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Register a new userStanley sequences with odd character
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Richard A. Moy
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Given a set of integers containing no 3-term arithmetic progressions, one constructs a Stanley sequence by choosing integers greedily without forming such a progression. Independent Stanley sequences are a well-structured class of Stanley sequences with two main parameters: the character $lambda(A)$ and the repeat factor $rho(A)$. Rolnick conjectured that for every $lambda in mathbb{N}_0backslash{1, 3, 5, 9, 11, 15}$, there exists an independent Stanley sequence $S(A)$ such that $lambda(A) =lambda$. This paper demonstrates that $lambda(A) otin {1, 3, 5, 9, 11, 15}$ for any independent Stanley sequence $S(A)$.
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Given a set of integers with no three in arithmetic progression, we construct a Stanley sequence by adding integers greedily so that no arithmetic progression is formed. This paper offers two main contributions to the theory of Stanley sequences. First, we characterize well-structured Stanley sequences as solutions to constraints in modular arithmetic, defining the modular Stanley sequences. Second, we introduce the basic Stanley sequences, where elements arise as the sums of subsets of a basis sequence, which in the simplest case is the powers of 3. Applications of our results include the construction of Stanley sequences with arbitrarily large gaps between terms, answering a weak version of a problem by ErdH{o}s et al. Finally, we generalize many results about Stanley sequences to $p$-free sequences, where $p$ is any odd prime.
For an integer $qge2$, a $q$-recursive sequence is defined by recurrence relations on subsequences of indices modulo some powers of~$q$. In this article, $q$-recursive sequences are studied and the asymptotic behavior of their summatory functions is analyzed. It is shown that every $q$-recursive sequence is $q$-regular in the sense of Allouche and Shallit and that a $q$-linear representation of the sequence can be computed easily by using the coefficients from the recurrence relations. Detailed asymptotic results for $q$-recursive sequences are then obtained based on a general result on the asymptotic analysis of $q$-regular sequences. Three particular sequences are studied in detail: We discuss the asymptotic behavior of the summatory functions of Sterns diatomic sequence, the number of non-zero elements in some generalized Pascals triangle and the number of unbordered factors in the Thue--Morse sequence. For the first two sequences, our analysis even leads to precise formulae{} without error terms.
We study two types of dynamical extensions of Lucas sequences and give elliptic solutions for them. The first type concerns a level-dependent (or discrete time-dependent) version involving commuting variables. We show that a nice solution for this system is given by elliptic numbers. The second type involves a non-commutative version of Lucas sequences which defines the non-commutative (or abstract) Fibonacci polynomials introduced by Johann Cigler. If the non-commuting variables are specialized to be elliptic-commuting variables the abstract Fibonacci polynomials become non-commutative elliptic Fibonacci polynomials. Some properties we derive for these include their explicit expansion in terms of normalized monomials and a non-commutative elliptic Euler--Cassini identity.
Let $G$ be a finite cyclic group of order $n ge 2$. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)cdot ... cdot (n_lg)$ where $gin G$ and $n_1,..., n_l in [1,ord(g)]$, and the index $ind (S)$ of $S$ is defined as the minimum of $(n_1+ ... + n_l)/ord (g)$ over all $g in G$ with $ord (g) = n$. In this paper we prove that a sequence $S$ over $G$ of length $|S| = n$ having an element with multiplicity at least $frac{n}{2}$ has a subsequence $T$ with $ind (T) = 1$, and if the group order $n$ is a prime, then the assumption on the multiplicity can be relaxed to $frac{n-2}{10}$. On the other hand, if $n=4k+2$ with $k ge 5$, we provide an example of a sequence $S$ having length $|S| > n$ and an element with multiplicity $frac{n}{2}-1$ which has no subsequence $T$ with $ind (T) = 1$. This disproves a conjecture given twenty years ago by Lemke and Kleitman.
Divide-and-conquer functions satisfy equations in F(z),F(z^2),F(z^4)... Their generated sequences are mainly used in computer science, and they were analyzed pragmatically, that is, now and then a sequence was picked out for scrutiny. By giving several classes of ordinary generating functions together with recurrences, we hope to help with the analysis of many such sequences, and try to classify a part of the divide-and-conquer sequence zoo.
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