A369410 Irregular triangle read by rows: row n lists the length of a "normal" proof (see comments) for each of the distinct derivable strings (theorems) in the MIU formal system that are n characters long.

1, 4, 4, 2, 13, 13, 8, 13, 8, 8, 11, 11, 6, 11, 6, 6, 11, 6, 6, 6, 3, 12, 12, 18, 12, 18, 18, 12, 18, 18, 18, 12, 12, 18, 18, 18, 12, 18, 12, 12, 12, 7, 10, 10, 16, 10, 16, 16, 10, 16, 16, 16, 10, 10, 16, 16, 16, 10, 16, 10, 10, 10, 5, 10, 16, 16, 16, 10, 16, 10, 10, 10, 5, 16, 10, 10, 10, 5, 10, 10, 5, 10, 5, 5

Offset

2,2

Comments

See A368946 for the description of the MIU formal system, A369173 for the triangle of the corresponding strings (theorems) and A369409 for the definition of "normal" proof.

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 = 2..10922 (rows 2..14 of the triangle, flattened).

Armando B. Matos and Luis Filipe Antunes, Short Proofs for MIU theorems, Technical Report Series DCC-98-01, University of Porto, 1998.

Wikipedia, MU Puzzle.

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

Formula

T(n,k) >= A369408(n,k).

If A369173(n,k) contains no zeros and 3+2^m ones (for m >= 0), then T(n,k) = 4*m + 3.

Example

Triangle begins:
  [2]  1;
  [3]  4  4  2;
  [4] 13 13  8 13  8  8;
  [5] 11 11  6 11  6  6 11  6  6  6  3;
  ...
For the theorem MIU (310), which is given by A369173(3,2), the "normal" proof is MI (31) -> MII (311) -> MIIII (31111) -> MIU (310), which consists of 4 lines: T(3,2) is therefore 4.

Mathematica

MIUDigitsW3[n_] := Select[Tuples[{0, 1}, n - 1], !Divisible[Count[#, 1], 3]&];
MIUProofLineCount[t_] := Module[{c = Count[t, 0], ni}, ni = Length[t] + 2*c; While[ni > 1, If[OddQ[ni], ni = (ni+3)/2; c += 4, ni/=2; c++]]; c+1];
Map[MIUProofLineCount, Array[MIUDigitsW3, 7, 2], {2}]

Crossrefs

Row lengths of A369409.

Cf. A368946, A369173, A369408, A369411, A369412.

Cf. A024495 (row lengths).

Sequence in context: A034933 A320148 A320147 * A272364 A188657 A021697

Adjacent sequences: A369407 A369408 A369409 * A369411 A369412 A369413

Keyword

nonn,tabf

Author

Paolo Xausa, Jan 23 2024

Status

approved