A369172 Irregular triangle read by rows: row n lists the lengths of 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.

2, 3, 3, 5, 4, 5, 9, 7, 6, 9, 3, 3, 17, 13, 11, 4, 4, 10, 17, 7, 7, 7, 7, 7, 7, 4, 5, 5, 33, 25, 21, 9, 9, 9, 9, 7, 7, 2, 19, 8, 8, 8, 8, 8, 8, 18, 33, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 8, 13, 5, 5, 5, 8, 13, 5, 5, 8, 13, 5, 8, 13, 5, 8, 13, 5, 5, 13, 5, 5, 5, 7, 6, 9, 9

Offset

0,1

Comments

See A368946 for the description of the MIU formal system and the triangle of corresponding strings.

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..3670 (rows 0..7 of the triangle, flattened).

Wikipedia, MU Puzzle.

Index entries for sequences from "Goedel, Escher, Bach".

Formula

T(n,k) = A055642(A368946(n,k)).

T(n,k) = A369206(n,k) + A369207(n,k) + 1.

Example

Triangle begins:
  [0]  2;
  [1]  3  3;
  [2]  5  4  5;
  [3]  9  7  6  9  3  3;
  [4] 17 13 11  4  4 10 17  7  7  7  7  7  7  4  5  5;
  ...

Mathematica

MIUStepOW3[s_] := Flatten[Map[{If[StringEndsQ[#, "1"], # <> "0", Nothing], # <> #, StringReplaceList[#, "111" -> "0"], StringReplaceList[#, "00" -> ""]}&, s]];
With[{rowmax = 5}, StringLength[NestList[MIUStepOW3, {"1"}, rowmax]]] + 1

Crossrefs

Cf. A055642, A368946, A368947 (row lengths), A369206, A369207.

Sequence in context: A086898 A123031 A271709 * A373815 A359042 A159070

Adjacent sequences: A369169 A369170 A369171 * A369173 A369174 A369175

Keyword

nonn,tabf

Author

Paolo Xausa, Jan 15 2024

Status

approved