[seqfan] Re: A193376 Tabl = 20 existing sequences
Ron Hardin
rhhardin at att.net
Mon Aug 1 00:21:40 CEST 2011
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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