binet formula
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2021 ◽  
Vol 27 (4) ◽  
pp. 257-266
Author(s):  
Fügen Torunbalcı Aydın ◽  

In this paper, k-Fibonacci hybrid numbers are defined. Also, some algebraic properties of k-Fibonacci hybrid numbers such as Honsberger identity, Binet Formula, generating functions, d’Ocagne identity, Cassini and Catalan identities are investigated. In addition, we also give 2 × 2 and 4 × 4 representations of the k-Fibonacci hybrid numbers HF_{k,n}.


2021 ◽  
Vol 21 (3) ◽  
pp. 625-638
Author(s):  
CAGLA CELEMOGLU

In this article, firstly, we have described new generalizations of generalized k - Horadam sequence and we named the generalizations as another generalized k - Horadam sequence {H k,n}nE, a different generalized k - Horadam sequence {qk,n} and an altered generalized k - Horadam sequence {Qk,n) , respectively. Then, we have studied properties of these new generalizations and we have obtained generating function and extended Binet formula for each generalization. Also, we have introduced a power sequence for an altered generalized k - Horadam sequence in order to be used in different applications like number theory, cryptography, coding theory and engineering.


2021 ◽  
Vol 27 (2) ◽  
pp. 70-78
Author(s):  
Renata Passos Machado Vieira ◽  
Milena Carolina dos Santos Mangueira ◽  
Francisco Regis Vieira Alves ◽  
Paula Maria Machado Cruz Catarino

In this article, a study is carried out around the Perrin sequence, these numbers marked by their applicability and similarity with Padovan’s numbers. With that, we will present the recurrence for Perrin’s polynomials and also the definition of Perrin’s complex bivariate polynomials. From this, the recurrence of these numbers, their generating function, generating matrix and Binet formula are defined.


2021 ◽  
Vol 27 (1) ◽  
pp. 134-137
Author(s):  
Helmut Prodinger ◽  

Balancing numbers possess, as Fibonacci numbers, a Binet formula. Using this, partial sums of arbitrary powers of balancing numbers can be summed explicitly. For this, as a first step, a power B_n^l is expressed as a linear combination of B_{mn}.


2021 ◽  
Vol 27 (1) ◽  
pp. 148-160
Author(s):  
Anthony G. Shannon ◽  
◽  
Özgür Erdağ ◽  
Ömür Deveci ◽  
◽  
...  

In this paper, we define the Fibonacci–Pell p-sequence and then we discuss the connection of the Fibonacci–Pell p-sequence with the Pell and Fibonacci p-sequences. Also, we provide a new Binet formula and a new combinatorial representation of the Fibonacci–Pell p-numbers by the aid of the n-th power of the generating matrix of the Fibonacci–Pell p-sequence. Furthermore, we derive relationships between the Fibonacci–Pell p-numbers and their permanent, determinant and sums of certain matrices.


2021 ◽  
Vol 52 ◽  
pp. 17-29
Author(s):  
Engin Özkan ◽  
Mine Uysal

We introduce Mersenne-Lucas hybrid numbers. We give the Binet formula, the generating function, the sum, the character, the norm and the vector representation of these numbers. We find some relations among Mersenne-Lucas hybrid numbers, Jacopsthal hybrid numbers, Jacopsthal-Lucas hybrid numbers and Mersenne hybrid numbers. Then we present some important identities such as Cassini identities for Mersenne-Lucas hybrid numbers


2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Helmut Prodinger

AbstractA new family of generalized Pell numbers was recently introduced and studied by Bród ([2]). These numbers possess, as Fibonacci numbers, a Binet formula. Using this, partial sums of arbitrary powers of generalized Pell numbers can be summed explicitly. For this, as a first step, a power P lnis expressed as a linear combination of Pmn. The summation of such expressions is then manageable using generating functions. Since the new family contains a parameter R = 2r, the relevant manipulations are quite involved, and computer algebra produced huge expressions that where not trivial to handle at times.


Author(s):  
Faruk Kaplan ◽  
Arzu Özkoç Öztürk

The main object of the present paper is to consider the binomial transforms for Horadam quaternion sequences. We gave new formulas for recurrence relation, generating function, Binet formula and some basic identities for the binomial sequence of Horadam quaternions. Working with Horadam quaternions, we have found the most general formula that includes all binomial transforms with recurrence relation from the second order. In the last part, we determined the recurrence relation for this new type of quaternion by working with the iterated binomial transform, which is a dierent type of binomial transform.


2020 ◽  
Vol 28 (3) ◽  
pp. 89-102
Author(s):  
Özgür Erdağ ◽  
Ömür Deveci ◽  
Anthony G. Shannon

AbstractIn this paper, we define the Pell-Pell p-sequence and then we discuss the connection of the Pell-Pell p-sequence with Pell and Pell p-sequences. Also, we provide a new Binet formula and a new combinatorial representation of the Pell-Pell p-numbers by the aid of the nth power of the generating matrix the Pell-Pell p-sequence. Furthermore, we obtain an exponential representation of the Pell-Pell p-numbers and we develop relationships between the Pell-Pell p-numbers and their permanent, determinant and sums of certain matrices.


2020 ◽  
Vol 77 (1) ◽  
pp. 27-38
Author(s):  
Dorota Bród ◽  
Anetta Szynal-Liana ◽  
Iwona Włoch

AbstractIn this paper, we introduce bihyperbolic balancing and Lucas-balancing numbers. We give some of their properties, among others the Binet formula, Catalan, Cassini, d’Ocagne identities and the generating function.


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