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A368953
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.
4
31, 310, 311, 31010, 3110, 31111, 301, 310, 310101010, 3110110, 311110, 311111111, 3010, 30101, 3011111, 3100, 31010, 31010101010101010, 3101111, 3110110110110, 3110111, 3111011, 3111101, 31111011110, 3111110, 3111111110, 31111111111111111
OFFSET
0,1
COMMENTS
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.
REFERENCES
Douglas R. Hofstadter, Gödel, Escher, Bach: an Eternal Golden Braid, Basic Books, 1979, pp. 33-41 and pp. 261-262.
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..16809 (rows 0..8 of the triangle, flattened).
Wikipedia, MU Puzzle
EXAMPLE
After recursively applying the rules three times, we get the following tree (cf. Hofstadter (1979), page 40, Figure 11).
.
MI
0 ---------------------- 31
/ \
1 2 <--- Rule applied
/ \
MIU MII
1 ---------------- 310 311
/ / \
2 1 2
/ / \
MIUIU MIIU MIIII
2 --------- 31010 3110 31111
/ / / | | \
2 2 1 2 3 3
/ / / | | \
MIUIUIUIU MIIUIIU MIIIIU | MUI MIU
3 --- 310101010 3110110 311110 | 301 310
MIIIIIIII
311111111
.
After ordering the encoded strings lexicographically within a tree level (and removing duplicates, if present), the triangle begins:
[0] 31;
[1] 310 311;
[2] 31010 3110 31111;
[3] 301 310 310101010 3110110 311110 311111111;
...
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.
MATHEMATICA
MIUStepL[s_] := Union[Flatten[Map[{If[StringEndsQ[#, "1"], # <> "0", Nothing], # <> StringDrop[#, 1], StringReplaceList[#, {"111" -> "0", "00" -> ""}]}&, s]]];
With[{rowmax = 4}, Map[FromDigits, NestList[MIUStepL, {"31"}, rowmax], {2}]]
CROSSREFS
Cf. A331536, A368946, A368954 (row lengths), A369173 (all MIU strings).
Sequence in context: A295975 A002225 A124335 * A368946 A042870 A152033
KEYWORD
nonn,tabf
AUTHOR
Paolo Xausa, Jan 10 2024
STATUS
approved