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arXiv · 1606.00835

An Improved Lower Bound for $n$-Brinkhuis $k$-Triples

Michael Sollami·Craig C. Douglas·Manfred Liebmann
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Abstract

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: $952^{n/53} \approx 1.1381531^n$.

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Michael Sollami, Craig C. Douglas, Manfred Liebmann. 2016-06-02. An Improved Lower Bound for $n$-Brinkhuis $k$-Triples. https://arxiv.org/abs/1606.00835

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