Tribonacci and Tribonacci-Lucas Hybrid Numbers

International Journal of Contemporary Mathematical Sciences Vol. 14, 2019, no. 4, 245 - 254 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijcms.2019.91124

Yasemin Taşyurdu

Department of Mathematics, Faculty of Science and Art Erzincan Binali Yıldırım University, 24100 Erzincan, Turkey

Abstract

In this paper, we define the Tribonacci and Tribonacci-Lucas hybrid numbers and obtain Binet’s formulas and generating functions for these numbers. Then we present some Hessenberg matrices with applications to the Tribonacci and Tribonacci-Lucas hybrid numbers and show that the determinants and permanents of these Hessenberg matrices are equal to the 푛th Tribonacci hybrid number and 푛th Tribonacci-Lucas hybrid number.

Keywords: Tribonacci number, Tribonacci-Lucas number, hybrid number, Hessenberg matrix

1 Introduction

Tribonacci and Tribonacci Lucas sequences which are sequences of integer number defined by recurrence relations are well-known third order recurrence sequences in all of mathematics and generalizations of the Fibonacci sequence. Most of the authors introduced Fibonacci pattern based sequences in many ways which are known as generalization of Fibonacci sequence (see, for example, [3], [5], [6], [12]).

In [4], the Tribonacci sequence originally was studied in 1963 by M. Feinberg. The Tribonacci sequence {푇푛}푛≥0 and Tribonacci-Lucas sequence {퐾푛}푛≥0 are defined by the recurrence relations as follows 푇푛 = 푇푛−1 + 푇푛−2 + 푇푛−3, 푇0 = 0, 푇1 = 1, 푇2 = 1 (1) and 퐾푛 = 퐾푛−1 + 퐾푛−2 + 퐾푛−3, 퐾0 = 3, 퐾1 = 1, 퐾2 = 3 (2) respectively. The first terms of the Tribonacci sequence are 0,1,1,2,4,7,13,24,44,81,149,274,504,927,1705, … and the first terms of the Tribonacci-Lucas sequence are 246 Yasemin Taşyurdu

3,1,3,7,11,21,39,71,131,241,443,815,1499,2757,5071, …

The Tribonacci sequence and Tribonacci-Lucas sequence for negative subscripts are defined by the recurrence relations as follows 푇−푛 = −푇−(푛−1) − 푇−(푛−2) + 푇−(푛−3) and 퐾−푛 = −퐾−(푛−1) − 퐾−(푛−2) + 퐾−(푛−3) for 푛 ≥ 1, respectively. The Binet’s formulas for the 푛th Tribonacci number and 푛th Tribonacci-Lucas number with positive and negative subscripts are given by 훼푛+1 훽푛+1 훾푛+1 푇 = + + (3) 푛 (훼−훽)(훼−훾) (훽−훼)(훽−훾) (훾−훼)(훾−훽)

훼−푛+1 훽−푛+1 훾−푛+1 푇 = + + (4) −푛 (훼−훽)(훼−훾) (훽−훼)(훽−훾) (훾−훼)(훾−훽) and 푛 푛 푛 퐾푛 = 훼 + 훽 + 훾 (5)

−푛 −푛 −푛 퐾−푛 = 훼 + 훽 + 훾 (6) respectively, where 훼, 훽 and 훾 are the roots of the cubic equation 푥3 − 푥2 − 푥 − 1 = 0 and 3 3 1+ √19+3 33+ √19−3 33 훼 = √ √ , 3 3 3 1+휔 √19+3 33+휔2 √19−3 33 훽 = √ √ , 3 3 3 1+휔2 √19+3 33+휔 √19−3 33 훾 = √ √ 3 −1+푖 3 2휋푖 where 휔 = √ = exp ( ) is a primitive cube rot of unity. Note that we have 2 3 the following identities 훼 + 훽 + 훾 = 1 훼훽 + 훼훾 + 훽훾 = −1 훼훽훾 = 1. (see, for example, [7]).

Many authors studied the sequences of integer number defined by recurrence relations and their generalizations. For instance, several authors have defined new classes of hybrid numbers associated with these sequences of integer number. Now we give information about some special classes of hybrid numbers were introduced and studied in the literature. A hybrid number is a generalization of complex numbers, dual numbers and hyperbolic numbers. Complex, dual and hyperbolic numbers are well known two-dimensional number systems. The sets of complex numbers, dual numbers and hyperbolic numbers are ℂ = {푎 + 풊푏 ∶ 푎, 푏 ∈ ℝ, 풊2 = −1}, 픻 = {푎 + 휺푏 ∶ 푎, 푏 ∈ ℝ, 휺2 = 0}, ℙ = {푎 + 풉푏 ∶ 푎, 푏 ∈ ℝ, 풉2 = 1}, Tribonacci and Tribonacci-Lucas hybrid numbers 247

respectively (see, for example, [13]). Özdemir [11] defined a new generalized of complex, dual and hyperbolic numbers different from known generalizations. In this generalization, the author gave a system of such numbers that consists of all three number systems together. This set was called hybrid numbers, denoted by 핂, is defined as 핂 = {푎 + 푏풊 + 푐휺 + 푑풉: 푎, 푏, 푐, 푑 ∈ ℝ, 풊2 = −1, 휺2 = 0, 풉2 = 1, 풊풉 = −풉풊 = 휺 + 풊}. Let 풁 = 푎 + 푏풊 + 푐휺 + 푑풉 be a hybrid number. The real number 푎 is called the scalar part and is denoted by 푆(풁). The part 푏푖 + 푐휀 + 푑ℎ is also called the vector part and is denoted by 푉(풁). Let 풁ퟏ = 푎1 + 푏1풊 + 푐1휺 + 푑1풉 and 풁ퟐ = 푎2 + 푏2풊 + 푐2휺 + 푑2풉 be any two hybrid numbers. The equality, addition, substraction and multiplication by scalar are defined as follows

Equality: 풁ퟏ = 풁ퟐ only if 푎1 = 푎2, 푏1 = 푏2, 푐1 = 푐2, 푑1 = 푑2 Addition: 풁ퟏ + 풁ퟐ = (푎1 + 푎2) + (푏1 + 푏2)풊 + (푐1 + 푐2)휺 + (푑1 + 푑2)풉 Substraction: 풁ퟏ − 풁ퟐ = (푎1 − 푎2) + (푏1 − 푏2)풊 + (푐1 − 푐2)휺 + (푑1 − 푑2)풉 Multiplication by scalar 휆 ∈ ℝ: 휆풁ퟏ = 휆푎1 + 휆푏1풊 + 휆푐1휺 + 휆푑1풉.

Table 1. The multiplication table for the basis of 핂. × 1 푖 휀 ℎ 1 1 푖 휀 ℎ 푖 푖 −1 1 − ℎ 휀 + 푖 휀 휀 1 + ℎ 0 − 휀 ℎ ℎ −휀 − 푖 휀 1

The Table 1 shows us that the multiplication operation in the hybrid numbers is not commutative. But it has the property of associativity. Addition operation in the hybrid numbers is both commutative and associative. Zero is the null element. With respect to the addition operation, the inverse element of 풁 is −풁 = −푎 − 푏푖 − 푐휀 − 푑ℎ . This implies that, (핂, +) is an Abelian group. Let 푛 ≥ 0 be an integer. The 푛th Horadam hybrid number 퐻푛 is defined as follows 퐻푛 = 푊푛 + 푖푊푛+1 + 휀푊푛+2 + ℎ푊푛+3 where 푊푛 is the 푛th Horadam number (see, for example, [8]). Special cases of Horadam hybrid numbers are definitions of the 푛th Fibonacci hybrid number and 푛th Pell hybrid number as follows 퐹퐻푛 = 퐹푛 + 푖퐹푛+1 + 휀퐹푛+2 + ℎ퐹푛+3 푃퐻푛 = 푃푛 + 푖푃푛+1 + 휀푃푛+2 + ℎ푃푛+3 where 퐹푛 is the 푛th Fibonacci number and 푃푛 is the 푛th Pell number, respectively. The 푛th Jacobsthal hybrid number and 푛th Jacobsthal-Lucas hybrid number are defined by

퐽퐻푛 = 퐽푛 + 퐽푛+1푖 + 퐽푛+2휀 + 퐽푛+3ℎ 푗퐻푛 = 푗푛 + 푗푛+1푖 + 푗푛+2휀 + 푗푛+3ℎ

248 Yasemin Taşyurdu

where 퐽푛 is the 푛th Jacobsthal number and 푗푛 is the 푛th Jacobsthal-Lucas number, respectively (see, for example, [9]). Catarino [2] introduced a new sequence of numbers called k-Pell hybrid numbers as follows 퐻푃푘,푛 = 푃푘,푛 + 푃푘,푛+1푖 + 푃푘,푛+2휀 + 푃푘,푛+3ℎ where 푃푘,푛 is the 푛th 푘-Pell number.

2 Tribonacci and Tribonacci-Lucas Hybrid Numbers

In this section, we define the Tribonacci and Tribonacci-Lucas hybrid numbers and give Binet’s formulas and generating functions for these numbers.

Definition 1. The 푛th Tribonacci hybrid number ℍ푇푛 and 푛th Tribonacci-Lucas hybrid number ℍ퐾푛 are defined by with the basis {1, 푖, 휀, ℎ} where 푖, 휀, ℎ satisfy the conditions 푖2 = −1, 휀2 = 0, ℎ2 = 1, 푖ℎ = −ℎ푖 = 휀 + 푖 as follows ℍ푇푛 = 푇푛 + 푇푛+1푖 + 푇푛+2휀 + 푇푛+3ℎ (7) and ℍ퐾푛 = 퐾푛 + 퐾푛+1푖 + 퐾푛+2휀 + 퐾푛+3ℎ (8) where 푇푛 is the 푛th Tribonacci number and 퐾푛 is the 푛th Tribonacci-Lucas number, respectively.

Using the equations (7) and (8) we can write the first Tribonacci and Tribonacci- Lucas hybrid numbers as follows 푖 + 휀 + 2ℎ, 1 + 푖 + 2휀 + 4ℎ, 1 + 2푖 + 4휀 + 7ℎ, 2 + 4푖 + 7휀 + 13ℎ, … and 3 + 푖 + 3휀 + 7ℎ, 1 + 3푖 + 7휀 + 11ℎ, 3 + 7푖 + 11휀 + 21ℎ, … respectively.

We now present the following theorem for the recurrence relations of the Tribonacci and Tribonacci-Lucas hybrid numbers.

Theorem 1. Let ℍ푇푛 be the 푛th Tribonacci hybrid number and ℍ퐾푛 be 푛th Tribonacci-Lucas hybrid number. Then we give the following recurrence relations ℍ푇푛 = ℍ푇푛−1 + ℍ푇푛−2 + ℍ푇푛−3 (9) and ℍ퐾푛 = ℍ퐾푛−1 + ℍ퐾푛−2 + ℍ퐾푛−3. (10) Proof. Using the equations (1) and (7) we have ℍ푇푛−1 + ℍ푇푛−2 + ℍ푇푛−3 = (푇푛−1 + 푇푛−2 + 푇푛−3) + (푇푛 + 푇푛−1 + 푇푛−2)푖 +(푇푛+1 + 푇푛 + 푇푛−1)휀 + (푇푛+2 + 푇푛+1 + 푇푛)ℎ = 푇푛 + 푇푛+1푖 + 푇푛+2휀 + 푇푛+3ℎ = ℍ푇푛. Similarly, we can obtain equation (10) by using equations (2) and (8). ∎ Using the equations (9) and (10) we obtain the recurrence relations of the Tribonacci and Tribonacci-Lucas hybrid numbers for negative subscripts as follows

Tribonacci and Tribonacci-Lucas hybrid numbers 249

ℍ푇−푛 = −ℍ푇−(푛−1) − ℍ푇−(푛−2) + ℍ푇−(푛−3) and ℍ퐾−푛 = −ℍ퐾−(푛−1) − ℍ퐾−(푛−2) + ℍ퐾−(푛−3) respectively. Thus the equations (9) and (10) holds for all integer 푛.

The following theorem gives the Binet’s formulas for the 푛th Tribonacci hybrid number and 푛th Tribonacci-Lucas hybrid number.

Theorem 2. Let ℍ푇푛 be the 푛th Tribonacci hybrid number and ℍ퐾푛 be 푛th Tribonacci-Lucas hybrid number. For 푛 ≥ 0, the 푛th Tribonacci hybrid number and 푛th Tribonacci-Lucas hybrid number are given by 훼̂훼푛+1 훽̂훽푛+1 훾̂훾푛+1 ℍ푇 = + + (11) 푛 (훼−훽)(훼−훾) (훽−훼)(훽−훾) (훾−훼)(훾−훽) and 푛 ̂ 푛 푛 ℍ퐾푛 = 훼̂훼 + 훽훽 + 훾̂훾 (12) respectively, where 훼̂ = 1 + 훼푖 + 훼2휀 + 훼3ℎ, 훽̂ = 1 + 훽푖 + 훽2휀 + 훽3ℎ, 훾̂ = 1 + 훾푖 + 훾2휀 + 훾3ℎ. Moreover, 훼, 훽 and 훾 are the roots of the cubic equation 푥3 − 푥2 − 푥 − 1 = 0 and 3 3 1+ √19+3 33+ √19−3 33 훼 = √ √ , 3 3 3 1+휔 √19+3 33+휔2 √19−3 33 훽 = √ √ , 3 3 3 1+휔2 √19+3 33+휔 √19−3 33 훾 = √ √ . 3 −1+푖 3 2휋푖 where 휔 = √ = exp ( ) is a primitive cube rot of unity. 2 3

Proof. Using equation (3) in equation (7) we have ℍ푇푛 = 푇푛 + 푇푛+1푖 + 푇푛+2휀 + 푇푛+3ℎ 훼푛+1 훽푛+1 훾푛+1 = + + (훼−훽)(훼−훾) (훽−훼)(훽−훾) (훾−훼)(훾−훽) 훼푛+2 훽푛+2 훾푛+2 + ( + + ) 푖 (훼−훽)(훼−훾) (훽−훼)(훽−훾) (훾−훼)(훾−훽) 훼푛+3 훽푛+3 훾푛+3 + ( + + ) 휀 (훼−훽)(훼−훾) (훽−훼)(훽−훾) (훾−훼)(훾−훽) 훼푛+4 훽푛+4 훾푛+4 + ( + + ) ℎ (훼−훽)(훼−훾) (훽−훼)(훽−훾) (훾−훼)(훾−훽) (1+훼푖+훼2휀+훼3ℎ)훼푛+1 (1+훽푖+훽2휀+훽3ℎ)훽푛+1 (1+훾푖+훾2휀+훾3ℎ)훾푛+1 = + + (훼−훽)(훼−훾) (훽−훼)(훽−훾) (훾−훼)(훾−훽) 훼̂훼푛+1 훽̂훽푛+1 훾̂훾푛+1 = + + . (훼−훽)(훼−훾) (훽−훼)(훽−훾) (훾−훼)(훾−훽) where 훼̂ = 1 + 훼푖 + 훼2휀 + 훼3ℎ, 훽̂ = 1 + 훽푖 + 훽2휀 + 훽3ℎ, 훾̂ = 1 + 훾푖 + 훾2휀 + 훾3ℎ. Similarly, we can obtain equation (12) by using equation (5) in equation (8). ∎ We can introduce Binet’s formulas for the 푛th Tribonacci hybrid number and 푛th Tribonacci-Lucas hybrid number with negative subscripts as follows 훼̂훼−푛+1 훽̂훽−푛+1 훾̂훾−푛+1 ℍ푇 = + + −푛 (훼−훽)(훼−훾) (훽−훼)(훽−훾) (훾−훼)(훾−훽) 250 Yasemin Taşyurdu

and −푛 ̂ −푛 −푛 ℍ퐾−푛 = 훼̂훼 + 훽훽 + 훾̂훾 respectively.

Next, we give generating functions for the Tribonacci and Tribonacci-Lucas hybrid numbers.

Theorem 3. Let ℍ푇푛 be the 푛th Tribonacci hybrid number and ℍ퐾푛 be 푛th Tribonacci-Lucas hybrid number. The generating functions for the Tribonacci and Tribonacci-Lucas hybrid numbers are

ℍ푇 +(ℍ푇 −ℍ푇 )푡+ℍ푇 푡2 𝑔(푡) = ∑∞ ℍ푇 푡푛 = 0 1 0 −1 (13) 푛=0 푛 1−푡−푡2−푡3 and ℍ퐾 +(ℍ퐾 −ℍ퐾 )푡+ℍ퐾 푡2 푟(푡) = ∑∞ ℍ퐾 푡푛 = 0 1 0 −1 (14) 푛=0 푛 1−푡−푡2−푡3 respectively.

∞ 푛 Proof. Let 𝑔(푡) = ∑푛=0 ℍ푇푛푡 be generating function for the Tribonacci hybrid numbers. On the other hand, since 2 푛 𝑔(푡) = ℍ푇0 + ℍ푇1푡 + ℍ푇2푡 + ⋯ + ℍ푇푛푡 + ⋯ 2 3 푛+1 𝑔(푡)푡 = ℍ푇0푡 + ℍ푇1푡 + ℍ푇2푡 + ⋯ + ℍ푇푛푡 + ⋯ 2 2 3 4 푛+2 𝑔(푡)푡 = ℍ푇0푡 + ℍ푇1푡 + ℍ푇2푡 + ⋯ + ℍ푇푛푡 + ⋯ 3 3 4 5 푛+3 𝑔(푡)푡 = ℍ푇0푡 + ℍ푇1푡 + ℍ푇2푡 + ⋯ + ℍ푇푛푡 + ⋯ we obtain that 2 3 2 2 2 (1 − 푡 − 푡 − 푡 )𝑔(푡) = ℍT0 + ℍ푇1푡 + ℍ푇2푡 − ℍT0푡 − ℍ푇1푡 − ℍT0푡 where ℍ푇푛 = ℍ푇푛−1 + ℍ푇푛−2 + ℍ푇푛−3 from equation (9). Here the coefficients of 푡푛 for 푛 ≥ 3 are equal to zero. Then generating function for the Tribonacci hybrid numbers is ℍ푇 +(ℍ푇 −ℍ푇 )푡+ℍ푇 푡2 ∑∞ ℍ푇 푡푛 = 0 1 0 −1 . 푛=0 푛 1−푡−푡2−푡3

Similarly, we can obtain equation (14). ∎

3 Tribonacci and Tribonacci-Lucas Hybrid Numbers by Hessenberg Matrices

In this section, we define four type lower Hessenberg matrices and investigate the relationships between these matrices and the Tribonacci and Tribonacci-Lucas hybrid numbers. We then show that the determinants and permanents of these Hessenberg matrices are the 푛th Tribonacci hybrid number and 푛th Tribonacci- Lucas hybrid number. A 푛푥푛 matrix 푀푛 = (푚푖푗) is called lower Hessenberg matrix if 푚푖푗 = 0 when 푗 − 푖 > 1, i.e.,

Tribonacci and Tribonacci-Lucas hybrid numbers 251

푚1,1 푚1,2 0 0 ⋯ 0 푚 푚 푚 0 ⋯ 0 2,1 2,2 2,3 푚 푚 푚 푚 ⋯ 0 푀 = 3,1 3,2 3,3 3,4 . 푛 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 푚푛−1,1 푚푛−1,2 푚푛−1,3 푚푛−1,4 ⋯ 푚푛−1,푛 ( 푚푛,1 푚푛,2 푚푛,3 푚푛,4 ⋯ 푚푛,푛 )

In [1], for 푛 ≥ 2, it was given the following formula 푛−1 푛−푖 푛−1 푑푒푡(푀푛) = 푚푛,푛 푑푒푡(푀n−1) + ∑푖=1 [(−1) 푚푛,푖 ∏푗=푖 푚푗,푗+1푑푒푡(푀i−1)] (15) with 푑푒t(푀0) = 1 and 푑푒푡(푀1) = 푚11. Also in [10], for 푛 ≥ 2, it was given the following formula 푛−1 푛−1 푝푒푟(푀푛) = 푚푛,푛 푝푒푟(푀n−1) + ∑푖=1 [푚푛,푖 ∏푗=푖 푚푗,푗+1푝푒푟(푀푖−1)] (16) with 푝푒푟(푀0) = 1 and 푝푒푟(푀1) = 푚11.

Theorem 4. Let 푛 × 푛 Hessenberg matrices 푃푛 = (푝푖푗) and 푄푛 = (푞푖푗) be as follows ℍ푇0 1 0 0 0 ⋯ 0 0 0 0 0 ℍ푇 ⁄ℍ푇 −1 0 0 ⋯ 0 0 0 0 1 0 −ℍ푇−1 1 1 −1 0 ⋯ 0 0 0 0 푃푛 = 0 1 1 1 −1 ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ ⋮ 0 0 0 0 0 ⋯ 1 1 1 −1 ( 0 0 0 0 0 ⋯ 0 1 1 1) and ℍ퐾0 1 0 0 0 ⋯ 0 0 0 0 0 ℍ퐾 ⁄ℍ퐾 −1 0 0 ⋯ 0 0 0 0 1 0 −ℍ퐾−1 1 1 −1 0 ⋯ 0 0 0 0 푄푛 = 0 1 1 1 −1 ⋯ ⋮ ⋮ ⋮ ⋮ . ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ ⋮ 0 0 0 0 0 ⋯ 1 1 1 −1 ( 0 0 0 0 0 ⋯ 0 1 1 1)

Then for 푛 ≥ 1, 푑푒푡푃푛 = ℍ푇푛−1 (17) and 푑푒푡푄푛 = ℍ퐾푛−1 (18) where ℍ푇푛 is the 푛th Tribonacci hybrid number and ℍ퐾푛 is the 푛th Tribonacci- Lucas hybrid number.

Proof. We can use the mathematical induction method on 푛 to prove 푑푒푡푃푛 = ℍ푇푛−1. Then 푛 = 1, 푑푒푡푃1 = ℍ푇0 252 Yasemin Taşyurdu

푛 = 2, 푑푒푡푃2 = ℍ푇0(ℍ푇1⁄ℍ푇0) − 0 = ℍ푇1 푛 = 3, 푑푒푡푃3 = ℍ푇0(ℍ푇1⁄ℍ푇0 + 1) − 1(−ℍ푇−1) = ℍ푇1+ℍ푇0 + ℍ푇−1 = ℍ푇2 ⋮ We assume that it is true for 푛 ∈ ℤ+, namely 푑푒푡푃푛 = ℍ푇푛−1, 푑푒푡푃푛−1 = ℍ푇푛−2, 푑푒푡푃푛−2 = ℍ푇푛−3, … and we shall show that it is true for 푛 + 1. Using the equation (15) we have 푛 푛+1−푖 푛 푑푒푡푃푛+1 = 푝푛+1,푛+1푑푒푡푃푛 + ∑푖=1[(−1) 푝푛+1,푖 ∏푗=푖 푝푗,푗+1푑푒푡푃푖−1] 푛−2 푛+1−푖 푛 = (1)푑푒푡푃푛 + ∑푖=1 [(−1) 푝푛+1,푖 ∏푗=푖 푝푗,푗+1푑푒푡푃푖−1] 2 +(−1)푝푛+1,푛푝푛,푛+1푑푒푡푃푛−1 + (−1) 푝푛+1,푛−1푝푛−1,푛푝푛,푛+1푑푒푡푃푛−2 2 = 푑푒푡푃푛 + 0 + (−1)(1)(−1)푑푒푡푃푛−1 + (−1) (1)(−1)(−1)푑푒푡푃푛−2 = 푑푒푡푃푛 + 푑푒푡푃푛−1 + 푑푒푡푃푛−2 = ℍ푇푛−1 + ℍ푇푛−2 + ℍ푇푛−3 = ℍ푇푛. Similarly, we can obtain equation (18). ∎ Theorem 5. Let 푛 × 푛 Hessenberg matrices 푅푛 = (푟푖푗) and 푆푛 = (푠푖푗) be as follows ℍ푇0 −1 0 0 0 ⋯ 0 0 0 0 0 ℍ푇 ⁄ℍ푇 1 0 0 ⋯ 0 0 0 0 1 0 −ℍ푇−1 1 1 1 0 ⋯ 0 0 0 0 푅푛 = 0 1 1 1 1 ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ ⋮ 0 0 0 0 0 ⋯ 1 1 1 1 ( 0 0 0 0 0 ⋯ 0 1 1 1) and ℍ퐾0 −1 0 0 0 ⋯ 0 0 0 0 0 ℍ퐾 ⁄ℍ퐾 1 0 0 ⋯ 0 0 0 0 1 0 −ℍ퐾−1 1 1 1 0 ⋯ 0 0 0 0 푆푛 = 0 1 1 1 1 ⋯ ⋮ ⋮ ⋮ ⋮ . ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ ⋮ 0 0 0 0 0 ⋯ 1 1 1 1 ( 0 0 0 0 0 ⋯ 0 1 1 1)

Then for 푛 ≥ 1, 푝푒푟푅푛 = ℍ푇푛−1 (19) and 푝푒푟푆푛 = ℍ퐾푛−1 (20) where ℍ푇푛 is the 푛th Tribonacci hybrid number and ℍ퐾푛 is the 푛th Tribonacci- Lucas hybrid number

Proof. We can use the mathematical induction method on 푛 to prove 푝푒푟푅푛 = ℍ푇푛−1. Then 푛 = 1, 푝푒푟푅1 = ℍ푇0 푛 = 2, 푝푒푟푅2 = ℍ푇0(ℍ푇1⁄ℍ푇0) + 0 = ℍ푇1 Tribonacci and Tribonacci-Lucas hybrid numbers 253

푛 = 3, 푝푒푟푅3 = ℍ푇0(ℍ푇1⁄ℍ푇0 + 1) − 1(−ℍ푇−1) = ℍ푇1+ℍ푇0 + ℍ푇−1 = ℍ푇2 ⋮ We assume that it is true for 푛 ∈ ℤ+, namely 푝푒푟푅푛 = ℍ푇푛−1, 푝푒푟푅푛−1 = ℍ푇푛−2, 푝푒푟푅푛−2 = ℍ푇푛−3, … and we shall show that it is true for 푛 + 1. Using the equation (16) we have 푛 푛 푝푒푟(푅푛+1) = 푟푛+1,푛+1 푝푒푟푅n + ∑푖=1[푟푛+1,푖 ∏푗=푖 푟푗,푗+1푝푒푟(푅푖−1)] 푛−2 푛 = (1)푝푒푟푅푛 + ∑푖=1 [푟푛+1,푖 ∏푗=푖 푟푗,푗+1푝푒푟푅푖−1] +푟푛+1,푛푟푛,푛+1푝푒푟푅푛−1 + 푟푛+1,푛−1푟푛−1,푛푟푛,푛+1푝푒푟푅푛−2 = 푝푒푟푅푛 + 0 + (1)(1)푝푒푟푅푛−1 + (1)(1)(1)푝푒푟푅푛−2 = 푝푒푟푅푛 + 푝푒푟푅푛−1 + 푝푒푟푅푛−2 = ℍ푇푛−1 + ℍ푇푛−2 + ℍ푇푛−3 = ℍ푇푛. Similarly, we can obtain equation (20). ∎

4 Discussion and Conclusions

In this paper, we define the Tribonacci and Tribonacci-Lucas hybrid numbers by using a new generalized of complex, dual and hyperbolic numbers and obtain their recurrence relations. We also states the expression for negative subscripts. Then we find Binet’s formulas for 푛th number and generating functions that plays an important role in the literature. Moreover, some Hessenberg matrices whose entries are the Tribonacci and Tribonacci-Lucas hybrid numbers was presented as a different way to obtain the 푛th Tribonacci hybrid number and 푛th Tribonacci- Lucas hybrid number and show that the determinants and permanents of these Hessenberg matrices are equal to the 푛th Tribonacci hybrid number and 푛th Tribonacci-Lucas hybrid number.

In the future, we consider all subsequences of the Tribonacci and Tribonacci- Lucas hybrid numbers of forms (ℍ푇푚푛+푠) and (ℍ퐾푚푛+푠) for arbitrary integers 푛, 푠 and 푚 with 0 ≤ 푠 < 푚. We intend to discuss Binet’s formulas, generating functions and the summation formulas for subsequences of the Tribonacci and Tribonacci-Lucas hybrid numbers of forms (ℍ푇푚푟+푠) and (ℍ퐾푚푟+푠) with positive and negative subscripts by using the new definitions presented in this paper.

Acknowledgements. We would like to express their sincere gratitude to the referees for their valuable comments, which have significantly improved the presentation of this paper. The author declares no conflict of interest.

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Received: November 19, 2019; Published: December 12, 2019