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A365624 a(n) is the length of the longest word w in the Thue-Morse sequence (A010060) in which every length-n factor of w is unique.

%I A365624 #39 Dec 19 2024 11:46:19
%S A365624 2,5,8,12,16,18,24,26,32,34,36,38,48,50,52,54,64,66,68,70,72,74,76,78,
%T A365624 96,98,100,102,104,106,108,110,128,130,132,134,136,138,140,142,144,
%U A365624 146,148,150,152,154,156,158,192,194,196,198,200,202,204,206
%N A365624 a(n) is the length of the longest word w in the Thue-Morse sequence (A010060) in which every length-n factor of w is unique.
%C A365624 Interestingly, 5 is the only odd number in the list so far.
%H A365624 Kevin Ryde, <a href="/A365624/b365624.txt">Table of n, a(n) for n = 1..8192</a>
%H A365624 Kevin Ryde, <a href="/A366408/a366408.gp.txt">PARI/GP Code</a>
%e A365624 The length of the longest word in Thue-Morse sequence that admits only unique length-2 factors is 5. For example, 11001 (which is not the only one). Hence a(2)=5.
%o A365624 (Walnut)
%o A365624 def tmfaceq "At (t<n) => T[i+t]=T[j+t]"; % Check if two length-n factors of Thue-Morse at positions i and j are equal; T is predefined in Walnut as the DFA that recognises Thue-Morse sequence. %
%o A365624 def tm_w_len_N_unique_factors "Ei (Aj,k (i<=j & j<(i+n-N) & j<k & k<(i+n-N+1)) => ~$tmfaceq(j,k,N))": % Find lengths of words with length-N unique factors; must replace N with a constant %
%o A365624 def longest_len_N "$tm_w_len_N_unique_factors(n) & Am (m>n) => ~$tm_w_len_N_unique_factors(m)"; % Check the longest of the lengths found in previous line; must replace N with the same constant %
%o A365624 (PARI) \\ See links.
%Y A365624 Cf. A010060, A005942 (subword complexity), A366408 (first location).
%K A365624 nonn
%O A365624 1,1
%A A365624 _Gandhar Joshi_, Sep 13 2023