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### $0\text{-}1$ sequences having the same numbers of $\left(1\text{-}1\right)$-couples of given distances

Mathematica Bohemica

Let $a$ be a $0-1$ sequence with a finite number of terms equal to 1. The distance sequence ${\delta }^{\left(a\right)}$ of $a$ is defined as a sequence of the numbers of $\left(1-1\right)$-couples of given distances. The paper investigates such pairs of $0-1$ sequences $a,b$ that a is different from $b$ and ${\delta }^{\left(a\right)}={\delta }^{\left(b\right)}$.

### 14-term arithmetic progressions on quartic elliptic curves.

Journal of Integer Sequences [electronic only]

### 3 arten von Linearverbindungen bei Bernoullipolynomen

Matematički Vesnik

### A basis of ℤₘ, II

Colloquium Mathematicae

Given a set A ⊂ ℕ let ${\sigma }_{A}\left(n\right)$ denote the number of ordered pairs (a,a’) ∈ A × A such that a + a’ = n. Erdős and Turán conjectured that for any asymptotic basis A of ℕ, ${\sigma }_{A}\left(n\right)$ is unbounded. We show that the analogue of the Erdős-Turán conjecture does not hold in the abelian group (ℤₘ,+), namely, for any natural number m, there exists a set A ⊆ ℤₘ such that A + A = ℤₘ and ${\sigma }_{A}\left(n̅\right)\le 5120$ for all n̅ ∈ ℤₘ.

### A basis of the set of sequences satisfying a given m-th order linear recurrence.

Elemente der Mathematik

### A basis of Zₘ

Colloquium Mathematicae

Let ${\sigma }_{A}\left(n\right)=|\left(a,{a}^{\text{'}}\right)\in A²:a+{a}^{\text{'}}=n|$, where n ∈ N and A is a subset of N. Erdős and Turán conjectured that for any basis A of order 2 of N, ${\sigma }_{A}\left(n\right)$ is unbounded. In 1990, Imre Z. Ruzsa constructed a basis A of order 2 of N for which ${\sigma }_{A}\left(n\right)$ is bounded in the square mean. In this paper, we show that there exists a positive integer m₀ such that, for any integer m ≥ m₀, we have a set A ⊂ Zₘ such that A + A = Zₘ and ${\sigma }_{A}\left(n̅\right)\le 768$ for all n̅ ∈ Zₘ.

Integers

Acta Arithmetica

### A Catalan transform and related transformations on integer sequences.

Journal of Integer Sequences [electronic only]

Integers

### A cellular automaton on a torus.

Portugaliae Mathematica

Acta Arithmetica

### A chaotic cousin of Conway's recursive sequence.

Experimental Mathematics

Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

Integers

### A characterization of sequences with the minimum number of k-sums modulo k

Colloquium Mathematicae

Let G be an additive abelian group of order k, and S be a sequence over G of length k+r, where 1 ≤ r ≤ k-1. We call the sum of k terms of S a k-sum. We show that if 0 is not a k-sum, then the number of k-sums is at least r+2 except for S containing only two distinct elements, in which case the number of k-sums equals r+1. This result improves the Bollobás-Leader theorem, which states that there are at least r+1 k-sums if 0 is not a k-sum.

### A characterization of the Bernoulli and Euler polynomials

Rendiconti del Seminario Matematico della Università di Padova