Leonardo’s three-dimensional relations and some identities

In this work, new results are explored in relation to the Leonardo sequence. With that, a study about this second order recursive sequence, little explored in the mathematical scope, is briefly presented, relating it to the Fibonacci sequence. Thus, its complexification process is carried out, where from its one-dimensional model, imaginary units are inserted, obtaining Leonardo’s three-dimensional numbers. In this way, the imaginary units i and j are inserted. Finally, some three-dimensional identities are presented for Leonardo’s numbers.

Qualitative Behavior of a Nonlinear Generalized Recursive Sequence with Delay

Difference equations are of growing importance in engineering in view of their applications in discrete time-systems used in association with microprocessors. We will check out the global stability and boundedness for a nonlinear generalized high-order difference equation with delay.

Dynamics and Solutions’ Expressions of a Higher-Order Nonlinear Fractional Recursive Sequence

The principle purpose of this article is to examine some stability properties for the fixed point of the below rational difference equation U n + 1 = ξ U n − 8 + ε U n − 8 2 / μ U n − 8 + κ U n − 17 where ξ , ε , μ , and κ are arbitrary real numbers. Moreover, solutions for some special cases of the proposed difference equation are introduced.

On Polynomial Recursive Sequences

We study the expressive power of polynomial recursive sequences, a nonlinear extension of the well-known class of linear recursive sequences. These sequences arise naturally in the study of nonlinear extensions of weighted automata, where (non)expressiveness results translate to class separations. A typical example of a polynomial recursive sequence is bn = n!. Our main result is that the sequence un = nn is not polynomial recursive.

Inequalities approach in determination of convergence of recurrence sequences

The paper proves convergence for three uniquely defined recursive sequences, namely, arithmetico-geometric sequence, the Newton-Raphson recursive sequence, and the nested/composite recursive sequence. The three main hurdles for this prove processes are boundedness, monotonicity, and convergence. Oftentimes, these processes lie in the predominant use of prove by mathematical induction and also require some bit of creativity and inspiration drawn from the convergence monotone theorem. However, these techniques are not adopted here, rather, as a novelty, extensive use of basic manipulation of inequalities and useful equations are applied in illustrating convergence for these sequences. Moreover, we established a mathematical expression for the limit of the nested recurrence sequence in terms of its leading term which yields favorable results.

LOOK, KNAVE

We examine a recursive sequence in which $s_n$ is a literal description of what the binary expansion of the previous term $s_{n-1}$ is not. By adapting a technique of Conway, we determine the limiting behaviour of $\{s_n\}$ and dynamics of a related self-map of $2^{\mathbb {N}}$ . Our main result is the existence and uniqueness of a pair of binary sequences, each the complement-description of the other. We also take every opportunity to make puns.

On Stability Analysis of Higher-Order Rational Difference Equation

In this paper, we study the equilibrium points, local asymptotic stability of equilibrium points, global behavior of equilibrium points, boundedness and periodicity of the rational recursive sequence wn+1=wn−pα+βwn/γwn+δwn−r, where γwn≠−δwn−r for r∈0,∞, α, β, γ, δ∈0,∞, and r>p≥0. With initial values w−p,w−p+1,…,w−r,w−r+1,…,w−1, and w0 are positive real numbers. Some numerical examples are given to verify our theoretical results.

Recursive sequence generation in monkeys, children, U.S. adults, and native Amazonians

The question of what computational capacities, if any, differ between humans and nonhuman animals has been at the core of foundational debates in cognitive psychology, anthropology, linguistics, and animal behavior. The capacity to form nested hierarchical representations is hypothesized to be essential to uniquely human thought, but its origins in evolution, development, and culture are controversial. We used a nonlinguistic sequence generation task to test whether subjects generalize sequential groupings of items to a center-embedded, recursive structure. Children (3 to 5 years old), U.S. adults, and adults from a Bolivian indigenous group spontaneously induced recursive structures from ambiguous training data. In contrast, monkeys did so only with additional exposure. We quantify these patterns using a Bayesian mixture model over logically possible strategies. Our results show that recursive hierarchical strategies are robust in human thought, both early in development and across cultures, but the capacity itself is not unique to humans.