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

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, 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 ≈ 1.1381531 n .