Related papers: Realizability of Some Combinatorial Sequences
For a sequence $\{a_n\}_{n\geq 0}$ of real numbers and for a parameter $0<p<1$, we define the sequence of its arithmetic means $\{a^*_n\}_{n\geq 0}$ and the sequence of its $p$-binomial means $\{a^p_n\}_{n\geq 0}$ as \begin{align*}…
An ordered triple $(s,p,n)$ is called admissible if there exist two different multisets $X=\{x_1,x_2,\dotsc,x_n\}$ and $Y=\{y_1,y_2,\dotsc,y_n\}$ such that $X$ and $Y$ share the same sum $s$, the same product $p$, and the same size $n$. We…
We consider self-averaging sequences in which each term is a weighted average over previous terms. For several sequences of this kind it is known that they do not converge to a limit. These sequences share the property that $n$th term is…
For x and y sequences of real numbers define the inner product (x,y) = x(0)y(0) + x(1)y(1)+ ... which may not be finite or even exist. We say that x and y are orthogonal iff (x,y) converges and equals 0. Define l_p to be the set of all real…
The Catalan number sequence is one of the most famous number sequences in combinatorics and is well studied in the literature. In this paper we further investigate its fundamental properties related to the moment problem and prove for the…
We identify pairs of positive integers $(t, d)$ with the property that the integer sequence with general term $\lfloor{n^t/d\rfloor}$ contains at most finitely many primes.
Suppose that we are given an infinite binary sequence which is random for a Bernoulli measure of parameter $p$. By the law of large numbers, the frequency of zeros in the sequence tends to~$p$, and thus we can get better and better…
A polynomial over a ring is called decomposable if it is a composition of two nonlinear polynomials. In this paper, we obtain sharp lower and upper bounds for the number of decomposable polynomials with integer coefficients of fixed degree…
An infinite binary sequence is deemed to be random if it has all definable properties that hold almost surely for the usual probability measure on the set of infinite binary sequences. There are only countably many such properties, so it…
One of the most popular and studied recursive series is the Fibonacci sequence. It is challenging to see how Fibonacci numbers can be used to generate other recursive sequences. In our article, we describe some families of integer…
It is shown that for two large subclasses of discrete-time nonlinear systems - analytic systems defined on a compact state space and rational systems - the minimum length $r^*$ for input sequences, called here accessibility index of the…
The proofs that the real numbers are denumerable will be shown, i.e., that there exists one-to-one correspondence between the natural numbers $N$ and the real numbers $\Re$. The general element of the sequence that contains all real numbers…
A sequence of nonzero integers $f = (f_1, f_2, \dots)$ is ``binomid'' if every $f$-binomid coefficient $\left[\! \begin{array}{c} n \\ k \end{array}\! \right]_f$ is an integer. Those terms are the generalized binomial coefficients: \[…
We prove the following conjecture of Zeilberger. Denoting by $C_n$ the Catalan number, define inductively $A_n$ by $(-1)^{n-1}A_n=C_n+\sum_{j=1}^{n-1} (-1)^{j} \binom{2n-1}{2j-1} A_j \,C_{n-j}$ and $a_n=2A_n/C_n$. Then $a_n$ (hence $A_n$)…
Given a graded ring $A$ and a homogeneous ideal $I$, the ideal is said to be of linear type if the Rees algebra of $I$ is isomorphic to the symmetric algebra of $I$. In general, $y$-regularity of Rees algebra of $I$ is $0 \Rightarrow$ $I$…
Given two closed oriented manifolds $M,N$ of the same dimension, we denote the set of degrees of maps from $M$ to $N$ by $D(M,N)$. The set $D(M,N)$ always contains zero. We show the following (non-)realisability results: (i) There exists an…
Consider the celebrated Lyness recurrence $x_{n+2}=(a+x_{n+1})/x_{n}$ with $a\in\Q$. First we prove that there exist initial conditions and values of $a$ for which it generates periodic sequences of rational numbers with prime periods…
We show that certain graphs of groups with cyclic edge groups are aTmenable. In particular, this holds when each vertex group is either virtually special or acts properly and semisimply on $\mathbb{H}^n$.
Nonlinear complexity, as an important measure for assessing the randomness of sequences, is defined as the length of the shortest feedback shift registers that can generate a given sequence. In this paper, the structure of n-periodic binary…
We consider the map X defined on the rational numbers given by x --> x * ceil(x), where ceil(x) denotes the smallest integer greater than or equal to x, and study the problem of finding, for each rational, the smallest number of iterations…