Related papers: Jumping sequences
We study sequences of random numbers {Z[1],Z[2],Z[3],...,Z[n]} -- which can be considered random walks with reflecting barriers -- and define their "types" according to whether Z[i] > Z[i+1], (a down-movement), or Z[i] < Z[i+1]…
The EKG or electrocardiogram sequence is defined by a(1) = 1, a(2) = 2 and, for n >= 3, a(n) is the smallest natural number not already in the sequence with the property that gcd {a(n-1), a(n)} > 1. In spite of its erratic local behavior,…
This paper describes a simple method for estimating lower bounds on the number of classes of equivalence for a special kind of integer sequences, called division sequences. The method is based on adding group structure to classes of…
An Ulam sequence U(1,n) is defined as the sequence starting with integers 1,n such that n > 1, and such that every subsequent term is the smallest integer that can be written as the sum of distinct previous terms in exactly one way. This…
An inversion sequence of length $n$ is an integer sequence $e=e_{1}e_{2}\dots e_{n}$ such that $0\leq e_{i}<i$ for each $i$. Corteel--Martinez--Savage--Weselcouch and Mansour--Shattuck began the study of patterns in inversion sequences,…
Let $\ell$ and $r$ be integers. A real number $\alpha \in [0,1)$ is a jump for $r$ if for any $\varepsilon > 0$ and any integer $m,\ m \geq r$, any $r$-uniform graph with $n > n_0(\varepsilon,m)$ vertices and at least \alpha+…
An ascent sequence is one consisting of non-negative integers in which the size of each letter is restricted by the number of ascents preceding it in the sequence. Ascent sequences have recently been shown to be related to (2+2)-free posets…
We introduce and investigate a new model of a finite number of particles jumping forward on the real line. The jump lengths are independent of everything, but the jump rate of each particle depends on the relative position of the particle…
The most common difference that occurs among the consecutive primes less than or equal to $x$ is called a jumping champion. Occasionally there are ties. Therefore there can be more than one jumping champion for a given $x$. In 1999 A.…
We derive weighted summation identities involving the second order recurrence sequence $\{w_n\} =\{ w_n(a,b; p, q)\}$ defined by $w_0 = a,\,w_1 = b;\,w_n = pw_{n - 1} - qw_{n - 2}\, (n \ge 2)$, where $a$, $b$, $p$ and $q$ are arbitrary…
We consider the problem of enumerating periodic $\sigma$-juggling sequences of length $n$ for multiplex juggling, where $\sigma$ is the initial state (or {\em landing schedule}) of the balls. We first show that this problem is equivalent to…
We introduce the notion of minimal inversion sequences for a pattern $\rho$, which form the smallest set of inversion sequences whose avoidance is equivalent to the avoidance of $\rho$ for inversion sequences. We give a characterization of…
For a fixed integer $k$, we define a sequence $A_k=(a_k(n))_{n\geq0}$ and a corresponding sparse subsequence $S_k$ using the cardinality of the $n$-th symmetric power of the set $\{1,2,\ldots, k\}$. For $k\in\{2,\dots,8\}$, we find…
This paper describes a class of sequences that are in many ways similar to Fibonacci sequences: given n, sum the previous two terms and divide them by the largest possible power of n. The behavior of such sequences depends on n. We analyze…
We introduce and investigate a new model of a finite number of particles jumping forward on the real line. The jump lengths are independent of everything, but the jump rate of each particle depends on the relative position of the particle…
The sequence a_1,...,a_m is a common subsequence in the set of permutations S = {p_1,...,p_k} on [n] if it is a subsequence of p_i(1),...,p_i(n) and p_j(1),...,p_j(n) for some distinct p_i, p_j in S. Recently, Beame and Huynh-Ngoc (2008)…
We define sequence patterns of length $n$ and level $\ell$ to be equivalence classes of sequences that have $n$ elements from the set of $\ell$ integer symbols $\{1,2,\ldots,\ell\}$ with no restriction on repetition, where the equivalence…
Inspired by the definition of modified ascent sequences, we introduce a new class of integer sequences called revised ascent sequences. These sequences are defined as Cayley permutations where each entry is a leftmost occurrence if and only…
Define a sequence of positive integers by the rule that a(n) = n for 1 <= n <= 3, and for n >= 4, a(n) is the smallest number not already in the sequence which has a common factor with a(n-2) and is relatively prime to a(n-1). We show that…
Aronson's sequence 1, 4, 11, 16, ... is defined by the English sentence ``t is the first, fourth, eleventh, sixteenth, ... letter of this sentence.'' This paper introduces some numerical analogues, such as: a(n) is taken to be the smallest…