3 citations · 8 across the 14 of their papers we have counts for
15 papers
Admissible Sequences, Semiconvergents, and Rational Catalan Numbers
Mike Winkler
Let be odd and put . We count parity sequences with exactly odd entries for which the coefficient in the iteration first drops belo…
Realized Rank Certificates for Matchstick Frameworks and Insertion Edges
Mike Winkler
We give an exact, checkable rank-certificate method for realized planar unit-distance frameworks. The method is motivated by Vogel's computations for matchstick graphs and by the i…
Thinned Wallis-type prime products in residue classes modulo
Mike Winkler
For odd primes we consider the factors , where is the quadratic Dirichlet character modulo , and study products of restricted to uni…
Aperiodic Sets of Prototiles Extracted From the Penrose Rhomb Tiling
Mike Winkler
We present aperiodic sets of prototiles whose shapes are based on the well-known Penrose rhomb tiling. Some decorated prototiles lead to an exact Penrose rhomb tiling without any m…
Approximate Solutions of 4-regular Matchstick Graphs with 50-62 Vertices
Mike Winkler
A 4-regular matchstick graph is a planar unit-distance graph whose vertices have all degree 4. Examples of 4-regular matchstick graphs are currently known for all number of vertice…
A 3-regular matchstick graph of girth 5 consisting of 54 vertices
Mike Winkler, Peter Dinkelacker, Stefan Vogel
In 2010 it was proved that a 3-regular matchstick graph of girth 5 must consist at least of 30 vertices. The smallest known example consisted of 180 vertices. In this article we co…
4-regular planar unit triangle graphs without additional triangles
Mike Winkler, Peter Dinkelacker, Stefan Vogel
In this article we proof the existence of 4-regular planar unit-distance graphs consisting only of unit triangles without additional triangles. It is shown that the smallest number…
Deterministic Structures in the Stopping Time Dynamics of the 3x+1 Problem
Mike Winkler
The problem concerns the iteration of the map defined by for even and for odd . We study the coefficient stoppi…
A catalog of 4-regular and (2;4)-regular matchstick graphs
Mike Winkler
The first part (page 1 - 7) of this article presents the currently known examples of 4-regular matchstick graphs with 63 - 70 vertices. The second part (page 8 - 15) presents the c…
On the existence of 4-regular matchstick graphs
Mike Winkler, Peter Dinkelacker, Stefan Vogel
A matchstick graph is a planar unit-distance graph. We call it \emph{4-regular} if every vertex has degree 4. While examples of 4-regular matchstick graphs with fewer than 63 verti…
Minimal completely asymmetric (4; n)-regular matchstick graphs
Mike Winkler, Peter Dinkelacker, Stefan Vogel
A matchstick graph is a graph drawn with straight edges in the plane such that the edges have unit length, and non-adjacent edges do not intersect. We call a matchstick graph $(m;n…
New minimal (4; n)-regular matchstick graphs
Mike Winkler, Peter Dinkelacker, Stefan Vogel
A matchstick graph is a graph drawn with straight edges in the plane such that the edges have unit length, and non-adjacent edges do not intersect. We call a matchstick graph ($m;n…
New results on the stopping time behaviour of the Collatz 3x + 1 function
Mike Winkler
Let . For the Collatz 3x + 1 function exists for each a set of different residue classes of starting num…
An Identity of Fillipi Related to Fermat-Type Equations
Mike Winkler, Andreas Fillipi
We give a structural proof of an algebraic identity proposed by Fillipi in 2014. For every integer , the identity represents an explicit multiple of as $\mathc…
Fibonacci Enumeration of Residue Classes via Forbidden Binary Factors
Mike Winkler
Binary words avoiding adjacent ones are counted by Fibonacci numbers. We identify a natural residue-class enumeration in which this classical word count arises from parity data of…