English

In this study, we define a new generalization of the Fibonacci sequence that gives the copper ratio, and we will call it the copper Fibonacci sequence. In addition, inspired by the copper Fibonacci definition, we also define copper Lucas sequences, and then we give the relationships between the terms of these sequences. We present some properties, such as the Binet formulas, special summation formulas, special generating functions, etc. We find the relationships between the roots of the characteristic equation of these sequences and the general terms of these sequences. What is interesting here is that the relationships obtained from that between the roots of the characteristic equation of these new sequences and the terms of the sequences are satisfied in both roots. In addition, we examine the relationships between these sequences with the classic Fibonacci and Lucas sequences. Moreover, we calculate some identities of these sequences, such as Cassini and Catalan. Then Catalan transformation is applied to these sequences, and their terms are found. Furthermore, we apply Hankel transform to the Catalan transform of these sequences. Besides, we associate the terms of the Hankel transformation of the Catalan copper Fibonacci sequence with the classical Fibonacci numbers and the terms of the Hankel transformation of the Catalan copper Lucas sequence with the terms of the copper Lucas sequence. We present the application of copper Fibonacci and copper Lucas sequences to hyperbolic quaternions. Finally, the terms of the copper Fibonacci and copper Lucas sequences are associated with their hyperbolic quaternion values.

It is shown that period, rank and order of the ) , ( b a -Fibonacci sequence remain invariant by extension to the generalized ) , , , ( d c b a -Lucas sequence provided a specific determinant is relatively prime to the modulus.Given an odd prime modulus p and the 1  p exceptional ) , ( b a -Lucas sequences, for which p divides the discriminant of these sequences, we determine the corresponding periods and relate them to those of the ) , ( b a -Fibonacci sequence.The results can be viewed as extended versions of classical results by Wall in case ) 1 , 1 ( ) , (  b a .

The famous problem of determining all perfect powers in the Fibonacci sequence and the Lucas sequence has recently been resolved by three of the present authors. We sketch the proof of this result, and we apply it to show that the only Fibonacci numbers Fn such that Fn ± 1 is a perfect power are 0, 1, 2, 3, 5 and 8. The proof of the Fibonacci Perfect Powers Theorem involves very deep mathematics, combining the modular approach used in the proof of Fermat’s Last Theorem with Baker’s Theory. By contrast, using the knowledge of the all perfect powers in the Fibonacci and Lucas sequences, the determination of the perfect powers among the numbers Fn ± 1 is quite elementary.

Efficient quantum key distribution using Fibonacci-number coding with a biased basis choice

It is well known that the companion sequence of the Fibonacci sequence is Lucas's sequence. For the generalized Fibonacci sequences, the companion sequence is not unique. Several authors proposed different definitions, and they are in a certain sense all good. Our purpose is to introduce a family of companion sequences for some generalized Fibonacci sequence: the $r$-Fibonacci sequence. We evaluate the generating functions and give some applications, and we exhibit convolution relations that generalize some known identities such as Cassini's. Afterwards, we calculate the sums of their terms using matrix methods. Next, we propose a $q$-analogue and extend the definition to negative $n$s. Also, we define the incomplete associated sequences using a Euler--Seidel-like approach.

The Horadam sequence Hn(a,b;p,q) unifies a number of well-known sequences, such as Fibonacci and Lucas sequences. We use the context-free grammars as a new tool to study Horadam sequences. By introducing a set of auxiliary basis polynomials (v1,v2,v3) and using the formal derivative associated with the Horadam grammar, we solve the convolution coefficients and provide a unified method to discover convolution formulas associated with binomial coefficients. These results are extended to subsequences with indices kn through a parameterized grammar Gk. Using the modified grammar Gk˜, we derive convolution formulas involving the weighting term (−q)n−i. Furthermore, applying the proposed framework to (p,q)-Fibonacci and (p,q)-Lucas sequences, we derive explicit convolution formulas with parameters (p,q). The framework is also applied to derive specific identities for Pell and Pell–Lucas numbers, as well as for Fermat and Fermat–Lucas numbers.

On the spacings between C-nomial coefficients

In this paper, we give explicit formulas for elements of the Fibonacci, and Lucas Pascal triangles. The structure of these objects and Pascal's original triangle coincide. Keeping the rule of addition, we replace both legs of the Pascal triangle by the Fibonacci sequence, and the Lucas sequence, respectively. At the end of the study we describe how to determine such a formula for any binary recurrence $\{G_n\}^{\infty}_{n=0}$ satisfying $G_n=G_{n−1}+G_{n−2}$. Other scattered results are also presented.

The Braxton-Shore house has been considered as an exemplar of 'space architecture' – an architecture of 'space forms' which R. M. Schindler believed distinguished his work from contemporary functionalists and internationalists. To achieve such' space architecture', Schindler utilized a proportional system of a space reference frame as a mental tool. This paper first outlines the spatial scheme of the Braxton-Shore house, explains Schindler's space reference frame, and finally interprets the proportional design of the house in analogy with the 'row' system, the Fibonacci and Lucas sequence, and musical ratios.

Let $(F_{n}^{(k)})$ and $(L_{n})$ be the $k$-generalized Fibonacci and Lucas sequences, respectively. In this paper, we look at $k$-generalized Fibonacci numbers which are perfect powers of exponent larger than $1$ of Lucas numbers. That is, we deal with the Diophantine equation $$ F_{n}^{(k)}=L_{m}^{a}$$ in non-negative integers $k$, $n$, $m$, $a$, with $k\geq 3$, $m\geq 4$ and $a\geq 2$. We show that this equation has no solution under these conditions. The proof depends on lower bounds for linear forms in logarithms and some tools from Diophantine approximation.

The Pell sequence is given by the recurrence Pn = 2Pn−1 + Pn−2 with initial condition P0 = 0, P1 = 1 and its associated Pell-Lucas sequence is given by the same recurrence relation but with initial condition Q0 = 2, Q1 = 2. Here we show that 6 is the only perfect number appearing in these sequences. This paper continues a previous work that searched for perfect numbers in the Fibonacci and Lucas sequences.

Two generalizations of Lucas sequence

We prove some Hyers‐Ulam stability results for an operator linear equation of the second order that is patterned on the difference equation, which defines the Lucas sequences (and in particular the Fibonacci numbers). In this way, we obtain several results on stability of some linear functional and differential and integral equations of the second order and some fixed point results for a particular (not necessarily linear) operator.

In this paper we present another characterization of (<TEX>${\pm}1$</TEX>)-invariant sequences. We also introduce truncated Fibonacci and Lucas sequences of the second kind and show that a sequence <TEX>$x\;{\in}\;R^{\infty}$</TEX> is (-1)-invariant(l-invariant resp.) if and only if <TEX>$D[_x^0]$</TEX> is perpendicular to every truncated Fibonacci(truncated Lucas resp.) sequence of the second kind where <TEX>$$D=diag((-1)^0,\; (-1)^1,\;(-1)^2,{\ldots})$$</TEX>.

The investigation of existence Wing sequence in Half Butterfly Method (HBM) is motivated by the used of sequences such as Fibonacci, Catalan and Lucas sequences for key generation to encrypt and decrypt data. In 2017, Half Butterfly Method (HBM) a work for decomposition complete graph in distinct circuits based on wings movement was introduced. The hidden idea for series of movement in HBM is an eye opener to unravel the hidden pattern associated with the flip-flops of the wings movement. Thus, this paper oriented to construct a new sequence, Wing sequence based on the hidden patterns of the associate iteration for the flip-flops of the wings movement in HBM. The sequence is constructed by fixing one element and creating direction from one element to another element. The total directions of each element will generate a set of series sequence that is called as the Wing sequence.