[seqfan] Re: A193376 Tabl = 20 existing sequences

[Ron Hardin]

Every row of T(n,k) for zX1 tiles is a polynomial in k, and has a binomial 
coefficient for every power of k.

Fooling around, it comes out the general T(n,k,z) = sum{s=0..[n/z]} 
(binomial(n-(z-1)*s,s)*k^s)
(Empirical)

which affects
http://oeis.org/A193376 z=2
http://oeis.org/A193515 z=3
http://oeis.org/A193516 z=4
http://oeis.org/A193517 z=5
http://oeis.org/A193518 z=6
 
 rhhardin at mindspring.com
rhhardin at att.net (either)



----- Original Message ----
> From: "israel at math.ubc.ca" <israel at math.ubc.ca>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Sent: Mon, July 25, 2011 1:18:16 PM
> Subject: [seqfan] Re: A193376 Tabl = 20 existing sequences
> 
> Yes, with z+1 tiles on an n x 1 grid (with n >= z), either there is a tile 
> (of any of the k colours) on the first spot, followed by any configuration 
> on the remaining (n-z) x 1 grid, or the first spot is vacant, followed by 
> any configuration on the remaining (n-1) x 1. So T(n,k) = T(n-1,k) + 
> k*T(n-z,k), with T(n,k) = 1 for n=0,1,...,z-1. The solution is T(n,k) = 
> sum_r r^(-n-1)/(1 + z k r^(z-1)) where the sum is over the roots of the 
> polynomial k x^z + x - 1.
> 
> Robert Israel                                  israel at math.ubc.ca
> Department of  Mathematics        http://www.math.ubc.ca/~israel 
> University of British  Columbia            Vancouver, BC,  Canada
> 
> On Jul 25 2011, Ron Hardin wrote:
> 
> >experimentally zX1  tiles give a table with the corresponding 
> >a(n)=a(n-1)+k*a(n-z) column  recurrences, taking a quick spot preview.
> >
> > rhhardin at mindspring.com
> >rhhardin at att.net (either)
> >
> >
> >
> >----- Original Message ----
> >>  From: Ron Hardin <rhhardin at att.net>
> >> To: Sequence  Fanatics Discussion list <seqfan at list.seqfan.eu>
> >>  Sent: Mon, July 25, 2011 7:33:50 AM
> >> Subject: [seqfan] Re: A193376  Tabl = 20 existing sequences
> >> 
> >> The same problem with 3X1  tiles apparently gives a n-1 n-3 recurrence 
> >>  (b-file
> >
> >> still in progress), needs a formula   too:
> >
> >_______________________________________________
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> >
> 
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