  Let $s_n$ be the number of words consisting of the ternary alphabet
consisting of the digits 0, 1, and 2 such that no subword (or factor) is a
square (a word concatenated with itself, e.g., $11$, $1212$, or $102102$). From
computational evidence, $s_n$ 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:
$952^{n/53} \approx 1.1381531^n$.
