A369173 -id:A369173 - cp's OEIS frontend

A369179 Irregular triangle read by rows: row n lists the number of I characters for each of the distinct derivable strings in the MIU formal system that are n characters long.

1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 4, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 4, 1, 2, 2, 2, 4, 2, 4, 4, 4, 5, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 4, 1, 2, 2, 2, 4, 2, 4, 4, 4, 5, 1, 2, 2, 2, 4, 2, 4, 4, 4, 5, 2, 4, 4, 4, 5, 4, 4, 5, 4, 5, 5

Offset: 2

Paolo Xausa, Jan 16 2024

See A368946 for the description of the MIU formal system and A369173 for the triangle of the corresponding derivable strings.

			Triangle begins:
  [2] 1;
  [3] 1 1 2;
  [4] 1 1 2 1 2 2;
  [5] 1 1 2 1 2 2 1 2 2 2 4;
  [6] 1 1 2 1 2 2 1 2 2 2 4 1 2 2 2 4 2 4 4 4 5;
  ...

Douglas R. Hofstadter, Gödel, Escher, Bach: an Eternal Golden Braid, Basic Books, 1979, pp. 33-41 and pp. 261-262.

Paolo Xausa, Table of n, a(n) for n = 2..10922 (rows 2..14 of the triangle, flattened).
Wikipedia, MU Puzzle.
Index entries for sequences from "Goedel, Escher, Bach".

Cf. A024495 (row lengths), A268643, A368946, A369173, A369174 (number of zeros).

A369179row[n_] := Select[Map[Count[#, 1]&, Tuples[{0, 1}, n - 1]], !Divisible[#, 3]&]; Array[A369179row, 6, 2]

T(n,k) = A268643(A369173(n,k)).
T(n,k) = n - 1 - A369174(n,k).
T(n,k) mod 3 > 0.

A369412 Maximum length of a "normal" proof (see comments) for strings (theorems) in the MIU formal system that are n characters long.

Original entry on oeis.org

1, 4, 13, 11, 18, 16, 25, 23, 24, 22, 26, 24, 34, 32, 33, 31, 35, 33, 34, 32, 39, 37, 49

Offset: 2

Paolo Xausa, Jan 23 2024

See A368946 for the description of the MIU formal system, A369410 for the triangle of the corresponding proof lengths and A369409 for the definition of "normal" proof.

Douglas R. Hofstadter, Gödel, Escher, Bach: an Eternal Golden Braid, Basic Books, 1979, pp. 33-41.

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".

Cf. A024495, A368946, A369173, A369409, A369410, A369413.

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[Max, Map[MIUProofLineCount, Array[MIUDigitsW3, 15, 2], {2}]]

a(n) = max_{k=1..A024495(n)} A369410(n,k).

A369413 Maximum number of symbols of a "normal" proof (see comments) for strings (theorems) in the MIU formal system that are n characters long.

Original entry on oeis.org

2, 13, 94, 75, 165, 139, 308, 269, 348, 299, 482, 423, 647, 581, 780, 701, 893, 807, 1064, 965, 1281, 1175, 1654

Offset: 2

Paolo Xausa, Jan 23 2024

See A368946 for the description of the MIU formal system, A369411 for the triangle of the corresponding symbol lengths and A369409 for the definition of "normal" proof.

Douglas R. Hofstadter, Gödel, Escher, Bach: an Eternal Golden Braid, Basic Books, 1979, pp. 33-41.

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".

Cf. A024495, A368946, A369173, A369409, A369411, A369412.

MIUDigitsW3[n_] := Select[Tuples[{0, 1}, n - 1], !Divisible[Count[#, 1], 3]&];
MIUProofSymbolCount[t_] := Module[{c = Length[t], nu = Count[t,0], ni}, ni = 2*nu+c; c += nu(nu+c+2); While[ni > 1, If[OddQ[ni], c += (7*ni+3)/2 + 13; ni = (ni+3)/2, c += ni/2 + 1; ni/=2]]; c+1];
Map[Max, Map[MIUProofSymbolCount, Array[MIUDigitsW3, 15, 2], {2}]]

a(n) = max_{k=1..A024495(n)} A369411(n,k).

A369148 In the MIU formal system, total number (including duplicates) of strings up to n steps distant from the MI string.

Original entry on oeis.org

1, 3, 6, 12, 28, 88, 444, 3671, 47981, 976631, 29554002, 1326494644

Offset: 0

Paolo Xausa, Jan 14 2024

See A368946 for the description of the MIU formal system.

Douglas R. Hofstadter, Gödel, Escher, Bach: an Eternal Golden Braid, Basic Books, 1979, pp. 33-41.

Wikipedia, MU Puzzle.
Index entries for sequences from "Goedel, Escher, Bach".

Cf. A331536 (without duplicates), A368946, A369173 (all MIU strings).

Partial sums of A368947.

MIUStepW3[s_] := Flatten[Map[{If[StringEndsQ[#, "1"], # <> "0", Nothing], # <> #, StringReplaceList[#, {"111" -> "0", "00" -> ""}]}&, s]];
With[{rowmax = 9}, Accumulate[Map[Length, NestList[MIUStepW3, {"1"}, rowmax]]]]

a(n) >= A331536(n+1).

cp's OEIS Frontend

A369179 Irregular triangle read by rows: row n lists the number of I characters for each of the distinct derivable strings in the MIU formal system that are n characters long.

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A369412 Maximum length of a "normal" proof (see comments) for strings (theorems) in the MIU formal system that are n characters long.

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A369413 Maximum number of symbols of a "normal" proof (see comments) for strings (theorems) in the MIU formal system that are n characters long.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A369148 In the MIU formal system, total number (including duplicates) of strings up to n steps distant from the MI string.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula