Narrow Results

In this paper, we present a new encoding and decoding method based on the recently introduced Minesweeper Model. Here, the main idea is to use Fibonacci–Lucas tree diagrams, and this proposed method uses a public key and a private key. To test this method, a new algorithm is constructed to enable a faster and more accurate control.

Hypercubes and Fibonacci cubes are classical models for interconnection networks with interesting graph theoretic properties. We consider $k$-Fibonacci cubes, which we obtain as subgraphs of Fibonacci cubes by eliminating certain edges during the fundamental recursion phase of their construction. These graphs have the same number of vertices as Fibonacci cubes, but their edge sets are determined by a parameter $k$. We obtain properties of $k$-Fibonacci cubes including the number of edges, the average degree of a vertex, the degree sequence and the number of hypercubes they contain.

One-parameter Darboux deformations are established for the simple ordinary differential equation (ODE) satisfied by the continuous generalizations of the Fibonacci sequence recently discussed by Faraoni and Atieh [Symmetry 13, 200 (2021)], who promoted a formal analogy with the Friedmann equation in the FLRW homogeneous cosmology. The method allows the introduction of deformations of the continuous Fibonacci sequences, hence of Darboux-deformed Fibonacci (noninteger) numbers. Considering the same ODE as a parametric oscillator equation, the Ermakov–Lewis invariants for these sequences are also discussed.

We consider a very simple Mealy machine (two nontrivial states over a two-symbol alphabet), and derive some properties of the semigroup it generates. It is an infinite, finitely generated semigroup, and we show that the growth function of its balls behaves asymptotically like ℓ^α, for ; that the semigroup satisfies the identity g⁶ = g⁴; and that its lattice of two-sided ideals is a chain.

Maximum possible lengths of short words in unavoidable sets of order no more than n have the form log n + O(log log n). The respective log bases of the upper and lower bounds of the shortest and second shortest words are (for a two-letter alphabet) 2 and τ, the Golden Ratio. The latter result comes through identifying certain bases of free monoids.

The main objective for the next generation wireless network is the offer of a high data rate when the user is on the move. The key element that offers continuous connectivity is the handoff. In this paper, we propose a handoff prediction model, which can predict handoff behavior of the user well in advance and reduce the latency in the handoff operation. The prediction model is validated with real life scenario both for the pedestrian user and the vehicle user, traveling at a speed of 80km/h. The experimental result verifies the capability of the proposed algorithm to predict the future sample with accuracy and minimum latency. Simulation results demonstrate the proposed system outperforming the existing system compared to the probability of the handoff detection and minimizing the false alarm probability. There is also the fact of the proposed algorithm not requiring any additional hardware for predicting the mobility of the user.

In this paper, we define the Fibonacci-norm of a natural number n to be the smallest integer k such that n|F_k, the kth Fibonacci number. We show that for m ≥ 5. Thus by analogy we say that a natural number n ≥ 5 is a local-Fibonacci-number whenever . We offer several conjectures concerning the distribution of local-Fibonacci-numbers. We show that , where provided F_m+k ≡ F_m (mod n) for all natural numbers m, with k ≥ 1 the smallest natural number for which this is true.

We show that the set of the numbers that are the sum of a prime and a Fibonacci number has positive lower asymptotic density.

For certain sequences of numbers, we show that their convolution identities come from a parametrization of a variety equipped with a vector field. A Weyl algebra and the universal enveloping algebra of a Lie algebra appear in the framework.

In this paper, we continue to investigate the properties of those sequences $\{a_n\}$ satisfying the condition ${\sum }_{k=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{k}\right){(-1)}^k{a}_k=\pm a_n$$(n\ge 0)$. As applications we deduce some recurrence relations and congruences for Bernoulli and Euler numbers.

In this paper, we show that the lower density of integers representable as the sum of a prime and a Fibonacci number is at least 0.0254905.

Let ${(F_n)}_{n\ge 1}$ be the sequence of Fibonacci numbers. Guy and Matiyasevich proved that

We prove that for every periodic sequence $s={(s_n)}_{n\ge 1}$ in $\{-1,+1\}$ there exists an effectively computable rational number $C_s>0$ such that

Let $F_n$ be the $n$th Fibonacci number and $L_n$ be the $n$th Lucas number. The order of appearance $z(n)$ of a natural number $n$ is defined as the smallest natural number $k$ such that $n$ divides $F_k$. For instance, $z(F_n)=n$, for all $n>2$. In this paper, among other things, we prove that $z(F_m-F_n)$ depends on $L_p$, where $p$ is the greatest common divisor of numbers $m$ and $n$, which fulfill the condition $m\equiv n(mod2)$.

We present a new family of Gauss $(k,t)$-Horadam numbers and obtain Binet formula of this family. We give the relationship between this family and the known Gauss $t$-Horadam number. Then we prove the Cassini and Catalan identities for this family. Furthermore, we investigate the sums, the recurrence relations and generating functions of this family.