn 1 tUn . (4.1) − Un sUn 1 +tUn 2 tUn 1 − − − For a proof of (4.1), see [3]. Matrix formulation of Vn can be given as

Vn+2 r s t n V2 Vn 1 = 1 0 0 V1 (4.2) + Vn 0 1 0 V0 The matrix M was deﬁned and used in [40]. For matrix formulation (4.2), see [49] and [52]. Now we deﬁne the matrice MV as

V4 V3 +V2 V3 MV = V V +V V . b3 b2 b1 b2 V V +V V b2 b1 b0 b1 This matrice MV can be called generalizedb b Tribonaccib b sedenion matrix. As special cases, Tribonacci and Tribonacci-Lucas sedenion matricies are

T4 T3 + T2 T3 K4 K3 + K2 K3 MT = T T + T T and MK = K K + K K b3 b2 b1 b2 b3 b2 b1 b2 T T + T T K K + K K b2 b1 b0 b1 b2 b1 b0 b1 respectively. b b b b b b b b ON GENERALIZED TRIBONACCI SEDENIONS 119

Theorem 4.3. For n 0, the following holds: ≥ n r s t Vn+4 sVn+3 +tVn+2 tVn+3 MV  1 0 0  =  Vn 3 sVn 2 +tVn 1 tVn 2  (4.3) b + b + b + b + 0 1 0 V sV +tV tV    bn+2 b n+1 b n bn+1  Proof. We prove by mathematical inductionb onbn. If n =b 0 thenb the result is clear. Now, we assume it is true for n = k, that is

Vk+4 sVk+3 +tVk+2 tVk+3 k MV M =  Vk 3 sVk 2 +tVk 1 tVk 2 . b + b + b + b + V sV +tV tV  bk+2 b k+1 b k bk+1  If we use (2.2), then for k 0,bwe havebVk+3 =brVk+2 b+ sVk+1 + tVk. Then by induction hypothesis, we obtain≥ b b b b k+1 k MV M =(MV M )M

Vk+4 sVk+3 +tVk+2 tVk+3 r s t = Vk 3 sVk 2 +tVk 1 tVk 2  1 0 0  b + b + b + b + V sV +tV tV 0 1 0  bk+2 b k+1 b k bk+1   rbVk+4 + sVbk+3 +tVbk+2 bsVk+4 +tVk+3 tVk+4 = rV + sV +tV sV +tV tV  bk+3 bk+2 bk+1 bk+3 bk+2 bk+3 rV + sV +tV sV +tV tV  b k+2 b k+1 b k bk+2 bk+1 bk+2  Vk+b5 sVk+b4 +tVk+b3 tVkb+4 b b = V sV +tV tV . bk+4 bk+3 bk+2 bk+3 V sV +tV tV  bk+3 bk+2 bk+1 bk+2  Thus, (4.3) holds for all non-negativeb b integersb n.b Corollary 4.1. For n 0, the following holds: ≥ Vn+2 = V2Un+2 + (V1 +V0)Un+1 +V1Un.

Proof. The proof canb be seenb by the coefﬁcientb b (4.1) ofb the matrix MV and (4.3). Note that we have similar results if we replace the matrix M with the matrices N and O deﬁned by r 1 0 0 1 0 N =  s 0 1  and O =  0 0 1 . t 0 0 r s t     Acknowledgment The authors express their sincere thanks to the referees for their careful reading and suggestions that helped to improve this paper. 120 YUKSEL¨ SOYKAN, INC˙ I˙ OKUMUS¸, AND ERKAN TAS¸DEMIR*˙

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(Received: May 13, 2019) Y¨uksel Soykan, Inci˙ Okumus¸ (Revised: Oktober 01, 2019) Zonguldak B¨ulent Ecevit University Department of Mathematics Art and Science Faculty 67100, Zonguldak, Turkey e-mail: yuksel [email protected] e-mail: inci okumus [email protected] and Erkan TAS¸DEMIR*˙ Corresponding Author Kırklareli University Pınarhisar Vocational School of Higher Education 39300, Kırklareli, Turkey e-mail: [email protected]