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%I #12 Aug 24 2021 18:08:29
%S 0,1,0,0,1,0,0,1,1,1,0,0,0,1,0,0,0,1,0,1,1,0,0,0,1,1,0,1,0,1,1,1,1,1,
%T 0,0,0,0,1,0,0,0,0,1,0,0,1,1,0,0,0,0,1,0,1,0,1,0,0,1,1,0,1,1,1,0,0,0,
%U 0,1,1,0,0,1,0,1,0,1,1,1,0,1,0,0,1,1
%N Concatenation of all 01-words in the order induced by A004526; see Comments.
%C Let s = (s(n)) be a strictly increasing sequence of positive integers with infinite complement, t = (t(n)). For n >=1, let s'(n) be the number of s(i) that are <= n-1 and let t'(n) be the number of t(i) that are <= n-1. Define w(1) = 0, w(t(1)) = 1, and w(n) = 0w(s'(n)) if n is in s, and w(n) = 1w(t'(n)) if n is in t. Then (w(n)) is the "s-induced ordering" of all 01-words.
%C ****
%C Guide to related sequences:
%C s 01-words ordered by s positions of palindromes
%C A092754 A076478 (lexicographic order) A341257
%C A004526 A341256 A341257
%C A000201 A341258 A341333
%C A016777 A341334 A342750
%C A001651 A342753 A342790
%C A032766 A342910 A344166
%F To generate successive words w(n), if n is in s, spell w(n) as 0 suffixed by the first w(k) that does not already suffix a word beginning with 0; otherwise, spell w(n) as 1 suffixed by the first w(k) that does not already suffix a word beginning with 1.
%e The first 20 words: 0, 1, 00, 10, 01, 11, 000, 100, 010, 110, 001, 101, 011, 111, 0000, 1000, 0100, 1100, 0010, 1010.
%t z = 80; s = Table[2 n - 1, {n, 1, z}]; (* A004526 *)
%t t = Complement[Range[Max[s]], s]; (* A005843 *)
%t s1[n_] := Length[Intersection[Range[n - 1], s]];
%t t1[n_] := n - 1 - s1[n]; (* A023416 *)
%t w[1] = {0}; w[t[[1]]] = {1};
%t w[n_] := If[MemberQ[s, n], Join[{0}, w[s1[n]]], Join[{1}, w[t1[n]]]]
%t tt = Table[w[n], {n, 1, z}] (* 01-words, in order *)
%t Flatten[tt] (* A341256 *)
%Y Cf. A004526, A005843, A023416, A092754, A341257.
%K nonn
%O 1
%A _Clark Kimberling_, Mar 16 2021
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