Related papers: Classification of normal sequences

The differences between the $N=0$ and $N=1$ standard models are emphasized in formulating their short distance extension. We sketch methods to reproduce many of the small numbers in the model in terms of scale ratios, applying see-saw like…

For Lucas sequences of the first kind (u_n) and second kind (v_n) defined as usual for positive n by u_n=(a^n-b^n)/(a-b), v_n=a^n+b^n, where a and b are either integers or conjugate quadratic integers, we describe the set of indices n for…

We give a characterization of all matrices $A,B,C \in \mathbb{F}_{2}^{m \times m}$ which generate a $(0,m,3)$-net in base $2$ and a characterization of all matrices $B,C\in\mathbb{F}_{2}^{\mathbb{N}\times\mathbb{N}}$ which generate a…

Batch normalization (BN) is a technique to normalize activations in intermediate layers of deep neural networks. Its tendency to improve accuracy and speed up training have established BN as a favorite technique in deep learning. Yet,…

We prove that digital sequences modulo $m$ along squares are normal, which covers some prominent sequences like the sum of digits in base $q$ modulo $m$, the Rudin-Shapiro sequence and some generalizations. This gives, for any base, a class…

We study the properties of the third order sequence $(w_n)=\left(w_n(a,b,c; r, s,t)\right)$ defined by the recurrence relation $w_n = rw_{n - 1} + sw_{n - 2} + tw_{n - 3}\, (n \ge 3)$ with $w_0 = a,\,w_1 = b,\,w_2=c$, where $a$, $b$, $c$,…

We classify the growth of a $k$-regular sequence based on information from its $k$-kernel. In order to provide such a classification, we introduce the notion of a growth exponent for $k$-regular sequences and show that this exponent is…

Let $H$ be a separable Hilbert space and let $\{x_n\}$ be a sequence in $H$ that does not contain any zero elements. We say that $\{x_n\}$ is a \emph{Bessel-normalizable} or \emph{frame-normalizable} sequence if the normalized sequence…

A balanced generalized de Bruijn sequence with parameters $(n,l,k)$ is a cyclic sequence of $n$ bits such that (a) the number of 0's equals the number of 1's, and (b) each substring of length $l$ occurs at most $k$ times. We determine…

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…

The Ulam sequence is given by $a_1 =1, a_2 = 2$, and then, for $n \geq 3$, the element $a_n$ is defined as the smallest integer that can be written as the sum of two distinct earlier elements in a unique way. This gives the sequence $1, 2,…

We classify all binary error correcting completely regular codes of length $n$ with minimum distance $\delta>n/2$.

Divisibility sequences are defined by the property that their elements divide each other whenever their indices do. The divisibility sequences that also satisfy a linear recurrence, like the Fibonacci numbers, are generated by polynomials…

Let $A,B\subseteq\mathbb Z_n$ be given and $S=(x_1,\ldots, x_k)$ be a sequence in $\mathbb Z_n$. We say that $S$ is an $(A,B)$-weighted zero-sum sequence if there exist $a_1,\ldots,a_k\in A$ and $b_1,\ldots,b_k\in B$ such that…

Denote by p_k the k-th power sum symmetric polynomial n variables. The interpretation of the q-analogue of the binomial coefficient as Hilbert function leads us to discover that n consecutive power sums in n variables form a regular…

A collection of complex sequences of length v is complementary if the sum of their periodic autocorrelation function values at all non-zero shifts is constant. For a complex sequence A=[a_0,a_1,...,a_{v-1}] of length v=dm we define the…

Let s be an integer greater than or equal to 2. A real number is simply normal to base s if in its base-s expansion every digit 0, 1, ..., s-1 occurs with the same frequency 1/s. Let X be the set of positive integers that are not perfect…

We study the sequences of numbers corresponding to lambda terms of given sizes, where the size is this of lambda terms with de Bruijn indices in a very natural model where all the operators have size 1. For plain lambda terms, the sequence…

Let $(u(n))_{n\in\mathbb{N}}$ be an arithmetic progression of natural integers in base $b\in\mathbb{N}\setminus \{0,1\}$. We consider the following sequences: $s(n)=\overline{u(0)u(1)\cdots u(n) }^b$ formed by concatenating the first $n+1$…

Let $L=(L_d)_{d \in \mathbb N}$ be any ordered probability sequence, i.e., satisfying $0 < L_{d+1} \le L_d$ for each $d \in \mathbb N$ and $\sum_{d \in \mathbb N} L_d =1$. We construct sequences $A = (a_i)_{i \in \mathbb N}$ on the…