On Gersenne Sequence: A Study of One Family in the Horadam-Type Sequence

Abstract

This study introduces an innovative approach to Mersenne-type numbers. This paper introduces a new class of numbers, which we call Gersenne numbers. The aim of this paper is to define the Gersenne sequence and to investigate some of their properties, such as the recurrence relation, the summation formula, and the generating function. Moreover, the classical identities are derived, such as the Tagiuri–Vajda, Catalan, Cassini, and d’Ocagne identities for Gersenne numbers.

1. Introduction and Background

The Mersenne sequence is composed of non-negative integers in the form of a power of two minus one, and it is best known for some of the prime numbers that make it up, which are called Mersenne primes. These numbers are defined by the recursion relation

$$M_n=3M_{n-1}-2M_{n-2}, \mathrm{for} \mathrm{all} n\ge 2$$

with initial condition $M_0=0$, and $M_1=1$, forming the sequence

$${\left\{M_n\right\}}_{n\ge 0}=\{0,1,3,7,15,31,63,127,255,\dots \},$$

which is referred to as sequence A000225 in the OEIS [1]. Considering the initial values $m_0=2$ and $m_1=3$, with the identical recurrence relation $m_n=3m_{n-1}-2m_{n-2}$, for all $n\ge 2$, we have the Mersenne–Lucas numbers. The terms of this sequence are called Mersenne–Lucas numbers and are expressed in the form $m_n=2^n+1$, which is identified as sequence A000051 in OEIS [1]. These two classes of numbers, which have important implications in areas such as cryptography and the identification of large prime numbers, are an indispensable concept in number theory.

Consider k, a and b as fixed constants (typically integers), with $k\ge 0$. We define the sequence ${\left\{G{M}_{(k,n)}\right\}}_{n\ge 0}$ as a Mersenne-type sequence that satisfies the recurrence relation

$$G{M}_{(k,n)}=(k+1)G{M}_{(k,n-1)}-kG{M}_{(k,n-2)};n\ge 2,$$

(1)

with initial values $G{M}_{(k,0)}=a$ and $G{M}_{(k,1)}=b$, where a and b are fixed constants. When $k=2$, $a=0$ and $b=1$, we have the classical Mersenne numbers. If we set $k=10$, $a=0$ and $b=1$, we obtain the ordinary Repunit numbers. Here, this sequence is referred to as the Gersenne sequence.

A straightforward calculation shows that the first five elements of the Gersenne sequence are

$$a,b,k(b-a)+b,(k^2+k)(b-a)+b,(k^3+k^2+k)(b-a)+b,\dots$$

(2)

If we take $k=0$, we have a constant sequence equal to b for every term n greater than 1, so

$$a,b,b,b,b,b,\dots$$

when $k=1$, we have the following sequence:

$$a,b,2b-a,3b-2a,4b-3a,5b-4a,\dots$$

Thus, by taking $k=1,a=0$ and $b=1$, we have the following sequence of non-negative integers: $0,1,2,3,4,\dots$. Therefore, throughout the text, unless otherwise stated, k is a positive integer greater than 1, that is, $k>1$.

The structure of the present paper is divided into four more sections, as follows. In Section 2, we present the recurrence of the Gersenne sequence, the generalized Binet formula, and the generating function. In Section 3, we derive several classical identities for the generalized Gersenne sequence for all integers n, including the Tagiuri–Vajda, Catalan, Cassini, d’Ocagnes, and Gelin–Cesàro identities. In the Section 4, we present results on partial sums of terms of the Gersenne sequence. We conclude with some final considerations and state some future work about this topic.

In the mathematical literature, there have been countless studies on the sequences of Mersenne and Mersenne–Lucas numbers. For example, in [2], the author offers a thorough and detailed examination of these two types of special numbers; in particular, they are intimately tied to classical problems in the theory of prime numbers (see [3,4,5]). It also considers practical applications and is particularly relevant in the specific context of cryptography (see [6,7]). In [8], the authors defined the generalized Mersenne number as in Equation (1), but with initial terms $a=0$ and $b=1$, and present their interpretations and matrix generators for this sequence. Other generalizations or extensions of the Mersenne or Mersenne–Lucas sequences, generating function and several identities, can be found in [9,10,11,12], among others. In [13,14], the authors studied the generalized Gaussian Mersenne numbers with arbitrary initial values and considered two particular cases, namely, Gaussian Mersenne and Gaussian Mersenne–Lucas numbers.

2. The Gersenne Numbers

We refer to Gersenne numbers as an abbreviation of the term ’generalized Mersenne numbers’ and they are denoted by $G{M}_{(k,n)}$. This name was inspired by the term Gibonacci numbers, a shortened form of generalized Fibonacci numbers, first used in [15], as well as ‘Geonardo numbers’, which was used in [16] to designate a generalization of the Leonardo numbers.

For a non-negative integer n, consider the Horadam sequence ${\left\{h_n\right\}}_{n\ge 0}$, which is defined by the second-order recurrence relation:

$$h_n=p{h}_{n-1}+q{h}_{n-2}\mathrm{for}\mathrm{all}n\ge 2,$$

(3)

where p and q are fixed integers, and the initial conditions are $h_0=a$ and $h_1=b$. This sequence was introduced by Horadam [17,18] and generalizes several well-known sequences that are ruled by a characteristic equation of the form $x^2-px-q=0$. For a more detailed discussion and further results on the Horadam sequence, see [18,19], as the authors of ref. [18] introduced and studied what is now referred to as the Horadam sequence, and ref. [19] notes that several other well-known sequences can also be expressed as Horadam sequences by selecting suitable initial terms and recurrence constants. In our study, we consider the class of Horadam-type sequences represented by the recurrence Equation (3), where we take $p=k+1$ and $q=-k$, for all integers $k\ge 2$. Then, for all k, a and b that are fixed constants, the Gersenne sequence is given by the recurrence relation given by Equation (1).

By looking at list (2), we can verify the following result.

Proposition 1. 

Let n denote arbitrary non-negative integers. For all $n\ge 2$, the following property holds:

$$G{M}_{(k,n)}=(k^{n-1}+k^{n-2}+\cdots +k)(b-a)+b$$

(4)

where ${\left\{G{M}_{(k,n)}\right\}}_{n\ge 0}$ is the Gersenne sequence, with initial values $G{M}_{(k,0)}=a$ and $G{M}_{(k,1)}=b$.

Proof is carried out by mathematical induction using the recurrence (1).

From Proposition 1, the next result follows.

Corollary 1. 

Let n denote arbitrary non-negative integers. For all $n\ge 2$, the following property holds:

$$G{M}_{(k,n)}={\displaystyle \frac{k^n-1}{k-1}}(b-a)+a,$$

(5)

where ${\left\{G{M}_{(k,n)}\right\}}_{n\ge 0}$ is the Gersenne sequence, with initial values $G{M}_{(k,0)}=a$ and $G{M}_{(k,1)}=b$.

Equation (5) can be rewritten in the form

$$G{M}_{(k,n)}={\displaystyle \frac{(k^n-1)(b-a)+(k-1)a}{k-1}}.$$

2.1. The Binet Formula

In this subsection, we present the characteristic equation of the Gersenne sequence and provide Binet’s formula. Additionally, we present the exponential generating function and the ordinary generating function of the Gersenne sequence.

Note that the Horadam-type characteristic equation associated with the Gersenne sequence ${\left\{G{M}_{(k,n)}\right\}}_{n\ge 0}$ is

$$x^2-(k+1)x+k=0,$$

(6)

and its real distinct roots are $x_1=k$ and $x_2=1$.

The Binet formula is presented in the next result.

Proposition 2. 

 (Binet’s formula) For all non-negative integers n, we have

$$G{M}_{(k,n)}={\displaystyle \frac{c{k}^n+d}{k-1}},$$

(7)

where $G{M}_{(k,0)}=a$, $G{M}_{(k,1)}=b$, $c=b-a$, and $d=ka-b$ with $k>1$.

The proof is immediate since the characteristic equation is (6).

The special case $a=0$ and $b=1$ is denoted as

$$G{M}_{(k,n)}={\upsilon }_{(k,n)}={\displaystyle \frac{k^n-1}{k-1}}.$$

(8)

are called the generalized k-Mersenne sequence.

Table 1. Some generalized k-Mersenne sequence.

k	Sequence	Name	ID Catalog
2	$0,1,3,7,15,31,63,127,255,\dots$	Mersenne	A000225 [1]
3	$0,1,4,13,40,121,364,1093,3280,\dots$	3-Mersenne	A003462 [1]
4	$0,1,5,21,85,341,1365,5461,\dots$	4-Mersenne	A002450 [1]
5	$0,1,6,31,156,781,3906,$ 19,531, …	5-Mersenne	A001047 [1]
6	$0,1,7,43,259,1555,9331,$ 55,987, …	6-Mersenne	A003464 [1]
7	$0,1,8,57,400,2801,$ 19,608, 137,257,…	7-Mersenne	A023000 [1]
8	$0,1,9,73,585,4681,$ 37,449, …	8-Mersenne	A023001 [1]
9	$0,1,10,91,820,7381,$ 66,430, …	9-Mersenne	A002452 [1]
10	$0,1,11,111,1111,1111,$ 11,111, …	repunit	A002275 [1]
11	$0,1,12,133,1464,$ 16,105, 177,156, …	11-Mersenne	A016123 [1]
⋮	⋮
k	$0,1,k+1,k^2+k+1,k^3+k^2+k+1,\dots$	k-Mersenne

shows some examples of the generalized k-Mersenne sequence.

According to Table 1, the 2-Mersenne sequence is the classic (or ordinary) Mersenne sequence, and the 10-Mersenne sequence is the repunit sequence.

See that for any integer $n>0$, the repunit or 10-Mersenne numbers are repunits in base 10, i.e., $r_n=[\underset{\mathrm{n}\dots \mathrm{d}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{t}\mathrm{s}}{\underbrace{11\dots 11}}{]}_{\mathrm{10}}$. Moreover, the representation of ordinary Mersenne or 2-Mersenne numbers in base 2 is of the form $M_n=[\underset{\mathrm{n} \mathrm{d}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{t}\mathrm{s}}{\underbrace{11\dots 11}}{]}_2$. A direct calculation using Equation (8) shows the result, as stated in the next proposition.

Proposition 3. 

Let $n,k$ be a non-negative integer. For all $n>0$ and $k>1$, the representation of ${\upsilon }_{(k,n)}$ numbers in base k is

$${\upsilon }_{(k,n)}=[\underset{n digits}{\underbrace{11\dots 11}}{]}_k,$$

where ${\upsilon }_{(k,n)}$ is the n-th k-Mersenne number given by Equation (8).

2.2. Generating Function

Now, we present the ordinary and exponential generating functions for the Gersenne sequence ${\left\{G{M}_{(k,n)}\right\}}_{n\ge 0}$. Taking into account the literature, the formal summation

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

(9)

is known as the ordinary generating function for the sequence $\{a_0,a_1,a_2,\dots \}$.

Our next result presents the ordinary generating function for the Gersenne sequence.

Proposition 4. 

The ordinary generating function for the Gersenne sequence ${\left\{G{M}_{(k,n)}\right\}}_{n\ge 0}$, denoted by $G_{G{M}_{(k,n)}}\left(x\right)$, is

$$G_{G{M}_{(k,n)}}\left(x\right)={\displaystyle \frac{a+\left(b-(k+1)a\right)x}{1-(k+1)x+k{x}^2}},$$

where $G{M}_{(k,0)}=a$ and $G{M}_{(k,1)}=b$.

Proof. 

According to Equation (9), the generating function for the Gersenne sequence is ${\displaystyle G_{G{M}_{(k,n)}}\left(x\right)=\sum _{n=0}^{\infty }G{M}_{(k,n)}{x}^n}$, and then, using the equations $-(k+1)x{G}_{G{M}_{(k,n)}}\left(x\right)$ and $k{x}^2{G}_{G{M}_{(k,n)}}\left(x\right)$, we obtain

$$\begin{array}{cc}\hfill G_{G{M}_{(k,n)}}\left(x\right)=& G{M}_{(k,0)}+G{M}_{(k,1)}x+G{M}_{(k,2)}{x}^2+\dots +G{M}_{(k,n)}{x}^n+\dots \hfill \\ \\ \hfill -(k+1)x{G}_{G{M}_{(k,n)}}\left(x\right)=& -(k+1)G{M}_{(k,0)}x-(k+1)G{M}_{(k,1)}{x}^2-(k+1)G{M}_{(k,2)}{x}^3\hfill \\ & -\dots -(k+1)G{M}_{(k,n)}{x}^{n+1}-\dots \hfill \\ \\ \hfill k{x}^2{G}_{G{M}_{(k,n)}}\left(x\right)=& kG{M}_{(k,0)}{x}^2+G{M}_{(k,1)}{x}^3+G{M}_{(k,2)}{x}^4+\dots +kG{M}_{(k,n)}{x}^{n+2}+\dots \hfill \end{array}$$

When we add to both sides of these equations and use Equation (1), we obtain the result. □

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

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

In the next result, we consider the Binet Equation (7), and obtain the exponential generating function for the Gersenne sequence ${\left\{G{M}_{(k,n)}\right\}}_{n\ge 0}.$

Proposition 5. 

For all $n\ge 0$, the exponential generating function for the Gersenne sequence ${\left\{G{M}_{(k,n)}\right\}}_{n\ge 0}$ is

$$E_{G{M}_{(k,n)}}\left(x\right)={\displaystyle \frac{1}{k-1}}(c{e}^{kx}+d{e}^x),$$

where $G{M}_{(k,0)}=a$, $G{M}_{(k,1)}=b$, $c=b-a$, and $d=ka-b$ with $k>1$.

The proof is immediate by Binet’s formula.

The Poisson generating function $P_{a_n}\left(x\right)$ for a sequence ${\left\{a_n\right\}}_{n\ge 0}$ is given by

$$P_{a_n}\left(x\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{a_n{x}^n}{n!}}{e}^{-x}=e^{-x}{E}_{a_n}\left(x\right),$$

where $E_{a_n}\left(x\right)$ is the exponential generating function of the sequence ${\left\{a_n\right\}}_{n\ge 0}$. Consequently, the corresponding Poisson generating function is derived.

Corollary 2. 

For all $n>0$, the Poisson generating function for the Gersenne sequence ${\left\{G{M}_{(k,n)}\right\}}_{n\ge 0}$ is

$$P_{G{M}_{(k,n)}}\left(x\right)={\displaystyle \frac{1}{k-1}}(c{e}^{(k-1)x}+d).$$

3. Some Identities for Gersenne Numbers

In this section, we establish some identities for the generalized Gersenne, some of which are classical identities, for example, the Tagiuri–Vajda, Catalan, Cassini, d’Ocagnes and Gelin–Cesàro identities.

The first result shows the addition and difference between two terms of the Gersenne sequence, and it is a direct consequence of Equation (7).

Proposition 6. 

Let ${\left\{G{M}_{(k,n)}\right\}}_{n\ge 0}$ be the Gersenne sequence. For all non-negative integers m and n with $m\ge n$, the following identities are verified:

$$\left(1\right)G{M}_{(k,m)}+G{M}_{(k,n)}=c{k}^n{\upsilon }_{(k,m-n)}+{\displaystyle \frac{2c{k}^n}{k-1}},$$

$$\left(2\right)G{M}_{(k,m)}-G{M}_{(k,n)}=c{k}^n{\upsilon }_{(k,m-n)},$$

where $G{M}_{(k,0)}=a$, $G{M}_{(k,1)}=b$, $c=b-a$, $d=ka-b$, and ${\left\{{\upsilon }_{(k,n)}\right\}}_{n\ge 0}$ is the generalized k-Mersenne sequence given by Equation (8), with $k>1$.

The next result establishes the linear combination for a product of two terms of the Gersenne sequence.

Proposition 7. 

Let ${\left\{G{M}_{(k,n)}\right\}}_{n\ge 0}$ be the Gersenne sequence. For all non-negative integers m and n, the following identity holds:

$$G{M}_{(k,m)}·G{M}_{(k,n+1)}-kG{M}_{(k,m-1)}·G{M}_{(k,n)}={\displaystyle \frac{c^2{k}^{m+n}-d^2}{k-1}},$$

where $G{M}_{(k,0)}=a$, $G{M}_{(k,1)}=b$, $c=b-a$, and $d=ka-b$, with $k>1$.

Proof. 

According to Binet’s Equation (7), we have

$$\begin{array}{cc}\hfill & G{M}_{(k,m)}·G{M}_{(k,n+1)}-kG{M}_{(k,m-1)}·G{M}_{(k,n)}\hfill \\ \hfill =& {\displaystyle \frac{c^2·k^{m+n+1}-c^2·k^{m+n}-k{d}^2+d^2}{{(k-1)}^2}}\hfill \\ \hfill =& {\displaystyle \frac{c^2·k^{m+n}(k-1)-d^2(k-1)}{{(k-1)}^2}}={\displaystyle \frac{c^2·k^{m+n}-d^2}{k+1}},\hfill \end{array}$$

as required. □

The following corollary follows directly from Proposition 7, considering $c=1$ and $d=-1$.

Corollary 3. 

Let ${\left\{{\upsilon }_{(k,n)}\right\}}_{n\ge 0}$ be the generalized k-Mersenne sequence given by Equation (8). For all non-negative integers m and n, we obtain the identity

$${\upsilon }_{(k,m)}·{\upsilon }_{n+1,k}-k{\upsilon }_{(k,m-1)}·{\upsilon }_{(k,n)}={\upsilon }_{(k,m+n)},$$

where k is an integer ($k>1)$.

As a direct consequence of Corollary 3, the following result is established.

Corollary 4. 

Let ${\left\{{\upsilon }_{(k,n)}\right\}}_{n\ge 0}$ be the generalized k-Mersenne sequence given by Equation (8). For all non-negative integers r, the following identities hold:

(1) 

${\upsilon }_{(k,2r-1)}={\left({\upsilon }_{(k,r)}\right)}^2-k{\left({\upsilon }_{(k,r-1)}\right)}^2$.

(2) 

${\upsilon }_{(k,2r)}=(k+1){\left({\upsilon }_{(k,r)}\right)}^2-2k{\upsilon }_{(k,r)}·{\upsilon }_{(k,r-1)}$ where k is an integer ($k>1)$.

Using the same technique employed in Proposition 7, we omit the proof of the next result in the interest of brevity.

Proposition 8. 

Let ${\left\{G{M}_{(k,n)}\right\}}_{n\ge 0}$ be the Gersenne sequence. For all non-negative integers $m,n$, if $m\ge n$, then

$$G{M}_{(k,m+1)}·G{M}_{(k,n)}-G{M}_{(k,m)}·G{M}_{(k,n+1)}=cd·k^n{\upsilon }_{(k,m-n)},$$

where $G{M}_{(k,0)}=a$, $G{M}_{(k,1)}=b$, $c=b-a$, $d=ka-b$, and ${\left\{{\upsilon }_{(k,m)}\right\}}_{m\ge 0}$ is the generalized k-Mersenne sequence given by Equation (8), with $k>1$.

The Tagiuri–Vajda identity for the Gersenne sequence is given below.

Theorem 1. 

Let ${\left\{G{M}_{(k,m)}\right\}}_{m\ge 0}$ be the Gersenne sequence, and $m,k,r,s$ non-negative integers. The following identity holds:

$$G{M}_{(k,m+r)}·G{M}_{(k,m+s)}-G{M}_{(k,m)}·G{M}_{(k,m+r+s)}=-cd{k}^m·{\upsilon }_{(k,r)}·{\upsilon }_{(k,s)},$$

where $G{M}_{(k,0)}=a$, $G{M}_{(k,1)}=b$, $c=b-a$, $d=ka-b$, and ${\left\{{\upsilon }_{(k,m)}\right\}}_{m\ge 0}$ is the generalized k-Mersenne sequence given by Equation (8), with $k>1$.

Proof. 

According to Binet’s Equation (7), we have

$$\begin{array}{cc}\hfill & G{M}_{(k,m+r)}·G{M}_{(k,m+s)}-G{M}_{(k,m)}·G{M}_{(k,m+r+s)}\hfill \\ \hfill =& {\displaystyle \frac{c·k^{m+r}+d}{k-1}}·{\displaystyle \frac{c·k^{m+s}+d}{k-1}}-{\displaystyle \frac{c·k^m+d}{k-1}}·{\displaystyle \frac{c·k^{m+r+s}+d}{k-1}}\hfill \\ \hfill =& -cd{k}^m·{\displaystyle \frac{k^r-1}{k-1}}·{\displaystyle \frac{k^s-1}{k-1}}\hfill \end{array}$$

and we have the validity of the result. □

The following identities will be derived as a consequence of the Tagiuri–Vajda identity, as demonstrated in Theorem 1.

Proposition 9. 

(d’Ocagne’s identity) Let ${\left\{G{M}_{(k,m)}\right\}}_{m\ge 0}$ be the Gersenne sequence and $m,n$ be non-negative integers with $m\ge n$. Then, the following identity holds:

$$G{M}_{(k,m+1)}·G{M}_{(k,n)}-G{M}_{(k,m)}·G{M}_{(k,n+1)}=cd·k^n·{\upsilon }_{(k,m-n)},$$

where $G{M}_{(k,0)}=a$, $G{M}_{(k,1)}=b$, $c=b-a$, $d=ka-b$, and ${\left\{{\upsilon }_{(k,m)}\right\}}_{m\ge 0}$ is the generalized k-Mersenne sequence given by Equation (8), with $k>1$.

Proof. 

Take $m=n$, $r=m-n$ and $s=1$ in Theorem 1, so

$$\begin{array}{cc}\hfill G{M}_{(k,n+1)}·G{M}_{(k,m)}-G{M}_{(k,m+1)}·G{M}_{(k,n)}=& -cd·k^n{\upsilon }_{m-n,k}·{\upsilon }_{1,k}.\hfill \end{array}$$

Since ${\upsilon }_{(1,k)}=1$, we obtain the result. □

Similar to Proposition 9, we have the Catalan identity.

Proposition 10. 

(Catalan’s identity) Let ${\left\{G{M}_{(k,n)}\right\}}_{n\ge 0}$ denote the Gersenene sequence, and let n and r be non-negative integers such that $n\ge r$. Under these conditions, the following identity holds:

$$G{M}_{(k,n+r)}·G{M}_{(k,n-r)}-{\left(G{M}_{(k,n)}\right)}^2=cd·k^{n-r}·{\left({\upsilon }_{(k,r)}\right)}^2,$$

(10)

where $G{M}_{(k,0)}=a$, $G{M}_{(k,1)}=b$, $c=b-a$, $d=ka-b$, and ${\left\{{\upsilon }_{(k,m)}\right\}}_{m\ge 0}$ is the generalized k-Mersenne sequence given by Equation (8), with $k>1$.

Proof. 

In Theorem 1, taking $r=s$ and $m+r=n$, we obtain

$$\begin{array}{cc}\hfill G{M}_{(k,n+r)}·G{M}_{(k,n-r)}-{\left(G{M}_{(k,n)}\right)}^2=& cd·k^{n-r}·{\left({\upsilon }_{(k,r)}\right)}^2.\hfill \end{array}$$

Thus, the result is obtained. □

Applying the Catalan identity yields the following result.

Corollary 5. 

Let ${\left\{G{M}_{(k,m)}\right\}}_{m\ge 0}$ represent the Gersenene sequence. For any non-negative integers m, the following identity is satisfied:

$$G{M}_{(k,m+2)}·G{M}_{(k,m-2)}={\left(G{M}_{(k,m)}\right)}^2+cd{k}^{m-2}{\left({\upsilon }_{k,2}\right)}^2,$$

where $G{M}_{(k,0)}=a$, $G{M}_{(k,1)}=b$, $c=b-a$, $d=ka-b$, and ${\left\{{\upsilon }_{(k,m)}\right\}}_{m\ge 0}$ is the generalized k-Mersenne sequence given by Equation (8), with $k>1$.

Other consequences from the Catalan identity are as follows:

Corollary 6.  

(Cassini’s identity) Let ${\left\{G{M}_{(k,m)}\right\}}_{m\ge 0}$ represent the Gersenene sequence. The following identity is satisfied for all integers $m\ge 1$

$$G{M}_{(k,m+1)}·G{M}_{(k,m-1)}-{\left(G{M}_{(k,m)}\right)}^2=cd·k^{m-1},$$

where $G{M}_{(k,0)}=a$, $G{M}_{(k,1)}=b$, $c=b-a$, and $d=ka-b$, with $k>1$.

The next result, which is one of the main results of this paper, gives a formula for calculating the multiplication of four consecutive elements of the Gersenne sequence, which generalizes the Gelin–Cesàro identity.

Theorem 2. 

Let ${\left\{G{M}_{(k,m)}\right\}}_{m\ge 0}$ denote the Gersenne sequence. The following identity holds for all non-negative integers $m>r$:

$$\begin{array}{cc}\hfill & G{M}_{(k,m-(r+1\left)\right)}·G{M}_{(k,m-r)}·G{M}_{(k,m+r)}·G{M}_{(k,m+(r+1\left)\right)}=\hfill \\ \hfill & {\left(G{M}_{(k,m)}\right)}^4-cd·k^{m-(r+1)}{\left(G{M}_{(k,m)}\right)}^2\left({\left({\upsilon }_{(k,r+1)}\right)}^2+k{\left({\upsilon }_{(k,r)}\right)}^2\right)\hfill \\ \hfill & +cd·k^{2(m-r)-1}{({\upsilon }_{(k,r+1)}·{\upsilon }_{(k,r)})}^2),\hfill \end{array}$$

where $G{M}_{(k,0)}=a$, $G{M}_{(k,1)}=b$, $c=b-a$, $d=ka-b$, and ${\left\{{\upsilon }_{(k,m)}\right\}}_{m\ge 0}$ is the generalized k-Mersenne sequence given by Equation (8), with $k>1$.

Proof. 

By substituting $n=r+1$ into Equation (10), we obtain the following:

$$G{M}_{(k,m-(r+1\left)\right)}·G{M}_{(k,m+(r+1\left)\right)}={\left(G{M}_{(k,m)}\right)}^2+cd{k}^{m-(r+1)}{\left({\upsilon }_{(k,r+1)}\right)}^2$$

(11)

By multiplying Equations (10) and (11), we obtain

$$\begin{array}{cc}\hfill & G{M}_{(k,m-(r+1\left)\right)}·G{M}_{(k,m-r)}·G{M}_{(k,m+r)}·G{M}_{(k,m+(r+1\left)\right)}\hfill \\ \hfill =& {\left(G{M}_{(k,m)}\right)}^4-cd·k^{m-(r+1)}{\left(G{M}_{(k,m)}\right)}^2\left({\left({\upsilon }_{(k,r+1)}\right)}^2+k{\left({\upsilon }_{(k,r)}\right)}^2\right)\hfill \\ \hfill & +cd·k^{2(m-r)-1}{({\upsilon }_{(k,r+1)}·{\upsilon }_{(k,r)})}^2),\hfill \end{array}$$

which completes the proof. □

By virtue of Theorem 2, and taking $r=1$, we have the following corollary.

Corollary 7.  

(Gelin-Cesàro’s identity) Let m be any natural number and ${\left\{G{M}_{(k,m)}\right\}}_{m\ge 0}$ denote the Gersenne sequence. Then, the identity holds

$$\begin{array}{cc}\hfill & G{M}_{(k,m-2)}·G{M}_{(k,m-1)}·G{M}_{(k,m+1)}·G{M}_{(k,m+2)}-{\left(G{M}_{(k,m)}\right)}^4\hfill \\ \hfill =& {\left(cd\right)}^2{k}^{2m-3}{\left({\upsilon }_{k,2}\right)}^2+cd·k^{m-2}\left({(k+1)}^2+k\right){\left(G{M}_{(k,m)}\right)}^2,\hfill \end{array}$$

where $G{M}_{(k,0)}=a$, $G{M}_{(k,1)}=b$, $c=b-a$, $d=ka-b$, and ${\left\{{\upsilon }_{(k,m)}\right\}}_{m\ge 0}$ is the generalized k-Mersenne sequence given by Equation (8), with $k>1$.

In the first example, we specify the last identity for the repunit sequence.

Example 1. 

Considering the repunit sequence ${\left\{r_n\right\}}_{n\ge 0}$ specified when we take $k=10,M_0=0$ and $M_1=1$ (see Table 1), $c=1$ and $d=-1$, and we have that the Gelin–Cesàro identity for the repunit sequence

$$r_{m-2}{r}_{m-1}{r}_{m+1}{r}_{m+2}-{\left(r_m\right)}^4={10}^{2m-3}·121-131·{10}^{m-2}{\left(r_m\right)}^2,$$

according to Proposition 3.6 in [20].

In the following example, we also illustrate the Gelin–Cesàro’s identity for the One-Zero sequence.

Example 2. 

Considering the One-Zero sequence ${\left\{U_n\right\}}_{n\ge 0}$ specified when we take $p={10}^2,U_0=0$ and $U_1=1$, $c=1$ and $d=1$, and we have that

$$U_{m-2}{U}_{m-1}{U}_{m+1}{U}_{m+2}-{\left(U_m\right)}^4={10}^{2(2m-3)}·10201-10301·{10}^{2(m-2)}{\left(U_m\right)}^2,$$

according to Proposition 3.1 in [21].

In concluding this section, we present a restricted result to the generalized k-Mersenne sequence. This result shows the difference between an even order and any n-th term.

Proposition 11. 

Let n be any non-negative integer. We have

$${\upsilon }_{(k,2n)}-2{\upsilon }_{(k,n)}=(k-1){\left({\upsilon }_{(k,n)}\right)}^2,$$

where ${\left\{{\upsilon }_{(k,n)}\right\}}_n$ is a k-Mersenne sequence given by Equation (8), with $k>1$.

Proof. 

A straightforward calculation, and making use of Equation (8), shows that

$$\begin{array}{ccc}\hfill {\upsilon }_{(k,2n)}& =& {\displaystyle \frac{k^{2n}-1}{k-1}}={\displaystyle \frac{(k^n-1)(k^n+1)}{k-1}}={\displaystyle \frac{(k^n-1)[(k^n-1)+2]}{k-1}}\hfill \\ & =& {\displaystyle \frac{(k^n-1)(k^n-1)}{k-1}}+2{\displaystyle \frac{k^n-1}{k-1}}=(k-1){{\upsilon }_{(k,n)}}^2+2{\upsilon }_{(k,n)},\hfill \end{array}$$

which proves the result. □

The above result generalizes Proposition 1 given in [22].

4. Sum of Terms Involving the Gersenne Sequence

In this section, we present results on partial sums of terms of the Gersenne numbers with n integers. In general, the partial sums ${\sum }_{k=0}^n{h}_k=h_0+h_1+\cdots +h_n$, for $n\ge 0$, where ${\left\{h_n\right\}}_{n\ge 0}$ is the Horadam sequence, can be obtained by applying Equations (3.5), (3.12) and (3.13) in [18], or the fundamental summation rule (Equation (33)) in [23], when $p-q\ne 1$, which we have not applied here.

So, we consider the sequence of partial sums

$$\sum _{i=0}^{n}G{M}_{(k,i)}=G{M}_{(k,0)}+G{M}_{(k,1)}+G{M}_{(k,2)}+\cdots +G{M}_{(k,n)},$$

for $n\ge 0$, where ${\left\{G{M}_{(k,n)}\right\}}_{n\ge 1}$ is the Gersenne sequence.

Proposition 12. 

Let ${\left\{G{M}_{(k,n)}\right\}}_{n\ge 1}$ be the Gersenne sequence; then, we have the following formulas:

(a) 

${\displaystyle \sum _{i=0}^{n}G{M}_{(k,i)}=\frac{c·{\upsilon }_{n+1,k}+d(n+1)}{k-1}}$;

(b) 

${\displaystyle \sum _{i=0}^{n}G{M}_{(k,2i)}=\frac{c(k^{n+1}+1){\upsilon }_{n+1,k}}{k^2-1}+\frac{d(n+1)}{k-1}}$;

(c) 

${\displaystyle \sum _{i=0}^{n}G{M}_{(k,2i+1)}=\frac{c(k^{n+2}+k){\upsilon }_{n+1,k}}{k^2-1}+\frac{d(n+1)}{k-1}}$;

(d) 

${\displaystyle \sum _{i=0}^n{\left(G{M}_{(k,i)}\right)}^2=\frac{c^2(k^{n+1}+1)·{\upsilon }_{n+1,k}}{k^3-k^2-k+1}+\frac{2cd·{\upsilon }_{n+1,k}+d^2(n+1)}{{(k-1)}^2}}$.

In the above, $G{M}_{(k,0)}=a$, $G{M}_{(k,1)}=b$, $c=b-a$, $d=ka-b$ and ${\left\{{\upsilon }_{(k,m)}\right\}}_{m\ge 0}$ is the generalized k-Mersenne sequence given by Equation (8), with $k>1$.

Proof. 

(a)

By Equation (7), we have

$$\begin{array}{cc}\hfill \sum _{i=0}^{n}G{M}_{(k,i)}=& G{M}_{(k,0)}+G{M}_{(k,1)}+G{M}_{(k,2)}+\cdots +G{M}_{(k,n)}\hfill \\ \hfill =& {\displaystyle \frac{c·k^0+d}{k-1}}+{\displaystyle \frac{c·k^1+d}{k-1}}+{\displaystyle \frac{c·k^2+d}{k-1}}+\cdots +{\displaystyle \frac{c·k^n+d}{k-1}}\hfill \\ \hfill =& {\displaystyle \frac{c·{\upsilon }_{n+1}+d(n+1)}{k-1}},\hfill \end{array}$$

as required.

Using the same approach as in (a), we obtain the other points. □

In order to better visualize Proposition 12, consider the following example.

Example 3. 

Considering the repunit sequence ${\left\{r_n\right\}}_{n\ge 0}$ specified when we take $k=10,r_0=0$ and $r_1=1$ (see Table 1), $c=1$, $d=-1$, the following formulas hold:

(a) 

${\displaystyle \sum _{i=0}^n{r}_i=\frac{r_{n+1}-(n+1)}{9}}$;

(b) 

${\displaystyle \sum _{i=0}^n{r}_{2i}=\frac{({10}^{n+1}+1)r_{n+1}}{99}-\frac{n+1}{9}}$;

(c) 

${\displaystyle \sum _{i=0}^n{r}_{2i+1}=\frac{({10}^{n+2}+10)r_{n+1}}{99}-\frac{n+1}{9}}$;

(d) 

${\displaystyle \sum _{i=0}^n{\left(r_i\right)}^2=\frac{({10}^{n+1}+1)r_{n+1}}{891}-\frac{2r_{n+1}-(n+1)}{81}}$.

In the above example, the items (a), (b) and (c) are direct consequences of Proposition 6.1, as presented in [20], while item (d) follows from Proposition 12.

To illustrate the previous result, we present in the next example the particular case for $k=4$.

Example 4. 

Considering the Gersenne sequence, specified when we take $k=4,G{M}_{(4,0)}=0$ and $G{M}_{(4,1)}=1$ (see Table 1), $c=1$ and $d=-1$, and $G{M}_{(4,n)}={\displaystyle \frac{4^n-1}{3}}$; therefore, the following formulas hold:

(a) 

${\displaystyle \sum _{i=0}^{n}G{M}_{(4,i)}=\frac{G{M}_{(4,n+1)}-(n+1)}{3}}$;

(b) 

${\displaystyle \sum _{i=0}^{n}G{M}_{(4,2i)}=\frac{(2^{2(n+1)}+1)G{M}_{(4,n+1)}}{15}-\frac{n+1}{3}}$;

(c) 

${\displaystyle \sum _{i=0}^{n}G{M}_{(4,2i+1)}=\frac{(2^{2(n+2)}+10)G{M}_{(4,n+1)}}{15}-\frac{n+1}{3}}$;

(d) 

${\displaystyle \sum _{i=0}^n{\left(G{M}_{(4,i)}\right)}^2=\frac{(2^{2(n+1)}+1)G{M}_{(4,n+1)}}{45}-\frac{2G{M}_{(4,n+1)}-(n+1)}{9}}$.

Consider now the sequence of alternating partial sums given by

$$\sum _{i=0}^n{(-1)}^{i}G{M}_{(k,i)}=G{M}_{(k,0)}-G{M}_{(k,1)}+G{M}_{(k,2)}-G{M}_{(k,3)}+\dots +{(-1)}^{n}G{M}_{(k,n)}$$

for $n\ge 0$, where ${\left\{G{M}_{(k,n)}\right\}}_{n\ge 0}$ denotes the Gersenne sequence.

Proposition 13. 

Let ${\left\{G{M}_{(k,n)}\right\}}_{n\ge 0}$ be the Gersenne sequence, and n be a non-negative integer. Hence, the following hold:

(a) 

${\displaystyle \sum _{i=0}^n{(-1)}^{i}G{M}_{(k,i)}=-\frac{c(k^{n+1}+1){\upsilon }_{n+1,k}}{k+1}}$, if n is odd;

(b) 

${\displaystyle \sum _{i=0}^n{(-1)}^{i}G{M}_{(k,i)}=c\frac{k^{n+2}({\upsilon }_{n+2,k}-{\upsilon }_{n+1,k})-(k{\upsilon }_{n+1,k}-{\upsilon }_{n+2,k})}{k^2-1}}$, if n is even.

In the above, $G{M}_{(k,0)}=a$, $G{M}_{(k,1)}=b$, $c=b-a$, $d=ka-b$, and ${\left\{{\upsilon }_{(k,m)}\right\}}_{m\ge 0}$ is the generalized k-Mersenne sequence given by Equation (8), with $k>1$.

Proof. 

(a)

First, suppose that n is an odd natural number, or equivalently, the last term is negative; thus,

$$\begin{array}{ccc}\hfill \sum _{i=0}^{2n+1}{(-1)}^{i}G{M}_{(k,i)}& =& G{M}_{(k,0)}-G{M}_{(k,1)}+G{M}_{(k,2)}-G{M}_{(k,3)}+\dots +G{M}_{(2n,k)}-G{M}_{(k,2n+1)}\hfill \\ & =& (G{M}_{(k,0)}+G{M}_{(k,2)}+\dots +G{M}_{(2n,k)})\hfill \\ & & -(G{M}_{(k,1)}+G{M}_{(k,3)}+\dots +G{M}_{(k,2n+1)})\hfill \\ & =& \sum _{i=0}^{n}G{M}_{(2i,k)}-\sum _{i=0}^{n}G{M}_{(k,2i+1)}.\hfill \end{array}$$

According to items (b) and (c) in Proposition 12, it follows that

$$\begin{array}{cc}\hfill \sum _{i=0}^{2n+1}{(-1)}^{i}G{M}_{(k,i)}=& -{\displaystyle \frac{c(k^{n+1}+1){\upsilon }_{n+1,k}}{k+1}}.\hfill \end{array}$$

(b)

The proof follows a similar approach to the one used in item (a).

□

5. Final Considerations

This study presents an innovative approach to the generalized Mersenne number, so that this generalization is essentially analogous to the generalized Horadam sequence for a second-order recurrence with $p-q=1$. Motivated by some previous work on Mersenne numbers, we generalize the Mersenne numbers by allowing for arbitrary initial values and a second-order linear recurrence that depends on a positive parameter k, thereby introducing a new family of Gersenne numbers. We explore their algebraic properties and derive several well-known identities, using explicit formulas.