##
**
An Improved Lower Bound on the Number of Ternary Squarefree Words
**

###
Michael Sollami

Ditto Labs, Inc.

1 Broadway, 14th Floor

Cambridge, MA 02142

USA

Craig C. Douglas

University of Wyoming

School of Energy Resources and Department of Mathematics

1000 E. University Ave., Dept. 3036

Laramie, WY 82072

USA

Manfred Liebmann

Technische Universität München

Center for Mathematical Sciences

Boltzmannstraße 3

85748 Garching bei München

Germany

**Abstract:**

Let *s*_{n}
be the number of words in the ternary alphabet Σ = {0, 1, 2} such
that no subword (or factor) is a square (a word concatenated with
itself, e.g., 11, 1212, and 102102). From computational evidence, the
sequence (*s*_{n}) grows exponentially at a rate of about 1.317277^{n}. While
known upper bounds are already relatively close to the conjectured
rate, effective lower bounds are much more difficult to obtain. In this
paper, we construct a 54-Brinkhuis 952-triple, which leads to an
improved lower bound on the number of *n*-letter ternary squarefree
words: 952^{n/53} ≈ 1.1381531^{n}.

**
Full version: pdf,
dvi,
ps,
latex
**
Accompanying files:
b0.txt,
b1.txt,
b2.txt,
code.tgz

**
**

(Concerned with sequences
A006156
A010060.)

Received March 14 2016; revised version received May 5 2016; June 9 2016.
Published in *Journal of Integer Sequences*, July 6 2016.

Return to
**Journal of Integer Sequences home page**