The Complex-Type <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M1" height="12.533pt" version="1.1" viewBox="-0.0657574 -12.2606 8.79534 12.533" width="8.79534pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-108" d="M480 416C480 431 465 448 438 448C388 448 312 383 252 330C217 299 188 273 155 237H153L257 680C262 700 263 712 253 712C240 712 183 684 97 674L92 648L126 647C166 646 172 645 163 606L23 -6L29 -12C51 -5 77 2 107 8C115 62 130 128 142 180C153 193 179 220 204 241C231 170 259 106 288 54C317 0 336 -12 358 -12C381 -12 423 2 477 80L460 100C434 74 408 54 398 54C385 54 374 65 351 107C326 154 282 241 263 299C296 332 351 377 403 377C424 377 436 372 445 368C449 366 456 368 462 375C472 386 480 402 480 416Z"/></g></svg>-</span>Pell Numbers and Their Applications

Aküzüm, Yeşim

doi:https://doi.org/10.1155/2023/6631659

Journal of Mathematics

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 6631659 | https://doi.org/10.1155/2023/6631659

The Complex-Type -Pell Numbers and Their Applications

Yeşim Aküzüm¹

Academic Editor: Xiaogang Liu

Received25 Jan 2023

Revised02 Mar 2023

Accepted04 Apr 2023

Published08 Jun 2023

Abstract

In this study, a new sequence called the complex-type -Pell number is defined. Also, we give properties of this sequence such as the generating matrix, the generating function, the combinatorial representations, the exponential representation, the sums, the permanental and determinantal representations, and the Binet formula. Then, we determine the periods of the recurrence sequence according to the modulo and produce cyclic groups with the help of the generating matrices of the sequence. We also get some findings about the ranks and periods of the complex-type -Pell sequence. Additionally, we create relations between the orders of the cyclic groups produced and the periods of the sequence. Then, this sequence is moved to groups and examined in detail in finite groups. As an application, we get the periods of the complex-type -Pell numbers in the polyhedral groups , , and and the quaternion group .

1. Introduction

Recurrence sequences are widely utilized to solve some problems in various scientific fields, or different problems in different scientific disciplines are directly created by taking the structural aspects of these sequences into account. As an example of such studies, the scientific outputs in [1–7] can be given. All recurrence sequences defined by recurrence relations and initial values are a special case of recurrence sequences defined by the relation. In this sense, recurrence sequences, such as Fibonacci, -step Fibonacci, Pell, generalized -order Pell, Jacobsthal, and generalized -order Jacobsthal, which were previously defined and associated with many scientific disciplines, were discussed and various properties were obtained [8–24]. Furthermore, the authors defined the new sequences in [25, 26] using quaternions and complex numbers, and then they gave various features. In Section 3, using the reduction relation of the generalized order- Pell numbers, a new sequence called the complex-type -Pell number is defined. Also, properties of this sequence such as the generating matrix, the generating function, the combinatorial representations, the exponential representation, the sums, permanental and determinantal representations, and the Binet formula are given.

Determining the periods of the recurrence sequences according to the modulo and reducing the elements of the generating matrices of the recurrence sequences according to the modulo , and producing cyclic and semigroups by choosing these matrices as generating are extremely interesting current study topics and are examples of recent studies in this sense [27–30] whose scientific outputs can be given. In addition, when determining the period of a sequence in the group, it is a common situation to use the period of this sequence according to the modulo . The scientific outputs in [31–34] can be given as an example of the studies in which such situations arise. With regard to the -step Fibonacci sequence, Lu and Wang studied the Wall number [35]. The -step Fibonacci sequence modulo is simply periodic according to Lu and Wang’s proof. In Section 4, the periods of the recurrence sequence were determined according to the modulo , and cyclic groups were produced with the help of the generating matrices of the sequence. Additionally, we get some findings about the ranks and periods of complex-type -Pell sequences for any and . Then, relations are created between the orders of the cyclic groups produced and the periods of the sequence, and special formulas are given for the orders of the cyclic groups and the periods of the sequence.

As the recurrence sequences are moved into groups, the elements of the group also appear as terms of the recurrence sequence moved to that group. Therefore, transferring recurrence sequences to groups is an extremely useful method in terms of examining the structures of groups. As an example of this situation, the study of [36] can be given and the theorem expressed as “ is a 2-generating group with and generating, and if a unit element occurs in either of the or Fibonacci sequences of , it is a commutative group.” Recurrence sequences were first transferred to algebraic structures by Wall’s work in [37]. In this study, Wall examined the standard Fibonacci sequences in cyclic groups. Later, Wilcox extended the theory to abelian groups with his work in [38]. In the next process, the concept was moved to different recurrence sequences and different algebraic structures. Some of the current ones of these studies are given in [39–44]. Therefore, in Section 5, the defined sequence is moved to groups and examined in detail in finite groups, both to test the usefulness of the defined reduction sequences and to better understand the structures of the groups. As an application, the periods of the complex-type -Pell numbers in the polyhedral groups , , and and the quaternion group are obtained.

2. Preliminaries

The generalized order- Pell numbers were defined by Kılıc and Tascı [17] as follows:for and , with initial conditionswhere is the ^th term of the ^th sequence. When , the generalized order- Pell sequence, , is reduced to the usual Pell sequence, .

The complex Fibonacci sequence is given [45] with the subsequent equation for such that and the Fibonacci number is designated as (cf. [46, 47]).

We assume that the ^th term in a sequence is defined recursively as the linear combination of the terms that came before itwhere are constants.

Number sequences can be derived from a matrix representation, as demonstrated by Kalman [15]. By using the companion matrix method, he arrived at the following closed-form formulas for the generalized sequence:and he demonstrated that

Definition 1. If after a particular point, a group of group elements only repeats a specified subsequence, then the sequence is periodic. The period of the sequence is determined by how many elements there are in the repeating subsequence. After the first element , for instance, the sequence is periodic and has period 3. If the first elements in a group element sequence form a repeating subsequence, then the group element sequence is simply periodic with period . For instance, the sequence is simply periodic with period 5.

Definition 2. The polyhedral group for has the presentation as follows:and the polyhedral group has the presentation for the generating pair , as follows:where (Coxeter and Moser) [48].

Definition 3. The generalized quaternion group is defined by the presentationfor every .

3. The Complex-Type -Pell Numbers

We next consider a new -step sequence to be called the complex-type -Pell numbers. This sequence is defined for any given and by the following recurrence relation:where , , and .

Using equation (10), we getwhere is a -square companion matrix as shown below:

The matrix is called the complex-type -Pell matrix.

Using an inductive argument for , it is simple to see that the ^th powers of the matrix iswhere is a matrix such as

Let us consider the following theorem, where is a companion matrix like this

Theorem 1 (see [49]). The entry in the matrix is given by the following formula:where the summing is over non-negative integers satisfying is a multinomial coefficient, and the coefficients in (16) are defined to be if .
Using this theorem, the next Corollary is given for complex-type -Pell numbers.

Corollary 1. Suppose that is the ^th complex-type -Pell number. Then,(i) where is the sum of non-negative numbers.(ii) where is the sum of non-negative numbers.

Proof. In Theorem 1, if we choose , for the case i. and , for the case ii., we can immediately view the outcomes from the matrix .
We will now deal with the exponential representation of the complex-type -Pell numbers. By calculating directly, we got the generating function of as shown below.

Theorem 2. The exponential representation for the complex-type -Pell numbers is as follows:

Proof. It is obvious thatUsing the function, the following relation is obtained:Thus, we obtainAs such, it complements the proof.
The sums of complex-type -Pell numbers are now being considered.
Letfor and let be the matrix as indicated below.Then, it can be shown by induction that

Definition 4. If exactly two nonzero entries can be found in the column or row of a real matrix , the matrix is said to be contractible in that column or row.
According to Brualdi and Gibson’s findings in citation [50], if is a real matrix of order and is contraction of , .
Let be a positive integer and let be the superdiagonal matrix, defined byfor .
Then, we give the following theorem:

Theorem 3. For and ,

Proof. An inductive method can be applied to for proof. Suppose the equation for is satisfied. Now let us prove that the equation satisfies . Then, expanding the with the Laplace expansion relative to the first row, we getSince , , and from definition of the complex-type -Pell numbers , the equation is easily obtainable as follows:This means that the proof is complete.
Let such that . Define the matrix as shown below.Suppose that the matrix is defined bywhere . Then, we can yield the next theorem using permanental representations.

Theorem 4. (i)For ,(ii)For ,

Proof. (i)Assuming that the equation is valid for , we now demonstrate that it is also valid for . When we expand the by the Laplace expansion of permanent with regard to the first row, we reach So, we achieve the conclusion.(ii)As we expand the with the Laplace expansion relative to the first row, we reachThe conclusion is reached by considering the result of part in Theorem 4 and the inductive argument.
If there is a -matrix such that , where stands for the Hadamard product of and , and then the matrix is said to be convertible.
We will now deal with the determinantal representations for the complex-type -Pell numbers. Let be the matrix, defined by

Corollary 2. For ,and

Proof. Since = , = , and = for , Theorems 3 and 4 give us the conclusion.
For the complex-type -Pell numbers, we now derive a generalized Binet formula. For this reason, we start by taking the following lemma.
From companion matrices, the characteristic equation of the complex-type -Pell matrix is known to be , which is also the characteristic equation of the complex-type -Pell numbers.

Lemma 1. The equation does not have multiple roots for .

Proof. Let , then . It is obvious that and for all . Let us assume that . Let be a multiple root of , then and . Given that is a multiple root, it is and . Considering the case and , we obtain . Thus, we obtainandfor , , and , which is a contradiction and with this contradiction, it is concluded.
If be the eigenvalues of the matrix , then by Lemma 1, ’s are known to be distinct. Define the Vandermonde matrix as shown below.We assume thatand is derived from by replacing the column of by the matrix .

Theorem 5. Let , thenfor .

Proof. The matrix may be diagonalized since are distinct. Then, we easily see that where . Since the matrix is invertible, we obtain . Then, the matrix is similar to ; so, we obtain . We therefore have the following linear system of equations:Then, for each , we get as follows:

Corollary 3. Suppose that is the ^th element of complex-type -Pell number, then

4. The Period and Rank of the Complex-Type -Pell Numbers Modulo

By a modulus reducing of the complex-type -Pell numbers , the repeating sequence is obtained as follows:where . Here, it is important to note that the recurrence relations in the sequences and are identical.

Theorem 6. For every value of , the sequence is a simply periodic sequence.

Proof. Let us think about the setIt is obvious that the set is finite. Let the cardinality of the set be indicated by the notation , then distinct -tuples of the complex-type -Pell numbers modulo are present in . Then, in the sequence , at least one -tuple is repeated twice. As a result, the subsequence that follows this -tuple repeats, proving that the sequence is periodic. Let , , and , then . We may easily determine the following by using the complex-type -Pell numbers:Therefore, it is simple to observe that and , indicating that the sequence is simply periodic.
Let us use the symbol to indicate the sequence ’s period.
An integer matrix shows that all of ’s elements are modulo , that is, for an integer matrix . Consider the set “” for a moment. The set “” is a semigroup if ; if , then the set “” is a cyclic group.
From companion matrices, we can easily obtain . Consequently, it follows that for each integer , “” is a cyclic group. It is simple to observe that from (13).

Definition 5. The rank of the sequence is the smallest positive integer such that . We designate the rank of by .
If , then the terms of begin with the index , that is, are exactly the initial terms of multiplied by a factor .
An easy arithmetic progression is formed by the exponents for which . Then, we haveSimilarly, an easy arithmetic progression is formed by the exponents for which for some , and henceAs a result, it is obvious that divides .
The formula yields the order of the sequence , and this order is shown by . Let , then is the least positive value of such that . Thus, it is established that is the least positive integer with . Consequently, we achieve . So, it is clear that is always a positive integer and that , the multiplicative order of .

Example 1. Sincewe get , , and .

Theorem 7. Let be a prime number. Then,(i)If is an element of , and is the smallest value such that , then for each integer , .(ii)If is an element of and is the smallest value such that , then for each integer , .

Proof. (i)We assume that is an element of and is the smallest value such that . Let be an element of . If , then . It follows that divides . Let be the smallest positive integer satisfying Additionally, we may write ; using the binomial theorem, we arrive at This suggests that divides . Consequently, or , and the latter is true if and only if there is that is not divisible by . Given that we presume to be an element of and is the smallest value such that , there exists that is not divisible by . This demonstrates that So, the proof is complete.(ii)The proof is omitted and is comparable to the evidence mentioned.

Theorem 8. If and are integers with , then for any . Similarly, .

Proof. Let . Then,andfor . It is seen that for by using the least common multiple operation. Consequently, and are obtained, indicating that divides . We also know that the following equivalences are satisfied:for . Then, we can writeand as a result of this, it follows that divides . The proof is finished.
It also uses a similar proof method for the period .

5. The Complex-Type -Pell Numbers in Groups

Let be a finite -generator group and let be the subset of such that if and only if is generated by . We call a generating -tuple for .

Definition 6. Let be a -generator group. For a generating -tuple , the complex-type -Pell orbit is defined as follows:The complex-type -Pell orbit of for generating -tuple is indicated by the notation .
The terms of a complex-type sequence which is defined by means of group elements determined by the following rules [25]:
For each elements of the group ,(i)Let be the identity of and consider , where and are the integers, then , , , .(ii)Given and , where , , , and are integers, .(iii)If , then .(iv) and .(v) and so and .It is important to note that we will obtain the terms of the complex-type -Pell orbit according to the above rules.

Theorem 9. Let be a -generator group. The complex-type -Pell orbit of is periodic if is finite.

Proof. We consider the following set: is a finite set since is also a finite set. If so, exists such that , for any . The sequence is periodic for all generating -tuples as a result of the repeating.
denotes the length of the complex-type -Pell orbit’s period .
The lengths of the periods of complex-type -Pell orbits in the polyhedral groups , and and the quaternion group will now be considered.

Theorem 10. For ,

Proof. Let us consider the group we consider the following sequence:We can obtain a repeating sequence by reducing this sequence by a modulus , which is indicated byLet us use the notation to denote the period of the sequence . Since , it is obvious that from equation (13). The sequence isIt is evident to say that ,and the complex-type -Pell orbit is in the form of four layers. Since and ,Then, we achieveThus, it is established that is the length of the period of the sequence .
A similar proof exists for .

Example 2. The sequence isThis demonstrates that . It is simply seen that .

Theorem 11. .

Proof. The direct calculation is used for proof. First, it should be noted that the present polyhedral group is as follows:Considering this group representation, the sequence isSo, the length of the period of the sequence for all values of is .

Theorem 12. For , .

Proof. Using the period , we consider the length of the period of the complex-type -Pell orbit in the quaternion group. Note that , , and . The complex-type -Pell orbit isand so, using the above equation, the complex-type -Pell orbit can be represented in the form below.where gcd and , and are the positive integers. So, we require the smallest integer such that , . If we choose , we getThe cycle begins again with the element after , since elements and depend on and for their values.
Thus, .

Example 3. The sequence iswhich implies that .

6. Conclusion

As it is known, while a problem is being constructed in all fields of modern science, even a small change to any concept to be used for a solution causes significant differences in the result to be achieved. Therefore, such changes can lead to shorter and more original results. In this sense, in this study, we defined a sequence called the complex-type -Pell number for the first time and gave its structural properties, such as the generating matrix, the generating function, the combinatorial representations, the exponential representation, the sums, the permanental and determinantal representations, and the Binet formula. Also, we determined the periods of the recurrence sequence according to the modulo and produced cyclic groups with the help of the generating matrices of the sequence. Then, this sequence was moved to groups and examined in detail in finite groups. Finally, we obtained the periods of the complex-type -Pell numbers in the polyhedral groups , , and and the quaternion group .

Data Availability

No data were used to support the findings of this study.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this article.

References

W. Adams and D. Shanks, “Strong Primality Tests that are not sufficient,” Mathematics of Computation, vol. 39, no. 159, pp. 255–300, 1982.
View at: Publisher Site | Google Scholar
P. G. Becker, “K-regular power series and mahler-type functional equations,” Journal of Number Theory, vol. 49, no. 3, pp. 269–286, 1994.
View at: Publisher Site | Google Scholar
M. Naschie, “Deriving the essential features of the standard model from the general theory of relativity,” Chaos, Solitons & Fractals, vol. 24, no. 4, pp. 941–946, 2005.
View at: Publisher Site | Google Scholar
A. S. Fraenkel and S. T. Kleinb, “Robust universal complete codes for transmission and compression,” Discrete Applied Mathematics, vol. 64, no. 1, pp. 31–55, 1996.
View at: Publisher Site | Google Scholar
D. Mandelbaum, “Synchronization of codes by means of Kautz’s Fibonacci encoding,” IEEE Transactions on Information Theory, vol. 18, no. 2, pp. 281–285, 1972.
View at: Publisher Site | Google Scholar
V. W. Spinadel, “The Metallic means family and forbidden symmetries,” Internet Mathematics J, vol. 2, pp. 279–288, 2002.
View at: Google Scholar
W. Stein, “Modeling the evolution of stelar architecture in vascular plants,” International Journal of Plant Sciences, vol. 154, no. 2, pp. 229–263, 1993.
View at: Publisher Site | Google Scholar
M. Bicknell, “Primer on Pell sequence and related sequences,” Fibonacci Quarterly, vol. 13, pp. 345–350, 1975.
View at: Google Scholar
O. Deveci and A. Shannon, “On the complex-type Catalan transform of the k-fibonacci numbers,” Journal of Integer Sequences, vol. 25, pp. 1352–1367, 2022.
View at: Google Scholar
O. Erdag and O. Deveci, “The arrowhead-Fibonacci-Random-type sequences,” Utilitas Mathematica, vol. 113, pp. 271–287, 2019.
View at: Google Scholar
O. Erdag and O. Deveci, “On the connections between Padovan numbers and Fibonacci p-numbers,” Jordan J. Math. Stat., vol. 15, pp. 507–521, 2022.
View at: Google Scholar
O. Erdag, S. Halıcı, and O. Deveci, “The complex-type Padovan-p sequences,” Mathematica Moravica, vol. 26, no. 1, pp. 77–88, 2022.
View at: Publisher Site | Google Scholar
O. Erdag, A. Shannon, and O. Deveci, “The arrowhead-Pell-Random-type sequences,” Notes on Number Theory and Discrete Mathematics, vol. 24, no. 1, pp. 109–119, 2018.
View at: Publisher Site | Google Scholar
G. Everest, A. V. D. Poorten, I. Shparlinski, and T. Ward, Recurrence Sequences, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, USA, 2003.
D. Kalman, “Generalized Fibonacci numbers by matrix methods,” Fibonacci Quarterly, vol. 20, pp. 73–76, 1982.
View at: Google Scholar
J. Hiller, Y. Akuzum, and O. Deveci, “The Adjacency-Pell-Hurwitz numbers,” Integers, vol. 18, pp. 1–16, 2018.
View at: Google Scholar
E. Kilic, “The generalized Pell (p,i)-numbers and their Binet formulas, combinatorial representations, sums,” Chaos, Solitons & Fractals, vol. 40, no. 4, pp. 2047–2063, 2009.
View at: Publisher Site | Google Scholar
E. Kilic and D. Tascı, “The generalized Binet Formula, representation and sums of the generalized order-k Pell numbers,” Taiwanese Journal of Mathematics, vol. 10, pp. 1661–1670, 2006.
View at: Google Scholar
M. Uysal and E. Ozkan, “Higher-order Jacobsthal-lucas quaternions,” Axioms, vol. 11, no. 12, p. 671, 2022.
View at: Publisher Site | Google Scholar
A. G. Shannon, P. G. Anderson, and A. F. Horadam, “Properties of cordonnier, perrin and van der Laan numbers,” Internat. J. Math. Ed. Sci. Tech, vol. 37, no. 7, pp. 825–831, 2006.
View at: Publisher Site | Google Scholar
A. P. Stakhov and B. Rozin, “Theory of Binet formulas for Fibonacci and Lucas p-numbers,” Chaos, Solitons & Fractals, vol. 27, no. 5, pp. 1162–1177, 2006.
View at: Publisher Site | Google Scholar
D. Tascı and E. Kilic, “On the order-k generalized Lucas numbers,” Applied Mathematics and Computation, vol. 155, no. 3, pp. 637–641, 2004.
View at: Publisher Site | Google Scholar
N. Tuglu, E. G. Kocer, and A. P. Stakhov, “Bivariate Fibonacci like p-polynomials,” Applied Mathematics and Computation, vol. 217, no. 24, pp. 10239–10246, 2011.
View at: Publisher Site | Google Scholar
F. Yılmaz and D. Bozkurt, “The generalized order-k Jacobsthal numbers,” Int. J. Contemp. Math. Sci., vol. 4, pp. 1685–1694, 2009.
View at: Google Scholar
O. Deveci and A. G. Shannon, “The complex-type k-Fibonacci sequences and their applications,” Communications in Algebra, vol. 49, no. 3, pp. 1352–1367, 2021.
View at: Publisher Site | Google Scholar
O. Deveci and A. G. Shannon, “The quaternion-Pell sequence,” Communications in Algebra, vol. 46, no. 12, pp. 5403–5409, 2018.
View at: Publisher Site | Google Scholar
H. Doostie and M. Hashemi, “Fibonacci lengths involving the wall number k(n),” J. Appl. Math. Comput., vol. 20, no. 1-2, pp. 171–180, 2006.
View at: Publisher Site | Google Scholar
S. Falcon and A. Plaza, “k-Fibonacci sequences modulo m,” Chaos, Solitons & Fractals, vol. 41, no. 1, pp. 497–504, 2009.
View at: Publisher Site | Google Scholar
E. Ozkan, “On truncated Fibonacci sequences,” Indian Journal of Pure and Applied Mathematics, vol. 38, pp. 241–251, 2007.
View at: Google Scholar
E. Ozkan, H. Aydin, and R. Dikici, “3-Step Fibonacci series modulo m,” Applied Mathematics and Computation, vol. 143, no. 1, pp. 165–172, 2003.
View at: Publisher Site | Google Scholar
H. Aydin and R. Dikici, “General Fibonacci sequences in finite groups,” Fibonacci Quarterly, vol. 36, pp. 216–221, 1998.
View at: Google Scholar
C. M. Campbell, H. Doostie, and E. F. Robertson, Fibonacci Length of Generating Pairs in Groups. Applications of Fibonacci Numbers, Springer, Dordrecht, 1990.
O. Deveci, Y. Akuzum, and E. Karaduman, “The Pell-Padovan p-sequences and its applications,” Utilitas Mathematica, vol. 98, pp. 327–347, 2015.
View at: Google Scholar
O. Deveci, S. Hulku, and A. G. Shannon, “On the co-complex-type k-Fibonacci numbers,” Chaos, Solitons & Fractals, vol. 153, Article ID 111522, 2021.
View at: Publisher Site | Google Scholar
K. Lu and J. Wang, “k-step Fibonacci sequence modulo m,” Utilitas Mathematica, vol. 71, pp. 169–178, 2006.
View at: Google Scholar
S. W. Knox, “Fibonacci sequences in finite groups,” Fibonacci Quarterly, vol. 30, pp. 116–120, 1992.
View at: Google Scholar
D. D. Wall, “Fibonacci series modulo m,” The American Mathematical Monthly, vol. 67, no. 6, pp. 525–532, 1960.
View at: Publisher Site | Google Scholar
H. J. Wilcox, “Fibonacci sequences of period n in groups,” Fibonacci Quarterly, vol. 24, pp. 356–361, 1986.
View at: Google Scholar
H. Aydin and G. C. Smith, “Finite p-Quotients of Some Cyclically Presented Groups,” Journal of the London Mathematical Society, vol. 49, no. 1, pp. 83–92, 1994.
View at: Publisher Site | Google Scholar
C. M. Campbell, P. P. Campbell, H. Doostie, and E. F. Robertson, “The Fibonacci lengths for certain Metacyclic groups,” Algebra Colloquium, vol. 11, pp. 215–222, 2004.
View at: Google Scholar
R. Dikici and G. C. Smith, “Recurrences in finite groups,” Turkish Journal of Mathematics, vol. 19, pp. 321–329, 1995.
View at: Google Scholar
E. Karaduman and H. Aydin, “k-nacci sequences in some special groups of finite order,” Mathematical and Computer Modelling, vol. 50, no. 1-2, pp. 53–58, 2009.
View at: Publisher Site | Google Scholar
T. Foguel and J. Hiller, “A note on abelian partitionable groups,” Communications in Algebra, vol. 48, no. 8, pp. 3268–3274, 2020.
View at: Publisher Site | Google Scholar
M. Hashemi and E. Mehraban, “On the generalized order 2-Pell sequence of some classes of groups,” Communications in Algebra, vol. 46, no. 9, pp. 4104–4119, 2018.
View at: Publisher Site | Google Scholar
A. F. Horadam, “A generalized Fibonacci sequence,” The American Mathematical Monthly, vol. 68, no. 5, pp. 455–459, 1961.
View at: Publisher Site | Google Scholar
G. Berzsenyi, “Sums of products of generalized Fibonacci numbers,” Fibonacci Quarterly, vol. 13, pp. 343-344, 1975.
View at: Google Scholar
A. F. Horadam, “Complex Fibonacci numbers and Fibonacci quaternions,” The American Mathematical Monthly, vol. 70, no. 3, pp. 289–291, 1963.
View at: Publisher Site | Google Scholar
H. S. M. Coxeter and W. O. J. Moser, Generator and Relations for Discrete Groups, Springer, Berlin, Germany, 3rd edition, 1972.
W. Y. C. Chen and J. D. Louck, “The combinatorial power of the companion matrix,” Linear Algebra and Its Applications, vol. 232, pp. 261–278, 1996.
View at: Publisher Site | Google Scholar
R. A. Brualdi and P. M. Gibson, “Convex polyhedra of doubly stochastic matrices I: applications of permanent function,” Journal of Combinatorial Theory - Series A, vol. 22, no. 2, pp. 194–230, 1977.
View at: Publisher Site | Google Scholar

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