Related papers: Remarks On General Fibonacci Numbers
Recently, Chu studied some properties of the partial sums of the sequence $P^k(F_n)$, where $P(F_n)=\big(\sum_{i=1}^nF_i\big)_{n\geq1}$ and $(F_n)_{n\geq1}$ is the Fibonacci sequence, and gave its combinatorial interpretation. We generalize…
In 2021, Guyer and Mbirika gave two equivalent formulas that computed the greatest common divisor (GCD) of all sums of $k$ consecutive terms in the generalized Fibonacci sequence $\left(G_n\right)_{n \geq 0}$ given by the recurrence $G_n =…
We consider various properties and manifestations of some sign-alternating univariate polynomials borne of right-triangular integer arrays related to certain generalizations of the Fibonacci sequence. Using a theory of the root geometry of…
Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. We use this fact to introduce a family of double inequalities involving the generating function for the number of…
Morphic sequences form a natural class of infinite sequences, typically defined as the coding of a fixed point of a morphism. Different morphisms and codings may yield the same morphic sequence. This paper investigates how to prove that two…
We discuss an equivalence relation on the set of square binary matrices with the same number of 1's in each row and each column. Each binary matrix is represented using ordered n-tuples of natural numbers. We give a few starting values of…
We study determinants of matrices whose entries are powers of Fibonacci numbers. We then extend the results to include entries that are powers of generalized Fibonacci numbers defined as a second-order linear recurrence relation. These…
For a set $A$ of positive integers with $\gcd(A)=1$, let $\langle A \rangle$ denote the set of all finite linear combinations of elements of $A$ over the non-negative integers. Then it is well known that only finitely many positive integers…
Fibonacci codes are self-synchronizing variable-length codes that are proven useful for their robustness and compression capability. Asymptotically, these codes provide better compression efficiency as the order of the underlying Fibonacci…
An interesting characterization of the Fibonacci numbers is that, if we write them as $F_1 = 1$, $F_2 = 2$, $F_3 = 3$, $F_4 = 5, ...$, then every positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers. This is…
The study of Diophantine triples taking values in linear recurrence sequences is a variant of a problem going back to Diophantus of Alexandria which has been studied quite a lot in the past. The main questions are, as usual, about existence…
We derive closed form expressions for finite and infinite Fibonacci-Lucas sums having products of Fibonacci or Lucas numbers in the denominator of the summand. Our results generalize and extend those obtained by pioneer Brother Alfred…
Motivated by a question of Venkataramana, we consider the greatest common divisor of $\phi(f(n))$ where $f$ is a primitive polynomial with integer coefficients, and $n$ ranges over all natural numbers. Assuming Schinzel's hypothesis, we…
In this paper, we generalize a lot of facts from John Conway and Alex Ryba's paper, \textit{The extra Fibonacci series and the Empire State Building}, where we replace the Fibonacci sequence with the Tribonacci sequence. We study the…
We study generalized Fibonacci sequences $F_{n+1}=PF_n-QF_{n-1}$ with initial values $F_0=0$ and $F_1=1$. Let $P,Q$ be nonzero integers such that $P^2-4Q$ is not a perfect square. We show that if $Q=\pm 1$ then the sequence…
The following article summarizes research where theorems and their respective demonstrations are postulated based on quadratic equations with special properties given by the Pythagorean triplets and the Fibonacci sequence given the second…
In this paper, we introduce the generalized Fibonacci-Lucas quaternions and we prove that the set of these elements is an order,in the sense of ring theory, of a quaternion algebra. Moreover, we investigate some properties of these…
We introduce the notion of Differential Sequences of ordinary differential equations. This is motivated by related studies based on evolution partial differential equations. We discuss the Riccati Sequence in terms of symmetry analysis,…
In this paper we introduce a family of infinite words that generalize the Fibonacci word and we study their combinatorial properties. Moreover, we associate to this family of words a family of curves, which have fractal properties, in…
We define generalized bivariate polynomials, from which upon specification of initial conditions the bivariate Fibonacci and Lucas polynomials are obtained. Using essentially a matrix approach we derive identities and inequalities that in…