Thek-nacci triangle and applications

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.

In this paper, we will see that some k -Fibonacci sequences are related to the classical Fibonacci sequence of such way that we can express the terms of a k -Fibonacci sequence in function of some terms of the classical Fibonacci sequence. And the formulas will apply to any sequence of a certain set of k' -Fibonacci sequences. Thus we find k -Fibonacci sequences relating to other k -Fibonacci sequences when σ'k is linearly dependent of .

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.

A History and Introduction to the Fibonacci Numbers The Fibonacci Numbers in Nature The Fibonacci Numbers and the Pascal Triangle The Fibonacci Numbers and the Golden Ratio The Fibonacci Numbers and Continued Fractions A Potpourri of Fibonacci Number Applications The Fibonacci Numbers Found in Art and Architecture The Fibonacci Numbers and Musical Form The Famous Binet Formula for Finding a Particular Fibonacci Number The Fibonacci Numbers and Fractals Epilogue Index.

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 Fibonacci sequence can be obtained by drawing diagonals in a Pascal’s triangle, and from this, we can obtain the Lucas identity. An investigation on the behavior of certain kinds of other diagonals inside a Pascal’s triangle identifies a new family of recursive sequences: the $k$-Padovan sequences. This family both contains the Fibonacci and the Padovan sequences. A general binomial identity for $k$-Padovan sequences which extends both the well-known Lucas identity and the less known Padovan identity is derived.

In this paper, we first establish several identities for the alternating sums in the Catalan triangle whose (n, p) entry is defined by Bn, p = \( \tfrac{p} {n}\left( {_{n - p}^{2n} } \right) \). Second, we show that the Catalan triangle matrix C can be factorized by C = FY = ZF, where F is the Fibonacci matrix. From these formulas, some interesting identities involving Bn, p and the Fibonacci numbers Fn are given. As special cases, some new relationships between the well-known Catalan numbers Cn and the Fibonacci numbers are obtained, for example: $$ C_n = F_{n + 1} + \sum\limits_{k = 3}^n {\left\{ {1 - \frac{{(k + 1)(k5 - 6)}} {{4(2k - 1)(2k - 3)}}} \right\}C_k F_{n - k + 1} } , $$ and $$ \begin{gathered} \frac{{n - 1}} {{n + 2}}C_n = \frac{1} {2}F_n + F_{n - 2} \hfill \\ + \sum\limits_{k = 4}^n {\left\{ {1 - \frac{{(k + 2)(5k^2 - 16k + 9)}} {{4(k - 1)(2k - 1)(2k - 3)}}} \right\}\frac{{k - 1}} {{k + 2}}C_k F_{n - k + 1} } . \hfill \\ \end{gathered} $$

Two new asymmetric generalizations of the Filbert and Lilbert matrices constructed by the products of two Fibonacci and Lucas numbers are considered, with additional parameter settings. Explicit formulae are derived for their LU-decompositions and inverses.

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.

Let $d\geq 2$ be an integer. In 2010, the second, third, and fourth authors gave necessary and sufficient conditions for the infinite products $$ \prod_{\textstyle {k=1\atop U_{d^k}\neq-a_i}}^{\infty}\biggl( 1+\frac{a_i}{U_{d^k}}\bigg)\quad (i=1,\dots,

In this paper, by means of independent sets in a graph with multiplicity assigned to each set, we introduce generalized Fibonacci–Leonardo numbers, which are the common generalization of the classical Fibonacci and Leonardo numbers. We give the Binet formula, the generating function, and we prove some identities for generalized Fibonacci–Leonardo numbers. We also define matrix generators for the introduced numbers.

Counting: Basic Subsets of a Set Pascal's Triangle Binomial Coefficient Identities Counting: Intermediate Finding a Polynomial The Upward-Extended Pascal's Triangle Recurrence Relations and Fibonacci Numbers Counting: Advanced Generating Functions and Making Change Integer Triangles Rook Paths and Queen Paths Discrete Probability Probability Spaces and Distributions Markov Chains Random Tournaments Number Theory Divisibility of Factorials and Binomial Coefficients Covering Systems Partitions of an Integer Information Theory What Is Surprise? A Coin-Tossing Game Shannon's Theorems Games A Little Graph Theory Background The Ramsey Game Tic-Tac-Toe and Animal Games Algorithms Counters Listing Permutations and Combinations Sudoku Solving and Polycube Packing Appendix A: Hints and Solutions to Exercises Appendix B: Notation Bibliography Index

Skew circulant matrices have various applications such as cryptography, signal processing, and many more. Their structure can potentially simplify their determinant and inverse computations. This study presents explicit formulas for the determinant and inverse of skew circulant matrices with entries from the alternating Fibonacci sequence. Elementary row and column operations are used to derive simple explicit formulas for the determinant and inverse. Computational tests using Wolfram Mathematica show that the algorithm built from these explicit formulas performs with much faster execution time than the built-in functions, especially for large matrix size. The proposed approach offers a practical method for the numerical computation of the determinant and inverse of these matrices

There is no homogeneous polynomial of degree two, \(\phi (x,y) = a{{x}^{2}} + bxy + c{{y}^{2}} \) for which the classical Fibonacci sequence {F n} satisfies an identity of the form $$\phi(F_{n},F_{n-1}) = constant$$ .

In this paper, we present a generalization of well-known k-Fibonacci sequence. Namely, we defined generalized k-Fibonacci sequence. This sequence generalizes others, k-Fibonacci sequence, classical Fibonacci sequence, Pell sequence and Jacobsthal sequence. We establish some of the interesting properties of generalized k-Fibonacci sequence. Also, we obtain a generating function for them.