Related papers: Discriminators and k-Regular Sequences

For each natural number $n$, we define a category whose objects are discriminant algebras in rank $n$, i.e. functorial means of attaching to each rank-$n$ algebra a quadratic algebra with the same discriminant. We show that the discriminant…

Let n be a non-null positive integer and $d(n)$ is the number of positive divisors of n, called the divisor function. Of course, $d(n) \leq n$. $d(n) = 1$ if and only if $n = 1$. For $n > 2$ we have $d(n) \geq 2$ and in this paper we try to…

The ``comma sequence'' starts with 1 and is defined by the property that if k and k' are consecutive terms, the two-digit number formed from the last digit of k and the first digit of k' is equal to the difference k'-k. If there is more…

Let $K$ be a field and $X$, $Y$ denote matrices such that, the entries of $X$ are either indeterminates over $K$ or $0$ and the entries of $Y$ are indeterminates over $K$ which are different from those appearing in $X$. We consider ideals…

Let $A$ and $B$ be complex numbers, and let $(w_n)_{n\ge0}$ be a sequence of complex numbers with $w_{n+1}=Aw_n-Bw_{n-1}$ for all $n=1,2,3,\ldots$. When $w_0=0$ and $w_1=1$, the sequence $(w_n)_{n\ge0}$ is just the Lucas sequence…

Although 10^230 terms of Recaman's sequence have been computed, it remains a mystery. Here three distant cousins of that sequence are described, one of which is also mysterious. (i) {A(n), n >= 3} is defined as follows. Start with n, and…

Orientable sequences of order n are infinite periodic sequences with symbols drawn from a finite alphabet of size k with the property that any particular subsequence of length n occurs at most once in a period in either direction. They were…

Let $S= \{ p_1, \ldots, p_s\}$ be a finite, non-empty set of distinct prime numbers and $(U_{n})_{n \geq 0}$ be a linear recurrence sequence of integers of order $r$. For any positive integer $k,$ we define $(U_j^{(k)})_{j\geq 1}$ an…

In this paper we study some sophisticated supercongruences involving dual sequences. For $n=0,1,2,\ldots$ define $$d_n(x)=\sum_{k=0}^n\binom nk\binom xk2^k$$ and $$s_n(x)=\sum_{k=0}^n\binom nk\binom xk\binom{x+k}k=\sum_{k=0}^n\binom…

Denote by $A_n$ the set of square $(0,1)$ matrices of order $n$. The set $A_n$, $n\le8$, is partitioned into row/column permutation equivalence classes enabling derivation of various facts by simple counting. For example, the number of…

Erd\H{o}s and Szekeres showed in 1978 that for any four positive integers satisfying m_1+m_2 = n_1+n_2, the two binomial coefficients (m_1+m_2)!/m_1! m_2! and (n_1+n_2)!/n_1! n_2! have a common divisor >1. The analogous statement for…

The n-th Delannoy number and the n-th Schr\"oder number given by $D_n=\sum_{k=0}^n\binom{n}{k}\binom{n+k}{k}$ and $S_n=\sum_{k=0}^n\binom{n}{k}\binom{n+k}{k}/(k+1)$ respectively arise naturally from enumerative combinatorics. Let p be an…

In 1949 Wall showed that $x = 0.d_1d_2d_3 \dots$ is normal if and only if $(0.d_nd_{n+1}d_{n+2} \dots)_n$ is a uniformly distributed sequence. In this article, we consider sequences which are slight variants on this. In particular, we show…

The modular discriminant $\Delta$ is known to structure the sequence of modular forms $(M_{2k}(SL_2(\mathbb{Z})))_{k\in \; \mathbb{N}^*}$ at level $1$.\\ For all positive integer $N$, we define a strong modular unit $\Delta_N$ at level $N$…

Define $$D_n(x)=\sum_{k=0}^n\binom nk^2x^k(x+1)^{n-k}\ \ \ \mbox{for}\ n=0,1,2,\ldots$$ and $$s_n(x)=\sum_{k=1}^n\frac1n\binom nk\binom n{k-1}x^{k-1}(x+1)^{n-k}\ \ \ \mbox{for}\ n=1,2,3,\ldots.$$ Then $D_n(1)$ is the $n$-th central Delannoy…

Given a digraph D, the minimum semi-degree of D is the minimum of its minimum indegree and its minimum outdegree. D is k-ordered Hamiltonian if for every ordered sequence of k distinct vertices there is a directed Hamilton cycle which…

For every $m \in {\C} \setminus \{0, -2\}$ and every nonnegative integer $k$ we define the vertex operator (super)algebra $D_{m,k}$ having two generators and rank $ \frac{3 m}{m + 2}$. If $m$ is a positive integer then $D_{m,k}$ can be…

The zero-dilation index $d(A) $ of a matrix $A$ is the largest integer $k$ for which $\begin{bmatrix}0_k& *\\ * & *\end{bmatrix}$ is unitarily similar to $A$. In this study, the zero-dilation indices of certain block matrices are…

The Fibonacci sequence is a sequence of numbers that has been studied for hundreds of years. In this paper, we introduce the new sequence S_{k,n} with initial conditions S_{k,0} = 2b and S_{k,1} = bk + a, which is generated by the…

Let the sequence S_m of nonnegative integers be generated by the following conditions: Set the first term a_0 = 0, and for all k \geq 0, let a_k+1 be the least integer greater than a_k such that no element of {a_0,...,a_k+1} is the average…