# [seqfan] Re: Sequence needing more terms, A6156, squarefree ternary words

Neil Sloane njasloane at gmail.com
Sat Jul 28 02:37:17 CEST 2012

```Concerning A006156, thanks to everyone who responded.
Ron, could you enter the triangle T(n,k) when you reach a stopping point,
and a second triangle giving the coefficients of
the polynomials that generate the rows (even if they are
only empirical)?
Also the columns giving the number of squarefree words
over alphabets of sizes 4, 5, 6, ... (assuming they aren't already
in the OEIS)

Thanks!

Neil

On Fri, Jul 27, 2012 at 2:50 PM, Ron Hardin <rhhardin at att.net> wrote:

> It's possible I can add a term, but the thing hits an exponential-looking
> cost
> wall somewhere around there.
>
> If you make a T(n,k) table, the rows are polynomials and maybe somebody
> can play
> guess the coefficients
>
> current state:
>
> T(n,k)=Number of square-free words of length n in a (k+1)-ary alphabet
>
> Table starts
>
> .2...3.....4......5.......6........7.........8..........9.........10.........11
>
> .2...6....12.....20......30.......42........56.........72.........90........110
>
> .2..12....36.....80.....150......252.......392........576........810.......1100
>
> .0..18....96....300.....720.....1470......2688.......4536.......7200......10890
>
> .0..30...264...1140....3480.....8610.....18480......35784......64080.....107910
>
> .0..42...696...4260...16680....50190....126672.....281736.....569520....1068210
>
> .0..60..1848..15960...80040...292740....868560....2218608....5062320...10575180
>
> .0..78..4848..59580..383520..1706250...5953248...17467128...44991360..104683590
>
> .0.108.12768.222600.1838160..9946020..40806528..137522448..399866400.1036270620
>
> .0.144.33480.830880.8807400.57970080.279692784.1082712960.3553806960...........
>
> Rows 1-10
> Empirical: a(k) = 1*k + 1
> Empirical: a(k) = 1*k^2 + 1*k
> Empirical: a(k) = 1*k^3 + 1*k^2
> Empirical: a(k) = 1*k^4 + 1*k^3 - 1*k^2 - 1*k
> Empirical: a(k) = 1*k^5 + 1*k^4 - 2*k^3 - 1*k^2 + 1*k
> Empirical: a(k) = 1*k^6 + 1*k^5 - 3*k^4 - 2*k^3 + 2*k^2 + 1*k
> Empirical: a(k) = 1*k^7 + 1*k^6 - 4*k^5 - 3*k^4 + 5*k^3 + 2*k^2 - 2*k
> Empirical: a(k) = 1*k^8 + 1*k^7 - 5*k^6 - 4*k^5 + 8*k^4 + 4*k^3 - 4*k^2 -
> 1*k
> Empirical: a(k) = 1*k^9 + 1*k^8 - 6*k^7 - 5*k^6 + 12*k^5 + 8*k^4 - 9*k^3 -
> 4*k^2
> + 2*k
> Empirical: a(k) = 1*k^10 + 1*k^9 - 7*k^8 - 6*k^7 + 17*k^6 + 12*k^5 -
> 17*k^4 -
> 7*k^3 + 6*k^2
>
>
>
>  rhhardin at mindspring.com
> rhhardin at att.net (either)
>
>
>
> ----- Original Message ----
> > From: Neil Sloane <njasloane at gmail.com>
> > To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> > Cc: R. H. Hardin <rhhardin at mindspring.com>
> > Sent: Fri, July 27, 2012 12:44:10 PM
> > Subject: [seqfan] Sequence needing more terms, A6156, squarefree ternary
> words
> >
> > A006156 is the subject of several research papers, so it would
> > be nice to  have a b-file. Ron, is this something that can be
> > handled by your  techniques?
> > Neil
> >
> > --
> > Dear Friends, I have now retired from  AT&T. New coordinates:
> >
> > Neil J. A. Sloane, President, OEIS  Foundation
> > 11 South Adelaide Avenue, Highland Park, NJ 08904, USA
> > Email: njasloane at gmail.com
> >
> > _______________________________________________
> >
> > Seqfan  Mailing list - http://list.seqfan.eu/
> >
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>

--
Dear Friends, I have now retired from AT&T. New coordinates:

Neil J. A. Sloane, President, OEIS Foundation
11 South Adelaide Avenue, Highland Park, NJ 08904, USA