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k-regular sequence - Wikipedia
In mathematics and theoretical computer science, a k-regular sequence is a sequence satisfying linear recurrence equations that reflect the base-k representations of the integers. The class of k-regular sequences generalizes the class of k-automatic sequences to alphabets of infinite size. See more
There exist several characterizations of k-regular sequences, all of which are equivalent. Some common characterizations are as follows. For each, we take R′ to be a See more
Ruler sequence
Let $${\displaystyle s(n)=\nu _{2}(n+1)}$$ be the $${\displaystyle 2}$$-adic valuation of $${\displaystyle n+1}$$. The ruler sequence $${\displaystyle s(n)_{n\geq 0}=0,1,0,2,0,1,0,3,\dots }$$ (OEIS: A007814 See moreThe notion of k-regular sequences was first investigated in a pair of papers by Allouche and Shallit. Prior to this, Berstel and Reutenauer studied the theory of rational series, … See more
Given a candidate sequence $${\displaystyle s=s(n)_{n\geq 0}}$$ that is not known to be k-regular, k-regularity can typically be proved directly from the definition by calculating elements of the kernel of $${\displaystyle s}$$ and proving that all elements of … See more
Wikipedia text under CC-BY-SA license The ring of k-regular sequences - ScienceDirect
May 18, 1992 · In this paper, we generalize the concept of automatic sequence to the case where the sequence can take its values in a (possibly infinite) ring R; we call such sequences k …
- Author: Jean-Paul Allouche, Jeffrey Shallit
- Publish Year: 2003
Section 10.68 (0AUH): Regular sequences—The Stacks project
If $I$ is an ideal of $R$ and $f_1, \ldots , f_ r \in I$ then we call $f_1, \ldots , f_ r$ an $M$-regular sequence in $I$. If $M = R$, we call $f_1, \ldots , f_ r$ simply a regular sequence (in $I$). …
Related searches for k regular sequences
\ (k\)-regular sequences - Combinatorics
An introduction and formal definition of k -regular sequences can be found, for example, on the Wikipedia article k-regular_sequence or in [AS2003]. The binary sum of digits S (n) of a …
We look at k-regular sequences taking on integer values. In this case, the Z-module M( {f (n)};k) is torsion free and hence isomorphic to a finite direct sum of copies of Z. We investigate the set …
Decidability and k-regular sequences - ScienceDirect
Mar 12, 2022 · In this paper we examine a number of natural decision problems involving k-regular sequences, and show that they are undecidable.
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In this paper we examine a number of natural decision problems involving k-regular sequences, and show that they are undecidable. 1.1. Recursively solvable decision problems. A decision …
regular sequences forms a ring under the operations of term-by-term addition and convolution. Hence the set of associated formal power series in R[[X]] also forms a ring. We show how k …
Mar 8, 2022 · k-regular sequences De nition (Allouche and Shallit, 1992) Let k > 2 be an integer. A real sequence f is called k-regular provided its k-kernel, ker k(f) := (f(k‘n + r)) n>0: ‘> 0;0 6 r …
Definition 1.1 A sequence is k-regular if M({f(n)};k) is finitely generated as a Z-module. Since the k-kernel of a sequence {f(n)} spans M({f(n)};k) as a Z-module, we see that an automatic …
In this paper, we generalize the concept of automatic sequence to the case where the sequence can take its values in a (possibly infinite) ring R; we call such sequences k-regular. (When R is …
What is an automatic sequence? Also called k-recognizable in the literature. The notion is base-dependent, so we speak of k-automatic sequences. Is the Mephisto Waltz sequence w …
Mar 1, 2022 · Equality of k-regular Sequences Theorem (Krenn{Shallit 2020) \Given two k-regular sequences (f(n)) n 0 and (g(n)) n 0 over Q , does f(n) = g(n) for all n …
The ring of k -regular sequences - Springer
Jan 1, 2005 · In this paper, we generalize the concept of automatic sequence to the case where the sequence can take its values in a (possibly infinite) ring R; we call such sequences k …
Some applications of $k$-regular sequences and arithmetic rank …
Some applications of k -regular sequences and arithmetic rank of an ideal with respect to modules. Kh. Ahmadi Amoli. Let R be a commutative Noetherian ring with identity, I be an …
The ring of k-regular sequences | Theoretical Computer Science
May 18, 1992 · We study the k-regular sequences introduced by Allouche and Shallit. We call a k-regular integer sequence s quasi-universal, if for every recursively enumerable set A of …
Characterizations of k-regularity (Allouche & Shallit 1992): s(n) is determined by finitely many linear recurrences in s(k e n+i) (along with finitely many initial conditions)
16 - k -Regular Sequences - Cambridge University Press
Oct 13, 2009 · In this chapter we examine one such possible generalization, called the class of k-regular sequences. Basics. Recall that a sequence is k -automatic if and only if its k -kernel is …
K regular sequence - Alchetron, The Free Social Encyclopedia
Nov 23, 2023 · A kregular sequence is an infinite sequence satisfying recurrence equations of a certain type, in which the nth term is a linear combination of mth terms for some integers m …
s is a Mathematica package for comput-ing with integer sequences. Its support for k-regular sequences includes basic.
Constructing k-ary orientable sequences with asymptotically …
Feb 28, 2025 · An orientable sequence of order n over an alphabet \(\{0,1,\ldots , k{-}1\}\) is a cyclic sequence such that each length-n substring appears at most once in either …
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