A369207 Irregular triangle read by rows: row n lists the number of I characters for each 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.

1, 1, 2, 2, 2, 4, 4, 4, 4, 8, 1, 1, 8, 8, 8, 1, 1, 8, 16, 5, 5, 5, 5, 5, 5, 1, 2, 2, 16, 16, 16, 5, 5, 5, 5, 2, 2, 1, 16, 5, 5, 5, 5, 5, 5, 16, 32, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 5, 10, 2, 2, 2, 5, 10, 2, 2, 5, 10, 2, 5, 10, 2, 5, 10, 2, 2, 10, 2, 2, 2, 2, 2, 4, 4

Offset

0,3

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) = A268643(A368946(n,k)).

T(n,k) = A369172(n,k) - A369206(n,k) - 1.

Example

Triangle begins:
  [0] 1;
  [1] 1 2;
  [2] 2 2 4;
  [3] 4 4 4 8 1 1;
  [4] 8 8 8 1 1 8 16 5 5 5 5 5 5 1 2 2;
  ...

Mathematica

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

Crossrefs

Cf. A268643, A368946, A368947 (row lengths), A369172, A369206 (number of zeros).

Sequence in context: A287155 A287654 A355911 * A113402 A238726 A240321

Adjacent sequences: A369204 A369205 A369206 * A369208 A369209 A369210

Keyword

nonn,base,tabf

Author

Paolo Xausa, Jan 16 2024

Status

approved