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A367184 Starting index in the Period doubling sequence (A096268) of the first maximum length word in which every subword of length n is distinct.

%I A367184 #31 Dec 19 2024 11:46:20
%S A367184 0,0,1,0,5,2,1,0,11,10,9,4,3,2,1,0,23,22,21,20,19,18,17,8,7,6,5,4,3,2,
%T A367184 1,0,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,16,15,14,13,12,11,
%U A367184 10,9,8,7,6,5,4,3,2,1,0,95,94,93,92,91,90,89,88,87,86,85,84,83,82,81,80
%N A367184 Starting index in the Period doubling sequence (A096268) of the first maximum length word in which every subword of length n is distinct.
%C A367184 a(2^m)=0; i.e. for all nonnegative integers m, the longest words w with no length-(2^m) subwords of w repeated are the prefixes of length A366462(2^m) of the Period doubling sequence.
%H A367184 Kevin Ryde, <a href="/A367184/b367184.txt">Table of n, a(n) for n = 1..8192</a>
%H A367184 Kevin Ryde, <a href="/A367184/a367184.gp.txt">PARI/GP Code</a>
%e A367184 For n=3, the first instance of one of the longest words w in A096268 with no repeated subwords of length 3 is w=1000101 which begins at index 1, so a(3)=1. The length of w is A366462(3) = 7.
%o A367184 (Walnut)
%o A367184 def pdfaceq "At (t<n) => PD[i+t]=PD[j+t]"; % Check if two length-n factors of Period doubling sequence at positions i and j are equal; PD is predefined in Walnut as the DFA that recognises the Period doubling sequence. %
%o A367184 def pd_w_len_N_unique_factors "Aj, k (i<=j & j<(i+n-N) & j<k & k<(i+n-N+1)) => ~$pdfaceq(j, k, N)": % Find lengths and positions of words with length-N unique factors; must replace N with a constant %
%o A367184 def pd_longest_len_N "$pd_w_len_N_unique_factors(i,n) & Am (m>n) => ~$pd_w_len_N_unique_factors(i,m)"; % Check the longest of the lengths of words defined in the line above; must replace N with the same constant %
%o A367184 def pd_longest_len_N_fpos "$pd_longest_len_N(i,M) & Aj (j<i) => ~$pd_longest_len_N(j,M)"; % This finds the first positions of the longest words required; must replace M with A366462(N).%
%o A367184 (PARI) \\ See links.
%Y A367184 Cf. A096268, A366462 (length of the longest word), A275202 (subword complexity).
%K A367184 nonn
%O A367184 1,5
%A A367184 _Gandhar Joshi_, Nov 08 2023