Zobrazeno 1 - 10
of 132
pro vyhledávání: '"Burns, Rob"'
Autor:
Burns, Rob
We investigate the running sums of some well-known automatic sequences to determine whether they are synchronised.
Comment: 21 pages, 7 figures
Comment: 21 pages, 7 figures
Externí odkaz:
http://arxiv.org/abs/2405.17536
Autor:
Burns, Rob
We provide a complete characterisation of the appearance function for paper-folding sequences for factors of any length. We make use of the software package {\tt Walnut} to establish these results.
Comment: 11 pages, 10 figures
Comment: 11 pages, 10 figures
Externí odkaz:
http://arxiv.org/abs/2210.14719
Autor:
Burns, Rob
Let $\bar{S}$ denote the set of integers $n$ such that $n!$ cannot be written as a sum of three squares. Let $\bar{S}(n)$ denote $\bar{S} \cap [1, n]$. We establish an exact formula for $\bar{S}(2^k)$ and show that $\bar{S}(n) = 1/8*n + \mathcal{O}(\
Externí odkaz:
http://arxiv.org/abs/2203.16469
Autor:
Burns, Rob
We provide a necessary and sufficient condition for $n!$ to be a sum of three squares. The condition is based on the binary representation of $n$ and can be expressed by the operation of an automaton.
Comment: 8 pages, 1 table, 1 figure
Comment: 8 pages, 1 table, 1 figure
Externí odkaz:
http://arxiv.org/abs/2101.01567
Autor:
Burns, Rob
We introduce extremely symmetric primes and provide some elementary properties of these.
Comment: 7 pages, 3 tables
Comment: 7 pages, 3 tables
Externí odkaz:
http://arxiv.org/abs/2005.02922
Autor:
Burns, Rob
We consider the sum of squares function in the ring $\mathbb{Z}_{n}$. We determine formulae in a number of cases when $n$ is a power of a prime.
Comment: 29 pages
Comment: 29 pages
Externí odkaz:
http://arxiv.org/abs/2004.10269
Autor:
Burns, Rob, McKenzie, Jen
We use partitions to provide some formulae for counting s-collisions and other events in various forms of the Birthday Problem.
Comment: 11 pages
Comment: 11 pages
Externí odkaz:
http://arxiv.org/abs/1906.08671
Autor:
Burns, Rob
We examine the representation of numbers as the sum of two squares in $\mathbb{Z}_n$ for a general positive integer $n$. Using this information we make some comments about the density of positive integers which can be represented as the sum of two sq
Externí odkaz:
http://arxiv.org/abs/1708.03930
Autor:
Burns, Rob
Let $C_n$ be the $n$th Catalan number. For any prime $p \geq 5$ we show that the set $\{C_n : n \in \mathbb{N} \}$ contains all residues mod $p$. In addition all residues are attained infinitely often. Any positive integer can be expressed as the pro
Externí odkaz:
http://arxiv.org/abs/1703.02705
Autor:
Burns, Rob
We establish a lower bound of 2/p(p-1) for the asymptotic density of the Motzkin numbers divisible by a general prime number p > 3. We provide a criteria for when this asymptotic density is actually 1. We also provide a partial characterisation of th
Externí odkaz:
http://arxiv.org/abs/1703.00826