[seqfan] Re: Sirag numbers.

[allouche at math.jussieu.fr]

2-automatic sequences (or more generally k-automatic sequences) are
instances of "automatic sequences": see, e.g., the MR classification
11B85, or ---without wanting to make exaggerated self-advertisement---
see http://www.cambridge.org/fr/knowledge/isbn/item1170556/?site_locale=fr_FR

Also note that a sequence is k-automatic if and only if it is both
k-regular and takes finitely many values.

For a question with slightly the same flavor as Sirag numbers, see, e.g.,
DOI: 10.1007/978-1-4419-6263-8_14

best
jean-paul


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Charles Greathouse <charles.greathouse at case.edu> a écrit :

>> Sirag numbers are 2-automatic, that is, there is a finite automaton that
>> accepts precisely their base-2 numerals.
>
> Where does that term come from?  I've used "2-regular" in the same way
> (cf. A038772).
>
> Charles Greathouse
> Analyst/Programmer
> Case Western Reserve University
>
> On Thu, Sep 29, 2011 at 11:38 PM, David Wilson   
> <davidwwilson at comcast.net> wrote:
>> On 9/29/2011 2:32 PM, franktaw at netscape.net wrote:
>>>
>>> From basic number theory, n is a Sirag number iff n*(n+1) = 4^j * (8k+7)
>>> for some integer j and k.
>>>
>>> Franklin T. Adams-Watters
>>
>> Ergo, Sirag numbers n are characterizable as one of the following forms:
>>
>> n == 12 (mod 32)
>> n == 19 (mod 32)
>> n = 4^j*(8k+1)-1 (j >= 2; k >= 0)
>> n = 4^j*(8k+7)  (j >= 2; k >= 0)
>>
>> and can be shown to have limit density 1/12 with respect to the integers.
>>
>> Sirag numbers are 2-automatic, that is, there is a finite automaton that
>> accepts precisely their base-2 numerals.
>>
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