[seqfan] Re: A193376 Tabl = 20 existing sequences

[Ron Hardin]

A little thought on the bicycle route

Putting s zX1 tiles in an nX1 grid, shrink the zX1 to 1X1 in a n-s*(z-1) grid.
Then distribute the k colors s^k ways.

Add up over all possible s number of tiles.

 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: Sun, July 31, 2011 3:10:07 PM
> Subject: [seqfan] Re: A193376 Tabl = 20 existing sequences
> 
> 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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