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pro vyhledávání: '"11B85"'
Let $(u_{n})_{n\in \mathbb{N}}$ be a $k$-automatic sequence with values in the set $\{0, 1\}$. In the paper, we consider properties of sequences $(f(n))_{n\in\mathbb{N}}$ governed by the recurrence relations of the form $f(n)=af(n-u_{n}-1)+bf(n-u_{n}
Externí odkaz:
http://arxiv.org/abs/2412.11319
Autor:
Krawczyk, Elżbieta
We study automatic sequences and automatic systems generated by general constant length (nonprimitive) substitutions. While an automatic system is typically uncountable, the set of automatic sequences is countable, implying that most sequences within
Externí odkaz:
http://arxiv.org/abs/2412.01974
Autor:
Kohen, Nadav
We give a formula for the density of $0$ in the sequence of generalized Motzkin numbers, $M^{a,b}_n$, modulo a prime, $p$, in terms of the first $p$ generalized central trinomial coefficients $T^{a,b}_n\mod p$ (with $n
Externí odkaz:
http://arxiv.org/abs/2411.03681
Autor:
Garrity, Thomas, Osterman, Otto Vaughn
We study the complexity of S-adic sequences corresponding to a family of 216 multi-dimensional continued fractions maps, called Triangle Partition maps (TRIP maps), with an emphasis on those with low upper bounds on complexity. Our main result is to
Externí odkaz:
http://arxiv.org/abs/2410.02032
Autor:
Dubbe, Thomas
The automaticity $A(x)$ of a set $\mathcal{X}$ is the size of the smallest automaton that recognizes $\mathcal{X}$ on all words of length $\leq x$. We show that the automaticity of the set of primes is at least $x\exp\left(-c(\log\log x)^2\log\log\lo
Externí odkaz:
http://arxiv.org/abs/2409.04314
In combinatorics on words, the well-studied factor complexity function $\rho_{\bf x}$ of a sequence ${\bf x}$ over a finite alphabet counts, for any nonnegative integer $n$, the number of distinct length-$n$ factors of $\mathbf{x}$. In this paper, we
Externí odkaz:
http://arxiv.org/abs/2406.09302
Let us consider an infinite word and $k\geq 1$ an integer. By steps of $k$, we substitute a letter ofthis infinite word by the power of an external letter. The new word obtaining by this process is called $k$ to $k$ substitution of a power letter. Af
Externí odkaz:
http://arxiv.org/abs/2405.20221
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
We study the number of indecomposable summands in tensor powers of the vector representation of SL2. Our main focus is on positive characteristic where this sequence of numbers and its generating function show fractal behavior akin to Mahler function
Externí odkaz:
http://arxiv.org/abs/2405.16786
Autor:
Garrett, Samuel, Robertson, Steven
In 2004, de Mathan and Teuli\'e stated the $p$-adic Littlewood Conjecture ($p$-$LC$) in analogy with the classical Littlewood Conjecture. Given a field $\mathbb{K}$ and an irreducible polynomial $p(t)$ with coefficients in $\mathbb{K}$, $p$-$LC$ admi
Externí odkaz:
http://arxiv.org/abs/2405.14454