Related papers: Studies on the Pea Pattern Sequence
We define recurrence matrices and study a few properties (links with automatic sequences, branch groups etc.) of them.
We call $i$ a fixed point of a given sequence if the value of that sequence at the $i$-th position coincides with $i$. Here, we enumerate fixed points in the class of restricted growth sequences. The counting process is conducted by…
We define a class of sequences ${a_n}$ by $a_1=a$ and $a_{n+1}=P(a_n)$, where $P(x)$ is a polynomial with real coefficients. We then find out for which values $a$ and for which polynomials $P(x)$ these sequences will be constant after a…
We study pattern densities in binary sequences, finding optimal limit sequences with fixed pattern densities.
We define a family of meta-Fibonacci sequences where the order of the of recursion at stage n is a variable r(n), and the n^{th} term of a sequence is the sum of the previous r(n) terms. For the terms of any such sequence, we give upper and…
All known terrestrial proteins are coded as continuous strings of ~20 amino acids. The patterns formed by the repetitions of elements in groups of finite sequences describes the natural architectures of protein families. We present a method…
We introduce a transformation of finite integer sequences, show that every sequence eventually stabilizes under this transformation and that the number of fixed points is counted by the Catalan numbers. The sequences that are fixed are…
In this note we classify sequences according to whether they are morphic, pure morphic, uniform morphic, pure uniform morphic, primitive morphic, or pure primitive morphic, and for each possibility we either give an example or prove that no…
Fixed points represent equilibrium states, stability, and solutions to a range of problems. It has been an active field of research. In this paper, we provide an overview of the main branches of fixed point theory. We discuss the key…
Regular sequences are natural generalisations of fixed points of constant-length substitutions on finite alphabets, that is, of automatic sequences. Using the harmonic analysis of measures associated with substitutions as motivation, we…
In this paper, we study the occurrence of patterns in the cycle structures of permutations.
We continue the work begun in OEIS sequence A332636 which presents recursive sequences that have triangles that appear embedded in them. This paper i) generalizes the main result presented in A332636, ii) provides a complete set of…
A consecutive pattern in a permutation $\pi$ is another permutation $\sigma$ determined by the relative order of a subsequence of contiguous entries of $\pi$. Traditional notions such as descents, runs and peaks can be viewed as particular…
Coordination sequences of periodic and quasiperiodic graphs are analysed. These count the number of points that can be reached from a given point of the graph by a number of steps along its bonds, thus generalising the familiar coordination…
In real-world scenario, many phenomena produce a collection of events that occur in continuous time. Point Processes provide a natural mathematical framework for modeling these sequences of events. In this survey, we investigate…
First we define a new kind of function over $\mathbb{N}$. For each $i\in\mathbb{N}$ we have an associated function, which will be called $S_i$ . Then we define a new kind of sequence, to be made from the functions $S_i$ . Finally, we will…
In this paper, we present formula solutions of a family of difference equations of higher order. We discuss the periodic nature of the solutions and we investigate the stability character of the equilibrium points. We utilize Lie symmetry…
A family of sequences produced by a non-homogeneous linear recurrence formula derived from the geometry of circles inscribed in lenses is introduced and studied. Mysterious ``underground'' sequences underlying them are discovered in this…
We show that the closure of the value set of a real linear recurrence sequence is the union of a countable set and a finite collection of intervals. Conversely, any finite collection of closed intervals is the closure of the value set of…
In this paper we elaborate on the structure of the semigroup tree and the regularities on the number of descendants of each node observed earlier. These regularites admit two different types of behavior and in this work we investigate which…