%I #29 Jan 18 2024 07:48:19
%S 31,310,311,31010,3110,31111,301,310,310101010,3110110,311110,
%T 311111111,3010,30101,3011111,3100,31010,31010101010101010,3101111,
%U 3110110110110,3110111,3111011,3111101,31111011110,3111110,3111111110,31111111111111111
%N Irregular triangle read by rows: row n lists (in lexicographical order and with duplicates removed) the strings of the MIU formal system at the n-th level of the tree generated by recursively applying the system rules, starting from the MI string.
%C This is a variant of A368946 (see there for the description of the MIU system) where, within a row, duplicates are removed and encoded strings are ordered lexicographically.
%D Douglas R. Hofstadter, Gödel, Escher, Bach: an Eternal Golden Braid, Basic Books, 1979, pp. 33-41 and pp. 261-262.
%H Paolo Xausa, <a href="/A368953/b368953.txt">Table of n, a(n) for n = 0..16809</a> (rows 0..8 of the triangle, flattened).
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/MU_puzzle">MU Puzzle</a>
%H <a href="/index/Go#GEB">Index entries for sequences from "Goedel, Escher, Bach"</a>
%e After recursively applying the rules three times, we get the following tree (cf. Hofstadter (1979), page 40, Figure 11).
%e .
%e MI
%e 0 ---------------------- 31
%e / \
%e 1 2 <--- Rule applied
%e / \
%e MIU MII
%e 1 ---------------- 310 311
%e / / \
%e 2 1 2
%e / / \
%e MIUIU MIIU MIIII
%e 2 --------- 31010 3110 31111
%e / / / | | \
%e 2 2 1 2 3 3
%e / / / | | \
%e MIUIUIUIU MIIUIIU MIIIIU | MUI MIU
%e 3 --- 310101010 3110110 311110 | 301 310
%e MIIIIIIII
%e 311111111
%e .
%e After ordering the encoded strings lexicographically within a tree level (and removing duplicates, if present), the triangle begins:
%e [0] 31;
%e [1] 310 311;
%e [2] 31010 3110 31111;
%e [3] 301 310 310101010 3110110 311110 311111111;
%e ...
%e Please note that some strings may be present in different rows: within the first four rows, the string MIU (310) is present in rows 1 and 3.
%t MIUStepL[s_] := Union[Flatten[Map[{If[StringEndsQ[#, "1"], # <> "0", Nothing], # <> StringDrop[#, 1], StringReplaceList[#, {"111" -> "0", "00" -> ""}]}&, s]]];
%t With[{rowmax = 4}, Map[FromDigits, NestList[MIUStepL, {"31"}, rowmax], {2}]]
%Y Cf. A331536, A368946, A368954 (row lengths), A369173 (all MIU strings).
%K nonn,tabf
%O 0,1
%A _Paolo Xausa_, Jan 10 2024