Related papers: The Square Trees in the Tribonacci Sequence
The conception of multi-alphabetical genetics is represented. Matrix forms of the representation of the multi-level system of molecular-genetic alphabets have revealed algebraic properties of this system. These properties are connected with…
Quasi-trees generalize trees in that the unique "path" between two nodes may be infinite and have any countable order type. They are used to define the rank-width of a countable graph in such a way that it is equal to the least upper-bound…
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…
We give a new characterization of generalized B\"uchi sequences (sequences whose sequence of squares has constant second difference $(a)$, for some fixed integer $a$) of length 3 over the integers and a strategy for attacking B\"uchi's n…
In this paper, we give quadratic approximation of generalized Tribonacci sequence $\{V_{n}\}_{n\geq0}$ defined by Eq. (\ref{eq:7}) and use this result to give the matrix form of the $n$-th power of a companion matrix of…
Tverberg's theorem says that a set with sufficiently many points in $\mathbb{R}^d$ can always be partitioned into $m$ parts so that the $(m-1)$-simplex is the (nerve) intersection pattern of the convex hulls of the parts. The main results…
The classes of tree permutations and forest permutations were defined by Acan and Hitczenko (2016). We study random permutations of a given length from these classes, and in particular the number of occurrences of a fixed pattern in one of…
Magic squares are well-known arrangements of integers with common row, column, and diagonal sums. Various other magic shapes have been proposed, but triangles have been somewhat overlooked. We introduce certain triangular arrangements of…
We introduce the notion of quota trees in directed graphs. Given a nonnegative integer ``quota'' for each vertex of a directed multigraph $G$, a quota tree is an immersed rooted tree which hits each vertex of $G$ the prescribed number of…
Fibonacci polynomials are generalizations of Fibonacci numbers, so it is natural to consider polynomial versions of the various results for Fibonacci numbers. According to Hong, Pongsriiam, Bulawa, and Lee, the generating function of the…
In the plane, three integer points ("fleas") are given. At every tick of time, two of them (say, P,Q) instantly jump to two vacant points R,S, so that PQRS is a square with that order of vertices. Description of all integers and…
Continued fractions are linked to Stern's diatomic sequence 0,1,1,2,1,3,2,3,1,4,... (given by the recursion relation a_2n=a_n and a_{2n+1} = a_n + a_{n+1}, where a_0=0 and a_1=1), which has long been known. Using a particular…
Neural networks with tree-based sentence encoders have shown better results on many downstream tasks. Most of existing tree-based encoders adopt syntactic parsing trees as the explicit structure prior. To study the effectiveness of…
According to a result of Richomme, Saari and Zamboni, the abelian complexity of the Tribonacci word satisfies $\rho^{\mathrm{ab}}(n)\in\{3,4,5,6,7\}$ for each $n\in\mathbb{N}$. In this paper we derive an automaton that evaluates the…
In the last 30 years, the mathematical theory of aperiodic order has developed enormously. Many new tilings and properties have been discovered, few of which are covered or anticipated by the early papers and books. Here, we start from the…
We provide an ergodic theory framework to study statistical properties of smooth sequences over the odd alphabet {1, 3}. The arithmetic nature of this alphabet yields a partition of the subshift of smooth sequences based on their local…
Common meadows are commutative and associative algebraic structures with two operations (addition and multiplication) with additive and multiplicative identities and for which inverses are total. The inverse of zero is an error term…
Only one three-term recurrence relation, namely, $W_{r}=2W_{r-1}-W_{r-4}$, is known for the generalized Tribonacci numbers, $W_r$, $r\in\mathbb{Z}$, defined by $W_{r}=W_{r-1}+W_{r-2}+W_{r-3}$ and \mbox{$W_{-r}=W_{-r+3}-W_{-r+2}-W_{-r+1}$},…
The Fibonacci cube of dimension n, denoted as $\Gamma\_n$, is the subgraph of the hypercube induced by vertices with no consecutive 1's. The irregularity of a graph G is the sum of |d(x)-d(y)| over all edges {x,y} of G. In two recent paper…
As well known the rotation distance D(S,T) between two binary trees S, T of n vertices is the minimum number of rotations of pairs of vertices to transform S into T. We introduce the new operation of chain rotation on a tree, involving two…