[seqfan] Re: A question about types of sequences
Allan Wechsler
acwacw at gmail.com
Sun Aug 11 20:29:31 CEST 2019
On one hand, a certain amount of categorization is useful. (One important
category that hasn't been mentioned yet in this thread is "list of positive
[or nonnegative] integers k such that <some predicate on k>, in numerical
order".)
On the other hand, these categories will be unavoidably hard-to-define. For
example, _all_ sequences can be shoehorned into the "a(n) depends on n"
category -- the word "depends" is hiding some deep philosophical problems.
On Sun, Aug 11, 2019 at 2:20 PM Antti Karttunen <antti.karttunen at gmail.com>
wrote:
> On 8/11/19, Ali Sada via SeqFan <seqfan at list.seqfan.eu> wrote:
> > Hi Everyone,
> > Are there any resources I can read about types of sequences? For example
> > there are sequences where a(n) depends on n, there are sequences where
> a(n)
> > depends on a(n-1), etc. I would appreciate any recommendations.
>
> You may start from
> https://oeis.org/wiki/User:Charles_R_Greathouse_IV/Properties
> but I guess it only lists a small part what is possible (with emphasis
> on number theory), because Z^N is uncountable, after all.
>
> Then there are also lots of sequences of type "Lexicographically
> earliest sequence that satisfies a <certain condition>".
>
> Of course a sequence may fulfill the criteria of several ad-hoc
> classes simultaneously.
>
> I have found it productive to consider recurrences of more general
> kind, like a(n) = f(a(Axxxxxx(n))), where f is some integer-valued
> function and Axxxxxx is some "meaningful" operation acting on n
> (usually already submitted as a sequence, like e.g., A003557, A028234
> or A064989), instead of just the common operations like n-1, n-k or
> floor(n/2).
>
>
> > Best,
> > Ali
>
> Best regards,
>
> Antti
>
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