k-regular sequence
A k-regular 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 whose base-k representations are close to that of n.
Regular sequences generalize automatic sequences to infinite alphabets.
Contents
Definition
Let . An integer sequence is k-regular if the set of sequences
- Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \{s(k^e n + r)_{n \geq 0} : e \geq 0 \text{ and } 0 \leq r \leq k^e - 1\}
is contained in a finite-dimensional vector space over the field of rational numbers.
Examples
Ruler sequence
Let be the -adic valuation of . The ruler sequence ( A007814) is -regular, and the set
- Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \{s(2^e n + r)_{n \geq 0} : e \geq 0 \text{ and } 0 \leq r \leq 2^e - 1\}
is contained in the -dimensional vector space generated by and the constant sequence . These basis elements lead to the recurrence equations
- Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \begin{align} s(2 n) &= 0 \\ s(4 n + 1) &= s(2 n + 1) - s(n) \\ s(4 n + 3) &= 2 s(2 n + 1) - s(n), \end{align}
which, along with initial conditions and , uniquely determine the sequence.
Thue–Morse sequence
Every k-automatic sequence is k-regular.[1] For example, the Thue–Morse sequence is -regular, and
- Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \{T(2^e n + r)_{n \geq 0} : e \geq 0 \text{ and } 0 \leq r \leq 2^e - 1\}
consists of the two sequences and .
Polynomial sequences
If is an integer-valued polynomial, then is k-regular for every .
Properties
The class of k-regular sequences is closed under termwise addition and multiplication.[2]
The nth term in a k-regular sequence grows at most polynomially in n.[3]
Generalizations
The notion of a k-regular sequence can be generalized to a ring as follows. Let be a commutative Noetherian subring of . A sequence with values in is -regular if the -module generated by
- Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \{s(k^e n + r)_{n \geq 0} : e \geq 0 \text{ and } 0 \leq r \leq k^e - 1\}
is finitely generated.
Notes
References
- Lua error in package.lua at line 80: module 'strict' not found..
- Lua error in package.lua at line 80: module 'strict' not found..
- Lua error in package.lua at line 80: module 'strict' not found.