[seqfan] Empirical Formula Hunt - Pseudopolynomials

Ron Hardin rhhardin at att.net
Fri Dec 12 23:28:33 CET 2014


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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