Related papers: Recursive Triangles Appearing Embedded in Recursiv…

In an automatic search, we found conjectural recurrences for some sequences in the OEIS that were not previously recognized as being D-finite. In some cases, we are able to prove the conjectured recurrence. In some cases, we are not able to…

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…

In this article one builds a class of recursive sets, one establishes properties of these sets, and one proposes applications.

In this paper, a theorem about similar triangles is proved. It shows that two small and four large triangles similar to the original triangle can appear if we choose well among several intersections of the perpendicular bisectors of the…

We define a sequence of positive integers recursively, where each term is determined as follows: starting with a given positive integer, if the term is odd, the next is the sum of its positive divisors; if the term is even, the subsequent…

We characterize a family of number triangles whose production matrices are closely related to the original number triangle. We study a number of such triangles that are of combinatorial significance. For a specific subfamily, these…

We define recurrence matrices and study a few properties (links with automatic sequences, branch groups etc.) of them.

The first author introduced a sequence of polynomials (\cite{8}, sequence A174531) defined recursively. One of the main results of this study is proof of the integrality of its coefficients.

If P is a point inside triangle ABC, then the cevians through P extended to the circumcircle of triangle ABC create a figure containing a number of curvilinear triangles. Each curvilinear triangle is bounded by an arc of the circumcircle…

In general graph theory, the only relationship between vertices are expressed via the edges. When the vertices are embedded in an Euclidean space, the geometric relationships between vertices and edges can be interesting objects of study.…

In this short note, we establish some identities containing sums of binomials with coefficients satisfying third order linear recursive relations. As a result and in particular, we obtain general forms of earlier identities involving…

In this note we show that the known relation between double groupoids and matched pairs of groups may be extended, or seems to extend, to the triple case. The references give some other occurrences of double groupoids.

Scattered sequences are a generalization of scattered polynomials. So far, only scattered sequences of order one and two have been constructed. In this paper an infinite family of scattered sequences of order three is obtained. Equivalence…

In this paper we formally define the family of sequences know as "Pea Pattern". We then analyse its behaviour and conditions for fixed and periodic points. The paper ends with a list of fixed points and cycles.

We study a new type of sequences whose elements are defined in terms of the position, sign and magnitude of another element of the sequence. The name ultra-recursive comes from the fact that these sequences possess terms that are generated…

A Circumconic passes through a triangle's vertices. We define the Circumbilliard, a circumellipse to a generic triangle for which the latter is a 3-periodic. We study its properties and associated loci.

We define a triangular array closely related to Stern's diatomic array and show that for a fixed integer $r\geq 1$, the sum $u_r(n)$ of the $r$th powers of the entries in row $n$ satisfy a linear recurrence with constant coefficients. The…

With the $[0,1,2]$-family of cyclic triangulations we introduce a rich class of vertex-transitive triangulations of surfaces. In particular, there are infinite series of cyclic $q$-equivelar triangulations of orientable and non-orientable…

In this work we resolve several conjectures stated in the On-Line Encyclopedia of Integer sequences.

We study families of rational curves on an algebraic variety satisfying incidence conditions. We prove an analogue of bend-and-break: that is, we show that under suitable conditions, such a family must contain reducibles. In the case of…