[seqfan] Re: Empirical Formula Hunt - Pseudopolynomials

[Neil Sloane]

Ron, there is a wonderful little book about
these recurrence relations, quasi polynomials, and their generating
functions, asymptotics, etc. : Manual Kauers and
Peter Paule, "The Concrete Tetrahedron".

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com


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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