A Study of the Symmetry of the Tricomplex Repunit Sequence with Repunit Sequence

Abstract

In this paper, we introduce a new family of sequences related to Horadam-type sequences. Specifically, we consider the repunit sequence ${\left\{r_n\right\}}_{n\ge 0}$, which is defined by the initial terms $r_0=0$ and $r_1=1$ and follows the Horadam recurrence relation given by $r_n=11r_{n-1}-10r_{n-2}$ for $n\ge 2$. Many studies have explored generalizations of integer sequences in different directions: some by preserving the initial terms, some by preserving the recurrence relation, and some by considering different numerical sets beyond positive integers. In this article, we take the third approach. Specifically, we study these sequences in the context of the tricomplex ring $\mathbb{T}$. We define the Tricomplex Repunit sequence ${\left\{t{r}_n\right\}}_{n\ge 0}$, with initial terms $t{r}_0=(0,1,11)$ and $t{r}_1=(1,11,111)$, and governed by the recurrence relation $t{r}_n=11t{r}_{n-1}-10t{r}_{n-2}$, for $n\ge 2$. This sequence is also a Horadam-type sequence but defined in the tricomplex ring $\mathbb{T}$. In this paper, we establish the properties of the Tricomplex Repunit sequence and establish several new as well as well-known identities associated with it, including Binet’s formula, Tagiuri–Vajda’s identity, d’Ocagne’s identity, and Catalan’s identity. We also derive the generating function for this sequence. Furthermore, we study various additional properties of these generalized sequences and establish results concerning the summation of terms related to the Tricomplex Repunit sequence, and one of our main goals is to determine analogous or symmetrical properties for the Tricomplex Repunit sequence to those we know for the ordinary repunit sequence.

1. Introduction

The repunit sequences ${\left\{r_n\right\}}_{n\ge 0}$ are the terms of the set $\{0,1,11,111,1111,1111,\cdots \}$, where each term satisfies the recursive formula $r_{n+1}=10r_n+1$ for all $n\ge 1$ and $r_0=0$: the sequence $A002275$ in OEIS [1]. In previous studies, Santos and Costa, in [2,3], demonstrated that the sequence in question also satisfied the Horadam recursive recurrence

$$r_{n+1}=11r_n-10r_{n-1},\mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{l}\mathrm{l} n\ge 1,$$

(1)

with initial terms $r_0=0$ and $r_1=1$.

Consider $\mathbb{T}$, the ring of tricomplex numbers, that is, the set of ordered triples of real numbers $(x,y,z)$, with the operations of addition $(+)$, and multiplication $(\times )$ given by

$$(x_1,y_1,z_1)+(x_2,y_2,z_2)=(x_1+x_2,y_1+y_2,z_1+z_2),$$

(2)

and

$$(x_1,y_1,z_1)\times (x_2,y_2,z_2)=(x_1{x}_2+y_1{z}_2+z_1{y}_2,z_1{z}_2+x_1{y}_2+y_1{x}_2,y_1{y}_2+x_1{z}_2+z_1{x}_2);$$

(3)

where $x_i,y_i$ and $z_i$ are real numbers with $i=1,2,3$.

Now, in ring $\mathbb{T}$, we introduce a novel type of sequence too, referred to as the Horadam sequence. In the literature, we can find generalizations of the well-known sequences, like Fibonacci, Lucas, Tribonacci, bicomplex Fibonacci, and so on. As an example, we would like to refer to the work of Kizilateş, Catarino, and Tuǧlu in [4], where the study of one of these generalizations was presented. The work of Costa et al. in [5] presented a study of the extension of the Fibonacci sequence in the tricomplex ring $\mathbb{T}$. So, in $\mathbb{T}$, we specify the triple Repunit element of order n as $t{r}_n=(r_n,r_{n+1},r_{n+2})$ for all $n\ge 0$. And, this study examines the symmetry of the Tricomplex Repunit sequence in relation to the ordinary repunit sequence ${\left\{r_n\right\}}_{n\ge 0}$.

Definition 1.

We define the Tricomplex Repunit sequence, denoted by ${\left\{t{r}_n\right\}}_{n\ge 0}$, as a sequence of triples $t{r}_n=(r_n,r_{n+1},r_{n+2})$ for all integers $n\ge 0$, where $r_n$ is a repunit number.

To illustrate, we may consider the following examples: $t{r}_0=(0,1,11)$ and $t{r}_1=(1,11,111)$, and furthermore, the value of $t{r}_2$ is given by $(11,111,1111)$. Note that,

$$11·t{r}_1-10·t{r}_0=11·(1,11,111)-10·(0,1,11)=(11,111,111)=t{r}_2.$$

See that this sequence is defined in three-dimensional space ${\mathbb{R}}^3$. So, our motivation is to introduce numerical sequences in the context of ${\mathbb{R}}^3$, something not yet common in the mathematical literature, in particular in the Tricomplex ring $\mathbb{T}$.

The aim of this paper is to determine symmetrical or analogous properties for the Tricomplex Repunit sequence to those we know for the ordinary repunit sequence. So, this article is organized as follows: In Section 2, we present a concise overview of the tricomplex ring $\mathbb{T}$ and a summary of the essential results related to the repunit sequence used in this work. In Section 3, we introduced the study of the Tricomplex Repunit sequence, explaining some of its characteristics and properties. Additionally, we presented the Binet formula, which provides an explicit expression for the terms of the sequence. Furthermore, we explore the generating function, which we express in its three-dimensional vectorial form. In Section 4, we present some basic properties of the Tricomplex Repunit numbers. In Section 5, we explore several fundamental identities for Tricomplex Repunit, such as the Tagiuri-Vajada, d’Ocagene, and associated identities. Finally, in Section 6, we present the sum of terms involving the Tricomplex Repunit numbers.

2. Background and Auxiliary Results

This section firstly offers a brief introduction to the tricomplex ring, denoted as $\mathbb{T}$, and a summary of selected results pertaining to the repunit sequence, which will be used throughout this work. Subsequently, the repunit sequence is addressed, characterized by a second-order linear recurrence and a Horadam-type recurrence, and a list of results involving some properties of the repunit sequence used throughout the work are presented.

2.1. The Tricomplex Ring

A considerable number of researchers have demonstrated a keen interest in the study of rings analogous to the ring of integers, wherein the foundations of arithmetic can also be developed. Among these rings, the ring of Gaussian integers merits particular attention due to its origins in Gauss’s investigations concerning cubic and biquadratic reciprocity. A notable property of this ring is its capacity to exhibit many arithmetic results that are analogous to those in integers, with additional geometric interpretations. For a comprehensive overview of the construction and historical background of the complex number field and the tricomplex ring, can be consulted in [6,7,8], among others.

In the context of mathematics, a tricomplex number represents an element of a number system that extends the complex numbers. In a manner comparable to complex numbers, which have a real part and an imaginary part, tricomplex numbers are defined by two imaginary parts in addition to a real part. Olariu in [9,10] introduced the concept of tricomplex numbers, which are expressed in the form $(x,y,z)=x+y\mathbf{i}+z\mathbf{j}$, where x, y, and z are real numbers, and $\mathbf{i}$ and $\mathbf{j}$ are imaginary units. Following [9,10], the multiplication rules for the tricomplex units $1,\mathbf{i}$, and $\mathbf{j}$ are given by

Table 1. The multiplication table for tricomplex units.

×	1	i	j
1	1	$\mathbf{i}$	$\mathbf{j}$
$\mathbf{i}$	$\mathbf{i}$	$\mathbf{j}$	1
$\mathbf{j}$	$\mathbf{j}$	1	$\mathbf{i}$

:

According to Equation (2), the sum of the tricomplex numbers $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ is the tricomplex number $(x_1+x_2,y_1+y_2,z_1+z_2)$. And, according to Equation (3), the product of the tricomplex numbers $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ is the tricomplex number $(x_1{x}_2+y_1{z}_2+z_1{y}_2,z_1{z}_2+x_1{y}_2+y_1{x}_2,y_1{y}_2+x_1{z}_2+z_1{x}_2)$. So, the tricomplex numbers and their operations can be represented by the notation $\mathbb{T}=(\mathbb{T},+,\times )$. It is straightforward to verify through a direct calculation that the tricomplex zero is represented by the vector $(0,0,0)$, which is denoted simply by the symbol $\mathbf{0}$. Similarly, the tricomplex unity is represented by the point $(1,0,0)$, which is denoted simply by the symbol $\mathbf{1}$. Furthermore, it can be demonstrated that the tricomplex ring is a commutative and unit ring; see [10,11,12] and references. So, the tricomplex ring $\mathbb{T}$ is a “symmetric” ring with respect to the ring of integers immersed in the three-dimensional space ${\mathbb{R}}^3$. By symmetric, we mean that the basic laws of arithmetic hold in $\mathbb{T}$. So, it makes sense to define numerical sequences in $\mathbb{T}$.

Tricomplex numbers can be a valuable tool in certain mathematical contexts, particularly in the study of three-dimensional systems. They can also have applications in physics and engineering, where three-dimensional systems are an inherent aspect of many phenomena. In addition, in [13], a study of the four-dimensional complex algebraic structure is given, and in four-dimensional complexes, there is an application for the construction of directional probability distributions.

2.2. The Repunit Sequence

The repunit sequence is constituted by a series of numbers, with the exception of the initial element, which is represented by the decimal system as the repetition of the unit. To obtain further details, kindly refer to the sources cited; see [14,15,16,17].

Remember that Equation (1) is of the type Horadam recurrence; see [18,19]. Specifically, the Horadam work in [18] focused on the generalized Fibonacci sequence. Later, in [19], he investigated what has since become widely recognized as the Horadam sequence. Their characteristic equation is $r^2=11r-10$. Solving it for r gives the roots of the characteristic equation. According to [2], we have $r_1=10$ and $r_2=1$, so the Binet formula provides a direct way to compute the n-th repunit number without iterating through the sequence

$$r_n={\displaystyle \frac{{10}^n-1}{9}}.$$

(4)

The exponential generating function $E_{a_n}\left(x\right)$ of a sequence ${\left\{a_n\right\}}_{n\ge 0}$ is a power series of the form

$$E_{a_n}\left(x\right)=a_0+a_{1}x+{\displaystyle \frac{a_2{x}^2}{2!}}+\cdots +{\displaystyle \frac{a_n{x}^n}{n!}}+\cdots =\sum _{n=0}^{\infty }{\displaystyle \frac{a_n{x}^n}{n!}}.$$

In the next result, we consider $a_n=r_n$ and make use of Equation (4), the Binet formula for repunit sequence, and then we obtain the classical exponential generating function for the repunit sequence ${\left\{r_n\right\}}_{n\ge 0}$.

Lemma 1

([20]). For all $n\ge 0$, the exponential generating function for the repunit sequence ${\left\{r_n\right\}}_{n\ge 0}$ is

$$E_{r_n}\left(x\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{r_n{x}^n}{n!}}={\displaystyle \frac{e^{10t}-e^t}{9}}.$$

The function $G{r}_{a_n}\left(x\right)$ is designated as the ordinary generating function for the sequence ${\left\{a_n\right\}}_{n\ge 0}$, with

$${\displaystyle G{r}_{a_n}\left(x\right)=\sum _{n=0}^{\infty }{a}_n{x}^n=a_0+a_{1}x+a_2{x}^2+a_3{x}^3+\dots +a_n{x}^n+\dots }$$

In addition, our next result presents the ordinary generating function for the repunit sequence.

Lemma 2

([20]). The ordinary generating function for the repunit sequence ${\left\{r_n\right\}}_{n\ge 0}$, denoted by $G{r}_{r_n}\left(x\right)$, is

$$G{r}_{r_n}\left(x\right)={\displaystyle \frac{x}{1-11x+10x^2}}.$$

Next, we present a compilation of identities pertaining to the repunit sequence ${\left\{r_n\right\}}_{n\ge 0}$, collated from a range of existing literature. These prove invaluable in establishing analogous identities for the Tricomplex Repunit sequence.

In all identities, $m,n$, and s are non-negative integers, and ${\left\{r_n\right\}}_{n\ge 0}$ is the repunit sequence.

$$\begin{array}{cc}\hfill r_{n+1}-r_n={10}^n;& \end{array}$$

(5)

$$\begin{array}{cc}\hfill r_{n+1}-10r_{n-1}={10}^n+1;& \end{array}$$

(6)

$$\begin{array}{cc}\hfill r_{2n+1}-r_{2n}={10}^{2n};& \end{array}$$

(7)

$$\begin{array}{cc}\hfill r_m{r}_{n+1}-10r_{m-1}{r}_n=r_{m+n};& \end{array}$$

(8)

$$\begin{array}{cc}\hfill r_n{r}_{m+1}-r_{n-1}{r}_m={10}^n{r}_{m-n};& \end{array}$$

(9)

$$\begin{array}{cc}\hfill r_{m+s}{r}_{m+n}-r_m{r}_{m+s+n}={10}^m{r}_s{r}_n& \left[\mathrm{T}\mathrm{a}\mathrm{g}\mathrm{i}\mathrm{u}\mathrm{r}\mathrm{i}\text{-}\mathrm{V}\mathrm{a}\mathrm{j}\mathrm{d}\mathrm{a}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{y}\right].\end{array}$$

(10)

Equations (5)–(7), can be consulted at [21], while Equations (8) and (9) can be obtained at [3] (Proposition 3 and 5). Finally, the Tagiuri-Vajda identity for the repunit sequence appeared in previous work [2], Theorem 3.1.

Now, consider the sequence of partial sums ${\sum }_{k=0}^n{r}_k=r_0+r_1+r_2+\cdots +r_n$, for $n\ge 0$, where ${\left\{r_n\right\}}_{n\ge 0}$ is the repunit sequence, which is related with these sums:

Lemma 3

([2], Proposition 6.1). Let ${\left\{r_n\right\}}_{n\ge 0}$ be the repunit sequence, then

$$\begin{array}{ccc}\hfill \left(a\right)& {\sum }_{k=0}^n{r}_k& ={\displaystyle \frac{10r_n-n}{9}},\hfill \\ \hfill \left(b\right)& {\sum }_{k=0}^n{r}_{2k}& ={\displaystyle \frac{{10}^2{r}_{2n}-n{r}_2}{99}},\hfill \\ \hfill \left(c\right)& {\sum }_{k=0}^n{r}_{2k+1}& ={\displaystyle \frac{r_{2n+3}-(n+1)r_2}{99}}.\hfill \end{array}$$

Finally, in [22] (Definition 1), the authors present an extension of the repunit sequence with a negative index, namely, the relations hold $r_{-n}=-{\displaystyle \frac{r_n}{{10}^n}}$ for all non-negative integers n.

3. Binet’s Formula and Generating Function for Tricomplex Repunit

In this section, we present the characteristic equation of the Tricomplex Repunit sequence and provide the Binet formula. Furthermore, we display the exponential generating function and the generating function of the Tricomplex Repunit sequence.

Making $I=(1,1,1)$, a direct calculation shows that

$$\begin{array}{c}\hfill 10t{r}_1+I=t{r}_2,10t{r}_2+I=t{r}_3,\dots ,10t{r}_{n-1}+I=t{r}_n,\end{array}$$

replacing n by $n+1$, we can rewrite this recurrence relation in the form $10t{r}_n+I=t{r}_{n+1}$. And still, subtracting $t{r}_{n+1}-t{r}_n$, we obtain another equivalent recurrence relation for this sequence. Challenging the ordinary, note that

$$\begin{array}{ccc}\hfill t{r}_{n+1}-t{r}_n& =& (10t{r}_n+I)-(10t{r}_{n-1}+I)\hfill \\ & =& 10t{r}_n-10t{r}_{n-1}.\hfill \end{array}$$

So, for all $n\ge 1$, we have $t{r}_{n+1}=11t{r}_n-10t{r}_{n-1}$, where $t{r}_0=(0,1,11)$ and $t{r}_1=(1,11,111)$ are initial terms.

This preliminary discussion guarantees the next result:

Proposition 1.

The Tricomplex Repunit sequence ${\left\{t{r}_n\right\}}_{n\ge 0}$ satisfies the recurrence relation

$$t{r}_{n+2}=11t{r}_{n+1}-10t{r}_n;n\ge 0,$$

(11)

with initial condition $t{r}_0=(0,1,11)$, and $t{r}_1=(1,11,111)$.

It can thus be demonstrated that a sequence derived from the repunit sequence, the Tricomplex Repunit, in vector form, satisfies the equation referenced in Equation (11), that is, in each coordinate, the same Horadam recurrence relation as that satisfied by the repunit recurrence, as referenced in Equation (1).

We see that the Equation (11) is a second-order characteristic equation in each coordinate, that is

$$\begin{array}{ccc}\hfill t{r}_{n+2}& =& 11t{r}_n-10t{r}_{n+1}\hfill \\ \hfill (r_{n+2},r_{n+3},r_{n+4})& =& (11r_n-10r_{n+1},11r_{n+1}-10r_{n+2},11r_{n+2}-10r_{n+3}).\hfill \end{array}$$

So, the vector Equation (11) is equivalent to the following vector characteristic equation:

$$(x^2-11x+10,y^2-11y+10,z^2-11z+10)=(0,0,0).$$

(12)

Note that the Equation (12) is again a second-order vector recurrence relation associated with the Tricomplex Repunit numbers ${\left\{t{r}_n\right\}}_{n\ge 0}$.

By employing the Binet formula for the n-th repunit number, Equation (4), in each coordinate, we have demonstrated the Binet formula for the n-th Tricomplex Repunit sequence.

Theorem 1

(Binet’s formula). For all $n\in \mathbb{N}$, we have

$$(r_n,r_{n+1},r_{n+2})=\left({\displaystyle \frac{{10}^n-1}{9}},{\displaystyle \frac{{10}^{n+1}-1}{9}},{\displaystyle \frac{{10}^{n+2}-1}{9}}\right).$$

(13)

Let ${\mathcal{E}}_{t{r}_n}\left(x\right)=\left(E_{r_n}\left(x\right),E_{r_{n+1}}\left(x\right),E_{r_{n+1}}\left(x\right)\right)$ be the exponential generating function for the Tricomplex Repunit sequence ${\left\{t{r}_n\right\}}_{n\ge 0}$. By combining Theorem 1 and Lemma 1, one deduces the next result.

Proposition 2.

For all $n\ge 0$, the exponential generating function for the Tricomplex Repunit sequence ${\left\{t{r}_n\right\}}_{n\ge 0}$ is

$${\mathcal{E}}_{t{r}_n}\left(x\right)=\left({\displaystyle \frac{e^{10x}-e^x}{9}},{\displaystyle \frac{10e^{10x}-e^x}{9}},{\displaystyle \frac{{10}^2{e}^{10x}-e^x}{9}}\right).$$

Let ${\mathcal{GF}}_{t{r}_n}\left(x\right)=\left(G{r}_{r_n}\left(x\right),G{r}_{r_{n+1}}\left(x\right),G{r}_{r_{n+1}}\left(x\right)\right)$ be the generating function for the Tricomplex Repunit sequence ${\left\{t{r}_n\right\}}_{n\ge 0}$.

We then proceed to present the ordinary generating function for the Tricomplex Repunit sequence.

Proposition 3.

The ordinary generating function for the Tricomplex Repunit ${\left\{t{r}_n\right\}}_{n\ge 0}$, denoted by ${\mathcal{GF}}_{t{r}_n}\left(x\right)$, is

$${\mathcal{GF}}_{t{r}_n}\left(x\right)=\left({\displaystyle \frac{x}{1-11x+10x^2}},{\displaystyle \frac{1}{1-11x+10x^2}},{\displaystyle \frac{11-x}{1-11x+10x^2}}\right).$$

(14)

Proof. 

The first coordinate of the Equation (14) is a direct application of the Lemma 2.

Note that ${\displaystyle G{r}_{r_{n+2}}\left(x\right)=\sum _{n=0}^{\infty }{r}_{n+2}{x}^n}$. Then, by expanding equation $x^{2}G{r}_{r_{n+1}}\left(x\right)$, we obtain

$$\begin{array}{ccc}\hfill x^{2}G{r}_{r_{n+1}}\left(x\right)& =& r_2{x}^2+r_3{x}^3+\cdots \hfill \\ \hfill x^{2}G{r}_{r_{n+1}}\left(x\right)& =& G{r}_{r_n}\left(x\right)-x\hfill \\ \hfill x^{2}G{r}_{r_{n+1}}\left(x\right)& =& {\displaystyle \frac{11x^2-x^3}{1-11x+10x^2}}\hfill \end{array}$$

This proves the third coordinate of the vector defined in Equation (14). Proof of the second coordinate of the vector defined in Equation (14) is performed similarly. □

4. Elementary Properties of Tricomplex Repunit Numbers

In this section, we establish a similar relation with some known identities for the repunit sequence, represented by the notation ${\left\{r_n\right\}}_{n\ge 0}$, where n is an integer. This allows us to derive some symmetrical identities for the Tricomplex Repunit sequence, represented by the notation ${\left\{t{r}_n\right\}}_{n\ge 0}$, where n is an integer.

It follows directly from Equation (5) that the difference between two successive Tricomplex Repunit sequence ${\left\{t{r}_n\right\}}_{n\ge 0}$ is

Proposition 4.

For all integers $n\ge 0$, we have $t{r}_{n+1}-t{r}_n=({10}^n,{10}^{n+1},{10}^{n+2})$, where ${\left\{t{r}_n\right\}}_{n\ge 0}$ is the Tricomplex Repunit sequence.

The next result states other differences between the two elements of ${\left\{t{r}_n\right\}}_{n\ge 0}$.

Proposition 5.

For all integers $n\ge 1$, we have

$$t{r}_{n+1}-10t{r}_{n-1}=({10}^n+1,{10}^{n+1}+1,{10}^{n+2}+1),$$

where ${\left\{t{r}_n\right\}}_{n\ge 0}$ is the Tricomplex Repunit sequence.

Proof. 

Making use of Equation (6), a direct calculation shows that

$$\begin{array}{ccc}\hfill t{r}_{n+1}-10t{r}_{n-1}& =& (r_{n+1},r_{n+2},r_{n+3})-10(r_{n-1},r_n,r_{n+1})\hfill \\ & =& (r_{n+1}-10r_{n-1},r_{n+2}-10r_n,r_{n+3}-10r_{n+1})\hfill \\ & =& ({10}^n+1,{10}^{n+1}+1,{10}^{n+2}+1),\hfill \end{array}$$

as required. □

The difference between an even index term and an odd index term of the Tricomplex Repunit sequence ${\left\{t{r}_n\right\}}_{n\ge 0}$ is:

Proposition 6.

For all integers $n\ge 0$, the difference $t{r}_{2n+1}-t{r}_{2n}$ is a vector of the form

$${10}^{2n}(1,{10}^2,{10}^4),$$

where ${\left\{t{r}_n\right\}}_{n\ge 0}$ is the Tricomplex Repunit sequence.

Proof. 

According to Equation (7), we have

$$\begin{array}{ccc}\hfill t{r}_{2n+1}-t{r}_{2n}& =& (r_{2n+1},r_{2n+2},r_{2n+3})-(r_{2n},r_{2n+1},r_{2n+2})\hfill \\ & =& (r_{2n+1}-r_{2n},r_{2n+2}-r_{2n+1},r_{2n+3}-r_{2n+2})\hfill \\ & =& ({10}^{2n},{10}^{2(n+1)},{10}^{2(n+2)}),\hfill \end{array}$$

as required. □

The next two results for Tricomplex Repunit numbers ${\left\{t{r}_n\right\}}_{n\ge 0}$ are obtained making use of Equation (8).

Proposition 7.

For any non-negative integers m and n, the Tricomplex Repunit sequence satisfies

$$t{r}_{m}t{r}_{n+1}-10t{r}_{m-1}t{r}_n=\left(r_{m+n}+2r_{m+n+3},r_{m+n+4}+2r_{m+n+1},3r_{m+n+2}\right),$$

(15)

where ${\left\{t{r}_n\right\}}_{n\ge 0}$ is the Tricomplex Repunit sequence.

Proof. 

We have

$$\begin{array}{cc}\hfill t{r}_{m}t{r}_{n+1}=& (r_m,r_{m+1},r_{m+2})\times (r_{n+1},r_{n+2},r_{n+3})\hfill \\ \hfill =& (r_m{r}_{n+1}+r_{m+1}{r}_{n+3}+r_{m+2}{r}_{n+2},\hfill \\ & r_{m+2}{r}_{n+3}+r_m{r}_{n+2}+r_{m+1}{r}_{n+1},\hfill \\ \hfill & r_{m+1}{r}_{n+2}+r_m{r}_{m+3}+r_{m+2}{r}_{n+1}),\hfill \end{array}$$

(16)

and

$$\begin{array}{cc}\hfill 10t{r}_{m-1}t{r}_n=& 10(r_{m-1}{r}_n+r_m{r}_{n+2}+r_{m+1}{r}_{n+1},\hfill \\ \hfill & r_{m+1}{r}_{n+2}+r_{m-1}{r}_{n+1}+r_m{r}_n,\hfill \\ \hfill & r_m{r}_{n+1}+r_{m-1}{r}_{n+2}+r_{m+1}{r}_n).\hfill \end{array}$$

(17)

To obtain the first coordinate of Equation (15), we subtract Equation (16) from Equation (17), that is,

$$\begin{array}{cc}\hfill & (r_m{r}_{n+1}-10·r_{m-1}{r}_n)+(r_{m+1}{r}_{n+3}-10·r_m{r}_{n+2})\hfill \\ & +(r_{m+2}{r}_{n+2}-10·r_{m+1}{r}_{n+1})\hfill \\ \hfill =& (r_m{r}_{n+1}-10·r_{m-1}{r}_n)+(r_{(m+1)}{r}_{(n+1)+2}-10·r_m{r}_{(n+2)})\hfill \\ \hfill & +(r_{(m+2)}{r}_{(n+1)+1}-10·r_{(m-1)+2}{r}_{n+1}).\hfill \end{array}$$

According to Equation (8), we have

$$\begin{array}{cc}\hfill & (r_m{r}_{n+1}-10·r_{m-1}{r}_n)+(r_{m+1}{r}_{n+3}-10·r_m{r}_{n+2})\hfill \\ & +(r_{m+2}{r}_{n+2}-10·r_{m+1}{r}_{n+1})\hfill \\ \hfill =& r_{m+n}+r_{(m+1)+(n+2)}+r_{(m+2)+(n+1)}\hfill \\ \hfill =& r_{m+n}+r_{m+n+3}+r_{m+n+3}\hfill \\ \hfill =& r_{m+n}+2r_{m+n+3}\hfill \end{array}$$

This proves the first coordinate of the vector defined in Equation (15). Proofs of the second and third coordinates of the vector defined in Equation (15) are performed similarly, also using Equation (8). □

Proposition 8.

For any non-negative integers $m,n$, we have

$$t{r}_{m+n}=r_{m}t{r}_{n+1}-10·r_{m-1}t{r}_n,$$

(18)

where ${\left\{r_n\right\}}_{n\ge 0}$ is the repunit sequence, and ${\left\{t{r}_n\right\}}_{n\ge 0}$ is the Tricomplex Repunit sequence.

Proof. 

Note that

$$\begin{array}{ccc}\hfill t{r}_{m+n}& =& \left(r_{m+n},r_{m+n+1},r_{m+n+2}\right)\hfill \\ & =& \left(r_m{r}_{n+1}-10r_{m-1}{r}_n,r_m{r}_{n+2}-10r_{m-1}{r}_{n+1},r_m{r}_{n+3}-10r_{m-1}{r}_{n+2}\right)\hfill \\ & =& r_m(r_{n+1},r_{n+2},r_{n+3})-10r_{m-1}(r_n,r_{n+1},r_{n+2}).\hfill \end{array}$$

Thus, we achieve the desired outcome. □

The next result follows directly from Proposition 8. In Equation (18), if we set $n=m-1$, it yields item (a), and setting $n=m$ yields item (b).

Corollary 1.

For all integers $m\ge 1$, we have

$$\begin{array}{ccc}\hfill \left(a\right)& t{r}_{2m-1}& =r_{m}t{r}_m-10r_{m-1}t{r}_{m-1},\hfill \\ \hfill \left(b\right)& t{r}_{2m}& =r_{m}t{r}_{m+1}-10r_{m-1}t{r}_m,\hfill \end{array}$$

where ${\left\{r_n\right\}}_{n\ge 0}$ is the repunit sequence, and ${\left\{t{r}_n\right\}}_{n\ge 0}$ is the Tricomplex Repunit sequence.

The next two results for Tricomplex Repunit numbers ${\left\{t{r}_n\right\}}_{n\ge 0}$ are obtained with the use of Equation (9). The first is similar to Proposition 8, and we omit the proof of the next result in the interest of brevity.

Proposition 9.

For any non-negative integers $m,n$, with $m\ge n$, we have

$${10}^{n}t{r}_{m-n}=r_{n}t{r}_{m+1}-r_{n-1}t{r}_m,$$

where ${\left\{r_n\right\}}_{n\ge 0}$ is the repunit sequence, and ${\left\{t{r}_n\right\}}_{n\ge 0}$ is the Tricomplex Repunit sequence.

Proposition 10.

For any non-negative integers m and n, we have the following identities

$$\begin{array}{cc}& t{r}_{n}t{r}_{m+1}-t{r}_{n-1}t{r}_m\hfill \\ =& {10}^n\left(r_{m-n}+110r_{m-n+3},100r_{m-n+4}+11r_{m-n+1},111r_{m-n+2}\right),\hfill \end{array}$$

(19)

where ${\left\{r_n\right\}}_{n\ge 0}$ is the repunit sequence, and ${\left\{t{r}_n\right\}}_{n\ge 0}$ is the Tricomplex Repunit sequence.

Proof. 

We have

$$\begin{array}{cc}\hfill t{r}_{n}t{r}_{m+1}=& \left(r_n,r_{n+1},r_{n+2}\right)\times \left(r_{m+1},r_{m+2},r_{m+3}\right)\hfill \\ \hfill =& (r_n{r}_{m+1}+r_{n+1}{r}_{m+3}+r_{n+2}{r}_{m+2},\hfill \\ \hfill & r_{n+2}{r}_{m+3}+r_n{r}_{m+2}+r_{n+1}{r}_{m+1},\hfill \\ \hfill & r_{n+1}{r}_{m+2}+r_n{r}_{m+3}+r_{n+2}{r}_{m+1})\hfill \end{array}$$

(20)

while,

$$\begin{array}{cc}\hfill t{r}_{n-1}t{r}_n=& \left(r_{n-1},r_n,r_{n+1}\right)\times \left(r_m,r_{m+1},r_{m+2}\right)\hfill \\ \hfill =& (r_{n-1}{r}_m+r_n{r}_{m+2}+r_{n+1}{r}_{m+1},\hfill \\ \hfill & r_{n+1}{r}_{m+2}+r_{n-1}{r}_{m+1}+r_n{r}_m,\hfill \\ \hfill & r_n{r}_{m+1}+r_{n-1}{r}_{m+2}+r_{n+1}{r}_m)\hfill \end{array}$$

(21)

To obtain the first coordinate of the vector defined in Equation (19), we subtract Equation (20) from Equation (21), that is,

$$\begin{array}{cc}\hfill & (r_n{r}_{m+1}-r_{n-1}{r}_m)+(r_{n+1}{r}_{m+3}-r_{n+2}{r}_{m+2})\hfill \\ & +(r_{n+2}{r}_{m+2}-r_{n+3}{r}_{m+1})\hfill \\ \hfill =& (r_n{r}_{m+1}-r_{n-1}{r}_m)+\left(r_{(n+1)}{r}_{(m+1)+2}-r_n{r}_{(m+2)}\right)\hfill \\ & +\left(r_{(n+2)}{r}_{(m+1)+1}-r_{(n-3)+2}{r}_{(m+1)}\right).\hfill \end{array}$$

By Equation (9), we have

$$\begin{array}{ccc}& & (r_n{r}_{m+1}-r_{n-1}{r}_m)+(r_{n+1}{r}_{m+3}-r_{n+2}{r}_{m+2})\hfill \\ & & +(r_{n+2}{r}_{m+2}-r_{n+3}{r}_{m+1})\hfill \\ & =& {10}^n·r_{m-n}+{10}^{n+1}·r_{(m+2)-(n-1)}+{10}^{n+2}·r_{(m+1)-(n-2)}\hfill \\ & =& {10}^n(r_{m-n}+10r_{m-n+3}+100r_{m-n+3})\hfill \\ & =& {10}^n(r_{m-n}+110r_{m-n+3})\hfill \end{array}$$

This proves the first coordinate of the vector defined in Equation (19). Proofs of the second and third coordinates of the vector defined in Equation (19) are performed similarly using Equation (9). □

5. Classical Identities for Tricomplex Repunit Numbers

This section presents some fundamental identities for the Tricomplex Repunit sequence, denoted by ${\left\{t{r}_n\right\}}_{n\ge 0}$. These include the Tagiuri–Vajda, Catalan, Cassini, and d’Ocagne identities.

First, we prove the Tagiuri–Vajda identity for the Tricomplex Repunit sequence ${\left\{t{r}_n\right\}}_{n\ge 0}$.

Theorem 2.

Let $m,s,k$ be any non-negative integers. We have

$$t{r}_{m+s}t{r}_{m+k}-t{r}_{m}t{r}_{m+s+k}={10}^m·r_3{r}_s{r}_k\left(1,1,1\right),$$

(22)

where ${\left\{r_n\right\}}_{n\ge 0}$ is the repunit sequence, and ${\left\{t{r}_n\right\}}_{n\ge 0}$ is the Tricomplex Repunit sequence.

Proof. 

We have

$$\begin{array}{ccc}\hfill t{r}_{m+s}t{r}_{m+k}& =& (r_{m+s},r_{m+s+1},r_{m+s+2})\times (r_{m+k},r_{m+k+1},r_{m+k+2})\hfill \\ \hfill & =& (r_{m+s}{r}_{m+k}+r_{m+s+1}{r}_{m+k+2}+r_{m+s+2}{r}_{m+k+1},\hfill \\ \hfill & & r_{m+s+2}{r}_{m+k+2}+r_{m+s}{r}_{m+k+1}+r_{m+s+1}{r}_{m+k},\hfill \\ \hfill & & r_{m+s+1}{r}_{m+k+1}+r_{m+s}{r}_{m+k+2}+r_{m+s+2}{r}_{m+k}),\hfill \end{array}$$

(23)

and

$$\begin{array}{ccc}\hfill t{r}_{m}t{r}_{m+s+k}& =& (r_m,r_{m+1},r_{m+2})\times (r_{m+s+k},r_{m+s+k+1},r_{m+s+k+2})\hfill \\ \hfill & =& (r_m{r}_{m+s+k}+r_{m+1}{r}_{m+s+k+2}+r_{m+2}{r}_{m+s+k+1},\hfill \\ \hfill & & r_{m+2}{r}_{m+s+k+2}+r_m{r}_{m+s+k+1}+r_{m+1}{r}_{m+s+k},\hfill \\ \hfill & & r_{m+1}{r}_{m+s+k+1}+r_m{r}_{m+s+k+2}+r_{m+2}{r}_{m+s+k}).\hfill \end{array}$$

(24)

To obtain the first coordinate of the vector defined in Equation (22), Let us make the difference between the Equations (23) and (24), that is,

$$\begin{array}{ccc}& & (r_{m+s}{r}_{m+k}-r_m{r}_{m+s+k})+(r_{m+s+1}{r}_{m+k+2}-r_{m+1}{r}_{m+s+k+2})\hfill \\ & & +(r_{m+s+2}{r}_{m+k+1}-r_{m+2}{r}_{m+s+k+1})\hfill \\ & =& \left(r_{m+s}{r}_{m+k}-r_m{r}_{m+s+k}\right)+\left(r_{(m+1)+s}{r}_{(m+1)+(k+1)}-r_{(m+1)}{r}_{(m+1)+s+(k+1)}\right)\hfill \\ & & +\left(r_{(m+2)+s}{r}_{(m+2)+(k-1)}-r_{(m+2)}{r}_{(m+2)+s+(k-1)}\right).\hfill \end{array}$$

According to Equation (10), we have

$$\begin{array}{ccc}& & (r_{m+s}{r}_{m+k}-r_m{r}_{m+s+k})+(r_{m+s+1}{r}_{m+k+2}-r_{m+1}{r}_{m+s+k+2})\hfill \\ & & +(r_{m+s+2}{r}_{m+k+1}-r_{m+2}{r}_{m+s+k+1})\hfill \\ & =& \left({10}^m·r_s{r}_k\right)+\left({10}^{m+1}·r_s{r}_{k+1}\right)+\left({10}^{m+2}·r_s{r}_{k-1}\right)\hfill \\ & =& {10}^m·r_s(r_k+10r_{k+1}+{10}^2{r}_{k-1})\hfill \\ & =& {10}^m·r_s\left(r_k+10(11r_k-10r_{k-1})+{10}^2{r}_{k-1}\right)\hfill \\ & =& {10}^m·111·r_s{r}_k.\hfill \end{array}$$

This proves the first coordinate of the vector defined in Equation (22). Proofs of the second and third coordinates of the vector defined in Equation (22) are performed similarly using Equation (10), the Tagiuri–Vajda identity for the repunit sequence. □

The remaining identities are derived as a consequence of the Tagiuri–Vajda identity, as demonstrated subsequently.

We now proceed by presenting the d’Ocagne identity.

Proposition 11

(d’Ocagne’s identity). Let $m,n$ be any non-negative integers. For $m\ge n$, we have

$$t{r}_{m}t{r}_{n+1}-t{r}_{m+1}t{r}_n=-{10}^n·r_3{r}_{m-n}\left(1,1,1\right),$$

where ${\left\{r_n\right\}}_{n\ge 0}$ is the repunit sequence, and ${\left\{t{r}_n\right\}}_{n\ge 0}$ is the Tricomplex Repunit sequence.

Proof. 

First, we consider $k=n-m$ and $s=1$ in Theorem 2, so

$$\begin{array}{cc}\hfill t{r}_{m+1}t{r}_n-t{r}_{m}t{r}_{n+1}=& {10}^m·r_3{r}_1{r}_{n-m}\left(1,1,1\right)\\ \hfill =& {10}^m·r_3{r}_1{r}_{-(m-n)}\left(1,1,1\right).\end{array}$$

Since $r_{-n}=-{\displaystyle \frac{r_n}{{10}^n}}$, for all n, we have

$$\begin{array}{cc}\hfill {10}^m·r_3{r}_1{r}_{-(m-n)}=& {10}^m·r_3{r}_1\left(-{\displaystyle \frac{r_{m-n}}{{10}^{m-n}}}\right)\\ \hfill =& -{10}^n·r_3{r}_1{r}_{m-n}.\end{array}$$

As $r_1=1$, we obtain the result. □

Similar to Proposition 11, we have the Catalan identity.

Proposition 12.

Let $m,n$ be any non-negative integers. For $m\ge n$, we have

$$t{r}_{m-n}t{r}_{m+n}-{\left(t{r}_m\right)}^2=-{10}^{m-n}·r_3{\left(r_n\right)}^2\left(1,1,1\right),$$

where ${\left\{r_n\right\}}_{n\ge 0}$ is the repunit sequence, and ${\left\{t{r}_n\right\}}_{n\ge 0}$ is the Tricomplex Repunit sequence.

Proof. 

Simply take $s=n$ and $k=-n$ in Theorem 2. We have

$$\begin{array}{ccc}\hfill t{r}_{m-n}t{r}_{m+n}-{\left(t{r}_m\right)}^2& =& {10}^m·r_3{r}_n{r}_{-n}\left(1,1,1\right),\hfill \end{array}$$

as

$$\begin{array}{ccc}\hfill {10}^m·r_3{r}_n{r}_{-n}& =& {10}^m·r_3{r}_n\left(-{\displaystyle \frac{r_n}{{10}^n}}\right),\hfill \\ & =& -{10}^{m-n}·r_3{\left(r_n\right)}^2,\hfill \end{array}$$

we obtain the result. □

The result for $n=1$ it can be derived with minimal effort from Proposition 12, which states that

Corollary 2

(Cassini’s identity). For all integers $m\ge 1$, we have

$$t{r}_{m-1}t{r}_{m+1}-{\left(t{r}_m\right)}^2=-{10}^{m-1}·r_3\left(1,1,1\right),$$

where ${\left\{r_n\right\}}_{n\ge 0}$ is the repunit sequence, and ${\left\{t{r}_n\right\}}_{n\ge 0}$ is the Tricomplex Repunit sequence.

By making the substitution of $m=2n$ in Corollary 2, we obtain

Corollary 3.

For all integers $m\ge 1$, we have

$$t{r}_{2n-1}t{r}_{2n+1}-{\left(t{r}_{2n}\right)}^2=-{10}^{2n-1}·r_3\left(1,1,1\right),$$

where ${\left\{r_n\right\}}_{n\ge 0}$ is the repunit sequence, and ${\left\{t{r}_n\right\}}_{n\ge 0}$ is the Tricomplex Repunit sequence.

6. Sum of Terms Involving the Tricomplex Repunit Numbers

In this section, we present the results of our investigation into the partial sums of the terms of the Tricomplex Repunit numbers with a variable integer number of terms. Consider the sequence of partial sums, defined as the sum of the terms of the Tricomplex Repunit sequence, for a given integer value of n,

$$\sum _{k=0}^{n}t{r}_k=t{r}_0+t{r}_1+t{r}_2+\cdots +t{r}_n,$$

for $n\ge 0$, where ${\left\{t{r}_n\right\}}_{n\ge 0}$ is the Tricomplex Repunit sequence.

Proposition 13.

Let ${\left\{t{r}_n\right\}}_{n\ge 0}$ be the Tricomplex Repunit sequence, we have the following identities:

$$\begin{array}{ccc}\hfill \left(a\right)\sum _{k=0}^{n}t{r}_k& =& \left({\displaystyle \frac{10r_n-n}{9}},{\displaystyle \frac{10r_{n+1}-(n+1)}{9}},{\displaystyle \frac{10r_{n+2}-(n+11)}{9}}\right),\hfill \\ \hfill \left(b\right)\sum _{k=0}^{n}t{r}_{2k}& =& \left({\displaystyle \frac{{10}^2{r}_{2n}-n{r}_2}{99}},{\displaystyle \frac{r_{2n+3}-(n+1)r_2}{99}},{\displaystyle \frac{{10}^2{r}_{2n+2}-(n+1)r_2}{99}}\right),\hfill \\ \hfill \left(c\right)\sum _{k=0}^{n}t{r}_{2k+1}& =& \left({\displaystyle \frac{r_{2n+3}-(n+1)r_2}{99}},{\displaystyle \frac{{10}^2{r}_{2n+2}-(n+1)r_2}{99}},{\displaystyle \frac{r_{2n+5}-(n+11)r_2}{99}}\right).\hfill \end{array}$$

where ${\left\{r_n\right\}}_{n\ge 0}$ is the repunit sequence.

Proof. 

(a) It follows from the definition of partial sum of terms of the Tricomplex Repunit numbers that

$$\begin{array}{ccc}\hfill \sum _{k=0}^{n}t{r}_k& =& t{r}_0+t{r}_1+\cdots +t{r}_n\hfill \\ & =& (r_0,r_1,r_2)+(r_1,r_2,r_3)+\cdots +(r_n,r_{n+1},r_{n+2})\hfill \\ & =& (r_0+r_1+\cdots +r_n,r_1+r_2+\cdots +r_{n+1},r_2+r_3+\cdots +r_{n+2})\hfill \\ & =& \left(\sum _{k=0}^n{r}_k,\sum _{k=1}^{n+1}{r}_k,\sum _{k=2}^{n+2}{r}_k\right)\hfill \\ & =& \left(\sum _{k=0}^n{r}_k,0+\sum _{k=1}^{n+1}{r}_k,0+1+\sum _{k=2}^{n+2}{r}_k-1\right)\hfill \\ & =& \left(\sum _{k=0}^n{r}_k,\sum _{k=0}^{n+1}{r}_k,\sum _{k=0}^{n+2}{r}_k\right)-(0,0,1).\hfill \end{array}$$

According to Lemma 3, item (a), we have

$$\begin{array}{ccc}\hfill \sum _{k=0}^{n}t{r}_k& =& \left({\displaystyle \frac{10r_n-n}{9}},{\displaystyle \frac{10r_{n+1}-(n+1)}{9}},{\displaystyle \frac{10r_{n+2}-(n+2)}{9}}-1\right),\hfill \end{array}$$

and we obtain the result.

(b) We have

$$\begin{array}{ccc}\hfill \sum _{k=0}^{n}t{r}_{2k}& =& t{r}_0+t{r}_2+\cdots +t{r}_{2n}\hfill \\ & =& (r_0,r_1,r_2)+(r_2,r_3,r_4)+\cdots +(r_{2n},r_{2n+1},r_{2(n+1)})\hfill \\ & =& (r_0+r_2+\cdots +r_{2n},r_1+r_3+\cdots +r_{2n+1},r_2+r_3+\cdots +r_{2(n+1)})\hfill \\ & =& \left(\sum _{k=0}^n{r}_k,\sum _{k=0}^n{r}_{2k+1},\sum _{k=1}^{n+1}{r}_{2k}\right)=\left(\sum _{k=0}^n{r}_k,\sum _{k=0}^n{r}_{2k+1},\sum _{k=0}^{n+1}{r}_{2k}\right).\hfill \end{array}$$

We obtain the result by making use of Lemma 3, items (b) and (c).

(c) Similarly, see that

$$\begin{array}{ccc}\hfill \sum _{k=0}^{n}t{r}_{2k+1}& =& t{r}_1+t{r}_3+\cdots +t{r}_{2n+1}\hfill \\ & =& (r_1,r_2,r_3)+(r_3,r_4,r_5)+\cdots +(r_{2n+1},r_{2n+2},r_{2n+3})\hfill \\ & =& (r_1+r_3+\cdots +r_{2n+1},r_2+r_4+\cdots +r_{2n+2},r_3+r_5+\cdots +r_{2(n+1)+1})\hfill \\ & =& \left(\sum _{k=0}^n{r}_{2k+1},\sum _{k=1}^{n+1}{r}_{2k},\sum _{k=1}^{n+1}{r}_{2k+1}\right)=\left(\sum _{k=0}^n{r}_{2k+1},\sum _{k=0}^{n+1}{r}_{2k},\sum _{k=0}^{n+1}{r}_{2k+1}-1\right).\hfill \end{array}$$

We obtain the result by making use of Lemma 3, items (b) and (c). □

A direct consequence of the Proposition 13 is the next result.

Proposition 14.

For all integers $n\ge 0$, we have the following identities:

$${\displaystyle \sum _{j=0}^n{(-1)}^{k}t{r}_k=\left(\frac{{10}^2{r}_{2n}+r_2-r_{2n+3}}{99},\frac{r_{2n+3}-{10}^2{r}_{2n+2}}{99},\frac{{10}^2{r}_{2n+2}+10r_2-r_{2n+5}}{99}\right);}$$

if the last term is negative, and

$${\displaystyle \sum _{j=0}^n{(-1)}^{k}t{r}_k=\left(\frac{r_{2n+3}-r_2-{10}^2{r}_{2n}}{99},\frac{{10}^2{r}_{2n+2}-r_{2n+3}}{99},\frac{r_{2n+5}-10r_2-{10}^2{r}_{2n+2}}{99}\right);}$$

and if the last term is positive, where ${\left\{r_n\right\}}_{n\ge 0}$ is the repunit sequence, ${\left\{t{r}_n\right\}}_{n\ge 0}$ is the Tricomplex Repunit sequence.

Proof. 

(a) It is first necessary to consider that the last term is negative, which gives rise to the following considerations:

$$\begin{array}{ccc}\hfill \sum _{k=0}^{2n+1}{(-1)}^{k}t{r}_k& =& t{r}_0-t{r}_1+t{r}_2-t{r}_3+\cdots +t{r}_{2n}-t{r}_{2n+1}\hfill \\ & =& (t{r}_0+t{r}_2+\cdots +t{r}_{2n})-(t{r}_1+t{r}_3+\cdots +t{r}_{2n+1})\hfill \\ & =& \sum _{k=0}^{n}t{r}_{2k}-\sum _{k=0}^{n}t{r}_{2k+1}\hfill \end{array}$$

According to Proposition 13, items (b) and (c), it follows that

$$\begin{array}{ccc}& & \sum _{k=0}^{2n+1}{(-1)}^{k}t{r}_k\hfill \\ & =& \left({\displaystyle \frac{{10}^2{r}_{2n}-n{r}_2}{99}},{\displaystyle \frac{r_{2n+3}-(n+1)r_2}{99}},{\displaystyle \frac{{10}^2{r}_{2n+2}-(n+1)r_2}{99}}\right)\hfill \\ & & -\left({\displaystyle \frac{r_{2n+3}-(n+1)r_2}{99}},{\displaystyle \frac{{10}^2{r}_{2n+2}-(n+1)r_2}{99}},{\displaystyle \frac{r_{2n+5}-(n+11)r_2}{99}}\right)\hfill \\ & =& \left({\displaystyle \frac{{10}^2{r}_{2n}+r_2-r_{2n+3}}{99}},{\displaystyle \frac{r_{2n+3}-{10}^2{r}_{2n+2}}{99}},{\displaystyle \frac{{10}^2{r}_{2n+2}+10r_2-r_{2n+5}}{99}}\right).\hfill \end{array}$$

(b) In which case that last term is positive, so

$$\begin{array}{ccc}\hfill \sum _{k=0}^{2(n+1)}{(-1)}^{k}t{r}_k& =& t{r}_0-t{r}_1+t{r}_2-t{r}_3+\cdots +t{r}_{2n}-t{r}_{2n+1}+t{r}_{2n+2}\hfill \\ & =& \sum _{k=0}^{n+1}t{r}_{2k}-\sum _{k=0}^{n}t{r}_{(2k+1)}\hfill \end{array}$$

As in item (a), the result is established by applying Proposition 13. □

The Repunit sequence is a Lucas (Horadam)-type sequence, so in each coordinate, the partial sum expressed here is a particular case of the fundamental summation rule, Equation 33, recently published in [23].

7. Conclusions

In this work, we proposed a novel sequence, a Horadam-type sequence in a three-coordinate vector, which represents a significant advance in the theoretical understanding of these sequences. This study introduced the concept of Tricomplex Repunit numbers and provides a comprehensive analysis of their properties and interrelations with ordinary repunit numbers. We established several identities that elucidate the behavior of these sequences, thus improving our understanding of their properties. Furthermore, we derive the generating function for this novel class of sequences and formulate a Binet-type formula in three-dimensional vector form. In addition, we investigated some classical identities associated with Tricomplex Repunit numbers, including Tagiuri–Vajda, d’Ocagne, Catalan, and Cassini. Our investigation not only deepens the understanding of Tricomplex Repunit sequences but also serves to provide a solid foundation for future research on the tricomplex ring $\mathbb{T}$. By elucidating these identities and providing a framework for Horadam-type recurrence, we aim to inspire further academic work and thus advance related mathematical fields. For instance, we aim to use also special tridiagonal determinants in order to study more about this sequence, motivated by the work of Kizilateş in [24]. We are interested in work about matrix representations and determinantal expansions of this sequence.