[seqfan] Re: Symmetric and Antisymmetric Linear Recurrences?

[Ron Hardin]

The (-1)^n is a way of saying that there are different polynomials for even and odd points.

Generalizing a little, I find that if you take row 3, and categorize points by their index being 0..11 modulo 12, you get a polynomial of degree 3 in each category; but not for fewer than 12 categories.  6 categories finds 3 polynomials and 3 nonpolynomials.
 
rhhardin at mindspring.com
rhhardin at att.net (either)


>________________________________
> From: Richard J. Mathar <mathar at mpia-hd.mpg.de>
>To: seqfan at seqfan.eu 
>Sent: Sunday, September 7, 2014 8:56 AM
>Subject: [seqfan] Re: Symmetric and Antisymmetric Linear Recurrences?
> 
>
>In partial answer to http://list.seqfan.eu/pipermail/seqfan/2014-September/013590.html :
>
>The recurrence
>  n=2: a(n)=4*a(n-1)-4*a(n-2)-4*a(n-3)+10*a(n-4)-4*a(n-5)-4*a(n-6)+4*a(n-7)-a(n-8)
>
>has a "signature" (4,-4,-4,10,-4,-4,4,-1), if we define the signature
>of a linear recurrence with constant coefficients as the coefficients in front
>of a(n-1), a(n-2) etc on the right hand side.
>
>If we consider sequences with rational generating functions with
>denominators (1+x)^k*(1-x)^l, the sequences can be written as a polynomial
>of order l in n plus (-1)^n times a polynomial of order k in n.
>....
>
>