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

Ron Hardin rhhardin at att.net
Fri Jul 27 20:50:46 CEST 2012

```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
> 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