A note on Modified Third-order Jacobsthal numbers

TL;DR

This paper introduces the modified third-order Jacobsthal sequence, explores its properties including a Binet-style formula and generating function, contributing to the mathematical understanding of this sequence.

Contribution

It defines a new variant of the third-order Jacobsthal sequence and derives its fundamental properties, such as formulas and generating functions.

Findings

Derived the Binet-style formula for the sequence

Established the generating function for the sequence

Provided properties and relations involving the sequence

Abstract

Modified third-order Jacobsthal sequence is defined in this study. Some properties involving this sequence, including the Binet-style formula and the generating function are also presented.

Equations

$$J_0=0,J_1=1,J_{n+2}=J_{n+1}+2J_n,n\ge 0.$$

$$J_{n+3}^{(3)}=J_{n+2}^{(3)}+J_{n+1}^{(3)}+2J_n^{(3)},J_0^{(3)}=0,J_1^{(3)}=J_2^{(3)}=1,n\ge 0,$$

$$j_{n+3}^{(3)}=j_{n+2}^{(3)}+j_{n+1}^{(3)}+2j_n^{(3)},j_0^{(3)}=2,j_1^{(3)}=1,j_2^{(3)}=5,n\ge 0,$$

$$3J_n^{(3)}+j_n^{(3)}=2^{n+1},$$

$$j_n^{(3)}-3J_n^{(3)}=2j_{n-3}^{(3)},n\ge 3,$$

$$J_{n+2}^{(3)}-4J_{n}^{(3)}=\left\{\begin{array}[]{ccc}-2&\textrm{if}&n\equiv 1\kern-5.0pt\pmod{3}\\ 1&\textrm{if}&n\not\equiv 1\kern-6.00006pt\pmod{3}\end{array}\right.,$$

$$j_{n}^{(3)}-4J_{n}^{(3)}=\left\{\begin{array}[]{ccc}2&\textrm{if}&n\equiv 0\kern-5.0pt\pmod{3}\\ -3&\textrm{if}&n\equiv 1\kern-5.0pt\pmod{3}\\ 1&\textrm{if}&n\equiv 2\kern-5.0pt\pmod{3}\end{array}\right.,$$

$$j_{n+1}^{(3)}+j_n^{(3)}=3J_{n+2}^{(3)},$$

$$j_{n}^{(3)}-J_{n+2}^{(3)}=\left\{\begin{array}[]{ccc}1&\textrm{if}&n\equiv 0\kern-5.0pt\pmod{3}\\ -1&\textrm{if}&n\equiv 1\kern-5.0pt\pmod{3}\\ 0&\textrm{if}&n\equiv 2\kern-5.0pt\pmod{3}\end{array}\right.,$$

$${\left(j_{n-3}^{(3)}\right)}^2+3J_n^{(3)}{j}_n^{(3)}=4^n,$$

$$\sum\limits_{k=0}^{n}J_{k}^{(3)}=\left\{\begin{array}[]{ccc}J_{n+1}^{(3)}&\textrm{if}&n\not\equiv 0\kern-6.00006pt\pmod{3}\\ J_{n+1}^{(3)}-1&\textrm{if}&n\equiv 0\kern-5.0pt\pmod{3}\end{array}\right.$$

$${\left(j_n^{(3)}\right)}^2-9{\left(J_n^{(3)}\right)}^2=2^{n+2}{j}_{n-3}^{(3)},n\ge 3.$$

$$x^3-x^2-x-2=0;x=2,\text{and}x=\frac{-1\pm i\sqrt{3}}{2}.$$

$$J_n^{(3)}=\frac{2}{7}{2}^n-\frac{3+2i\sqrt{3}}{21}{\omega }_1^n-\frac{3-2i\sqrt{3}}{21}{\omega }_2^n$$

$$j_n^{(3)}=\frac{8}{7}{2}^n+\frac{3+2i\sqrt{3}}{7}{\omega }_1^n+\frac{3-2i\sqrt{3}}{7}{\omega }_2^n,$$

$$V_{n}^{(2)}=\frac{A\omega_{1}^{n}-B\omega_{2}^{n}}{\omega_{1}-\omega_{2}}=\left\{\begin{array}[]{ccc}2&\textrm{if}&n\equiv 0\kern-5.0pt\pmod{3}\\ -3&\textrm{if}&n\equiv 1\kern-5.0pt\pmod{3}\\ 1&\textrm{if}&n\equiv 2\kern-5.0pt\pmod{3}\end{array}\right.,$$

$$V_{n+2}^{(2)}=-V_{n+1}^{(2)}-V_n^{(2)},V_0^{(2)}=2\text{and}{V}_1^{(2)}=-3.$$

$$J_n^{(3)}=\frac{1}{7}\left(2^{n+1}-V_n^{(2)}\right)\text{and}{j}_n^{(3)}=\frac{1}{7}\left(2^{n+3}+3V_n^{(2)}\right).$$

$$K_{n+3}^{(3)}=K_{n+2}^{(3)}+K_{n+1}^{(3)}+2K_n^{(3)},$$

$$g_{K_n^{(3)}}(t)=\sum _{n\ge 0}{K}_n^{(3)}{t}^n.$$

$$M_{n}^{(2)}=\left\{\begin{array}[]{ccc}2&\textrm{if}&n\equiv 0\kern-5.0pt\pmod{3}\\ -1&\textrm{if}&n\equiv 1,2\kern-5.0pt\pmod{3}\end{array}\right.$$

$$M_n^{(2)}=-\frac{1}{7}\left(4V_{n+1}^{(2)}-V_n^{(2)}\right)$$

$$147J_n^{(3)}=13K_n^{(3)}+48K_{n-1}^{(3)}+20K_{n-2}^{(3)},$$

$$6K_n^{(3)}=5j_n^{(3)}+3j_{n-1}^{(3)}-5j_{n-2}^{(3)},$$

$$49j_n^{(3)}=43K_n^{(3)}+8K_{n-1}^{(3)}+36K_{n-2}^{(3)},$$

$$K_{n}^{(3)}K_{m}^{(3)}+K_{n+1}^{(3)}K_{m+1}^{(3)}+K_{n+2}^{(3)}K_{m+2}^{(3)}=\left\{\begin{array}[]{c}21\cdot 2^{n+m}\\ +2^{n}\left(M_{m+1}^{(2)}+3M_{m+2}^{(2)}\right)\\ +2^{m}\left(M_{n+1}^{(2)}+3M_{n+2}^{(2)}\right)\\ +3(\omega_{1}^{n}\omega_{2}^{m}+\omega_{1}^{m}\omega_{2}^{n})\end{array}\right\},$$

$${\left(K_n^{(3)}\right)}^2+{\left(K_{n+1}^{(3)}\right)}^2+{\left(K_{n+2}^{(3)}\right)}^2=21\cdot 2^{2n}+2^{n+1}\left(M_{n+1}^{(2)}+3M_{n+2}^{(2)}\right)+6,$$

$$147J_2^{(3)}=147=13\cdot 3+48\cdot 1+20\cdot 3=13K_2^{(3)}+48K_1^{(3)}+20K_0^{(3)}.$$

$$147J_{n+1}^{(3)}$$

$$=13K_n^{(3)}+48K_{n-1}^{(3)}+20K_{n-2}^{(3)}$$

$$+13K_{n-1}^{(3)}+48K_{n-2}^{(3)}+20K_{n-3}^{(3)}$$

Full text

Departamento de Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Casilla 160-C, Concepción, Chile.

E-mails: gcerda@udec.cl

1. Introduction

The Jacobsthal numbers have many interesting properties and applications in many fields of science (see, e.g., [1, 6, 7]). The Jacobsthal numbers $J_{n}$ are defined by the recurrence relation

[TABLE]

Another important sequence is the Jacobsthal-Lucas sequence. This sequence is defined by the recurrence relation $j_{n+2}=j_{n+1}+2j_{n}$, where $j_{0}=2$ and $j_{1}=1$ (see, [7]).

In [5] the Jacobsthal recurrence relation is extended to higher order recurrence relations and the basic list of identities provided by A. F. Horadam [7] is expanded and extended to several identities for some of the higher order cases. For example, the third-order Jacobsthal numbers, $\{J_{n}^{(3)}\}_{n\geq 0}$, and third-order Jacobsthal-Lucas numbers, $\{j_{n}^{(3)}\}_{n\geq 0}$, are defined by

[TABLE]

and

[TABLE]

respectively.

Some of the following properties given for third-order Jacobsthal numbers and third-order Jacobsthal-Lucas numbers are used in this paper (for more details, see [2, 3, 4, 5]). Note that Eqs. (1.8) and (1.12) have been corrected in this paper, since they have been wrongly described in [5].

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Using standard techniques for solving recurrence relations, the auxiliary equation, and its roots are given by

[TABLE]

Note that the latter two are the complex conjugate cube roots of unity. Call them $\omega_{1}$ and $\omega_{2}$, respectively. Thus the Binet formulas can be written as

[TABLE]

and

[TABLE]

respectively. Now, we use the notation

[TABLE]

where $A=-3-2\omega_{2}$ and $B=-3-2\omega_{1}$. Furthermore, note that for all $n\geq 0$ we have

[TABLE]

From the Binet formulas (1.13), (1.14) and Eq. (1.15), we have

[TABLE]

Motivated essentially by the recent works [5], [2] and [4], in this paper we introduce the Modified third-order Jacobsthal sequences and we give some properties, including the Binet-style formula and the generating functions for these sequences. Some identities involving these sequences are also provided.

2. The Modified Third-order Jacobsthal sequence, Binet’s formula and the generating function

The principal goals of this section will be to define the Modified third-order Jacobsthal sequence and to present some elementary results involving it.

First of all, we define the Modified third-order Jacobsthal sequence, denoted by $\{K_{n}^{(3)}\}_{n\geq 0}$, which first terms are $\{3,1,3,10,15,31,66,...\}$. This sequence is defined recursively by

[TABLE]

with initial conditions $K_{0}^{(3)}=3$, $K_{1}^{(3)}=1$ and $K_{2}^{(3)}=3$. Note that $K_{n}^{(3)}=J_{n}^{(3)}+2J_{n-1}^{(3)}+6J_{n-2}^{(3)}$, where $J_{n}^{(3)}$ is the $n$-th third-order Jacobsthal number.

In order to find the generating function for the Modified third-order Jacobsthal sequence, we shall write the sequence as a power series where each term of the sequence correspond to coefficients of the series. As a consequence of the definition of generating function of a sequence, the generating function associated to $\{K_{n}^{(3)}\}_{n\geq 0}$, denoted by $\{g_{K_{n}^{(3)}}(t)\}$, is defined by

[TABLE]

Consequently, we obtain the following result:

Theorem 2.1**.**

The generating function for the Modified third-order Jacobsthal numbers $\{K_{n}^{(3)}\}_{n\geq 0}$ is $g_{K_{n}^{(3)}}(t)=\frac{3-2t-t^{2}}{1-t-t^{2}-2t^{3}}$.

Proof.

Using the definition of generating function, we have $g_{K_{n}^{(3)}}(t)=K_{0}^{(3)}+K_{1}^{(3)}t+K_{2}^{(3)}t^{2}+\cdots+K_{n}^{(3)}t^{n}+\cdots$. Multiplying both sides of this identity by $-t$, $-t^{2}$ and by $-2t^{3}$, and then from (2.1), we have $(1-t-t^{2}-2t^{3})g_{K_{n}^{(3)}}(t)=K_{0}^{(3)}+(K_{1}^{(3)}-K_{0}^{(3)})t+(K_{2}^{(3)}-K_{1}^{(3)}-K_{0}^{(3)})t^{2}$ and the result follows. ∎

The following result gives the Binet-style formula for $K_{n}^{(3)}$.

Theorem 2.2**.**

For $n\geq 0$, we have $K_{n}^{(3)}=2^{n}+\omega_{1}^{n}+\omega_{2}^{n}=2^{n}+M_{n}^{(2)}$, where

[TABLE]

and $\omega_{1},\omega_{2}$ are the roots of the characteristic equation associated with the respective recurrence relations $x^{2}+x+1=0$.

Proof.

Since the characteristic equation has three distinct roots, the sequence $K_{n}^{(3)}=a2^{n}+b\omega_{1}^{n}+c\omega_{2}^{n}$ is the solution of the Eq. (2.1). Considering $n=0,1,2$ in this identity and solving this system of linear equations, we obtain a unique value for $a$, $b$ and $c$, which are, in this case, $a=b=c=1$. So, using these values in the expression of $K_{n}^{(3)}$ stated before, we get the required result. ∎

Using the fact that $\omega_{1}+\omega_{2}=-\omega_{1}\omega_{2}=-1$, we have

[TABLE]

and $V_{n}^{(2)}$ as in Eq. (1.15). Then, $M_{n}^{(2)}=-M_{n-1}^{(2)}-M_{n-2}^{(2)}$, $M_{0}^{(2)}=2$ and $M_{1}^{(2)}=-1$. Furthermore,we easily obtain the identities stated in the following result:

Proposition 2.3**.**

For a natural number $n$ and $m$, if $K_{n}^{(3)}$, $j_{n}^{(3)}$, and $K_{n}^{(3)}$ are, respectively, the $n$-th third-order Jacobsthal, third-order Jacobsthal-Lucas and Modified third-order Jacobsthal numbers, then the following identities are true:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and $M_{n}^{(2)}$ as in Eq. (2.3).

Proof.

First, we will just prove Eqs. (2.4) and (2.7) since Eqs. (2.5), (2.6) and (2.8) can be dealt with in the same manner.

(2.4): To prove Eq. (2.4), we use induction on $n$. Let $n=2$, we get

[TABLE]

Let us assume that $147J_{m}^{(3)}=13K_{m}^{(3)}+48K_{m-1}^{(3)}+20K_{m-2}^{(3)}$ is true for all values $m$ less than or equal $n$. Then,

[TABLE]

(2.7): Using the the Binet formula of $K_{n}^{(3)}$ in Theorem 2.2, we have

[TABLE]

Then,

[TABLE]

Then, we obtain the Eq. (2.8) if $m=n$ in Eq. (2.7). ∎

3. Some identities involving this type of sequence

In this section, we state some identities related with these type of third-order sequence. As a consequence of the Binet formula of Theorem 2.2, we get for this sequence the following interesting identities.

Proposition 3.1** (Catalan’s identities).**

For a natural numbers $n$, $s$, with $n\geq s$, if $K_{n}^{(3)}$ is the $n$-th Modified third-order Jacobsthal numbers, then the following identity

[TABLE]

is true, where $M_{n}^{(2)}$ as in Eq. (2.3), $U_{n}^{(2)}=\frac{\omega_{1}^{n}-\omega_{2}^{n}}{\omega_{1}-\omega_{2}}$ and $\omega_{1}$, $\omega_{2}$ are the roots of the characteristic equation associated with the recurrence relation $x^{2}+x+1=0$.

Proof.

Using the Eq. (2.3) of Proposition 2.3 and the Binet formula of $K_{n}^{(3)}$ in Theorem 2.2, we have

[TABLE]

Using that $M_{n+s}^{(2)}=U_{s}^{(2)}M_{n+1}^{(2)}-U_{s-1}^{(2)}M_{n}^{(2)}$, $U_{s}^{(2)}=\frac{\omega_{1}^{s}-\omega_{2}^{s}}{\omega_{1}-\omega_{2}}$ and $U_{-s}^{(2)}=-U_{s}^{(2)}$. Then, we obtain

[TABLE]

Hence the result. ∎

Note that for $s=1$ in Catalan’s identity obtained, we get the Cassini identity for the Modified third-order Jacobsthal sequence. In fact, for $s=1$, the identity stated in Proposition 3.1, yields

[TABLE]

and using one of the initial conditions of the sequence $\{U_{n}^{(2)}\}$ in Proposition 3.1 we obtain the following result.

Proposition 3.2** (Cassini’s identities).**

For a natural numbers $n$, if $K_{n}^{(3)}$ is the $n$-th Modified third-order Jacobsthal numbers, then the identity

[TABLE]

is true.

The d’Ocagne identity can also be obtained using the Binet formula and in this case we obtain

Proposition 3.3** (d’Ocagne’s identities).**

For a natural numbers $m$, $n$, with $m\geq n$ and $K_{n}^{(3)}$ is the $n$-th Modified third-order Jacobsthal number, then the following identity

[TABLE]

is true.

Proof.

Using the Eq. (2.3) of Proposition 2.3 and the Eq. (2.2) of Theorem 2.2, we get the required result. ∎

In addition, some formulae involving sums of terms of the Modified third-order Jacobsthal sequence will be provided in the following proposition.

Proposition 3.4**.**

For a natural numbers $m$, $n$, with $n\geq m$, if $j_{n}^{(3)}$ and $K_{n}^{(3)}$ are, respectively, the $n$-th third-order Jacobsthal-Lucas and Modified third-order Jacobsthal numbers, then the following identities are true:

[TABLE]

[TABLE]

[TABLE]

Proof.

(3.1): Using Eq. (2.1), we obtain

[TABLE]

Then, the result in Eq. (3.1) is completed.

(3.2): As a consequence of the Eq. (2.2) of Theorem 2.2 and

[TABLE]

we have

[TABLE]

Hence we obtain the result.

(3.3): Using Eqs. (2.6) and (2.1), we have $49j_{n}^{(3)}=43K_{n}^{(3)}+8K_{n-1}^{(3)}+36K_{n-2}^{(3)}=18K_{n+1}^{(3)}+25K_{n}^{(3)}-10K_{n-1}^{(3)}$. Then, from Eq. (3.1), we obtain

[TABLE]

So, the proof is completed. ∎

For negative subscripts terms of the sequence of Modified third-order Jacobsthal we can establish the following result:

Proposition 3.5**.**

For a natural number $n$ the following identities are true:

[TABLE]

[TABLE]

Proof.

(3.4): Since $M_{-n}^{(2)}=M_{n}^{(2)}$, using the Binet formula stated in Theorem 2.2 and the fact that $\omega_{1}\omega_{2}=1$, all the results of this Proposition follow. In fact,

[TABLE]

So, the proof is completed.

(3.5): The proof is similar to the proof of Eq. (3.1) using the Eq. (3.4). ∎

4. Conclusion

Sequences of numbers have been studied over several years, with emphasis on the well known Tribonacci sequence and, consequently, on the Tribonacci-Lucas sequence. In this paper we have also contributed for the study of Modified third-order Jacobsthal sequence, deducing some formulae for the sums of such numbers, presenting the generating functions and their Binet-style formula. It is our intention to continue the study of this type of sequences, exploring some their applications in the science domain. For example, a new type of sequences in the quaternion algebra with the use of this numbers and their combinatorial properties.