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.

2, 5, 8, 12, 16, 18, 24, 26, 32, 34, 36, 38, 48, 50, 52, 54, 64, 66, 68, 70, 72, 74, 76, 78, 96, 98, 100, 102, 104, 106, 108, 110, 128, 130, 132, 134, 136, 138, 140, 142, 144, 146, 148, 150, 152, 154, 156, 158, 192, 194, 196, 198, 200, 202, 204, 206

Offset: 1

Author

Gandhar Joshi, Sep 13 2023

Keywords

• nonn

Comments

Interestingly, 5 is the only odd number in the list so far.

Examples

			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.
		

Links

• Kevin Ryde, Table of n, a(n) for n = 1..8192
• Kevin Ryde, PARI/GP Code

Crossrefs

Cf. A010060, A005942 (subword complexity), A366408 (first location).

Programs

PARI

\\ See links.

Walnut

def tmfaceq "At (t 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. %
def tm_w_len_N_unique_factors "Ei (Aj,k (i<=j & j<(i+n-N) & j ~$tmfaceq(j,k,N))": % Find lengths of words with length-N unique factors; must replace N with a constant %
def longest_len_N "$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 %