[seqfan] Re: Empirical Formula Hunt - Pseudopolynomials

[Charles Greathouse]

I've only heard these referred to as quasipolynomials. Who calls them
pseudopolynomials? (I know pseudopolynomial time in analysis of algorithms,
but that's different.)
 You get these when the characteristic polynomial is a product of roots of
unity (I think).

I certainly agree that these are interesting and worth searching for, and I
hope appropriate comments will be added to the OEIS when these are
recognized.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Fri, Dec 12, 2014 at 5:28 PM, Ron Hardin <rhhardin at att.net> wrote:

> I've only recently realized that an empirical linear recurrence that's
> symmetric or antisymmetric has a good chance of being a pseudopolynomial,
> ie a polynomial on every P'th point, a different polynomial depending on
> starting point, so that the coefficients can be said to have period P.
>
> A polynomial plus (-1)^n times another polynomial is often reported, but
> any period besides 2 can turn up.  3,6,12,24,60,360 and 720 are common, and
> one was 27720.
>
> I don't have the software necessary to search high degrees and high
> periods (10,000 point array limit in bc(1)), but maybe somebody would find
> looking for them entertaining.
>
> If you have the empirical recurrence, you can generate as many points as
> you want to check for polynomials at any spacing.
>
> Incidentally there's a philosophical pause that comes up from that: you're
> pretty confident of the linear recurrence because it generates a lot of
> empirical points successfully, many more points than there are
> coefficients; but if that recurrence then gives you a pseudopolynomial with
> period 100000, that seems to predict specific behavior pretty far beyond
> what you're entitled to extrapolate.
>
>
> rhhardin at mindspring.com
> rhhardin at att.net (either)
>
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