This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A309623 #40 Jan 04 2025 13:48:24
%S A309623 25,41,48,50,63
%N A309623 Numbers k for which there is an extremal ternary word of length k.
%C A309623 A ternary word is one over a three-letter alphabet, such as {0,1,2}. Such a word is called "squarefree" if it contains no subblock of the form XX, where X is a nonempty contiguous block. A word x is extremal if it is squarefree, but every possible insertion of a single letter, that is, every word of the form x' a x'' with x = x' x'', a in {0,1,2}, is not squarefree.
%C A309623 The Grytczuk paper proves there are arbitrarily long extremal words.
%H A309623 J. Grytczuk, H. Kordulewski, and A. Niewadomski, <a href="https://arxiv.org/abs/1910.06226">Extremal square-free words</a>, arxiv preprint arXiv:1910.06226v1 [math.CO], October 14 2019.
%H A309623 Jaroslaw Grytczuk, Hubert Kordulewski, and Artur Niewiadomski, <a href="https://doi.org/10.37236/9264">Extremal Square-Free Words</a>, Electronic J. Combinatorics, 27 (1), 2020, #1.48.
%e A309623 The smallest extremal word is of length 25, which is 0120102120121012010212012 and is unique up to renaming of the letters. The next smallest are of length 41, and there are two (up to renaming), namely 01021012021020121021201021012021020121021 and 02102012102120102101202102012102120102101. The next is of length 48, and is unique (up to renaming): 010212012102010212012101202120121020102120121020. The next is of length 50 and is unique (up to renaming): 01021201021012021020121012021201021012021020121020.
%e A309623 The next smallest are of length 63, and there are two (up to renaming): 010210120210201021202102012102120102101202102010212021020121021, 012010212012101202120121020120210120102120121012021201210201202. - _Michael S. Branicky_, May 06 2022
%e A309623 For lengths 25, 41, 48, 50, and 63, there is a unique extremal word up to both renaming and reversal. - _Pontus von Brömssen_, May 07 2022
%Y A309623 Cf. A006156, A332605.
%K A309623 nonn,more
%O A309623 1,1
%A A309623 _Jeffrey Shallit_, Oct 20 2019
%E A309623 a(5) from _Michael S. Branicky_, May 06 2022