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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.
5
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.
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. A024495 (row lengths).
Sequence in context: A034933 A320148 A320147 * A272364 A188657 A021697
KEYWORD
nonn,tabf
AUTHOR
Paolo Xausa, Jan 23 2024
STATUS
approved