A368954 Row lengths of A368953: in the MIU formal system, number of distinct strings n steps distant from the MI string.

1, 2, 3, 6, 15, 48, 232, 1544, 14959, 203333, 3919437, 105126522

Offset: 0

Author

Paolo Xausa, Jan 10 2024

Keywords

• nonn
• hard
• more

Comments

See A368946 for the description of the MIU formal system and A368953 for the variant where duplicates within a row are removed.

References

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

Links

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

Crossrefs

Cf. A331536, A368946, A368947, A368953, A369408.

Programs

Mathematica

MIUStepDW3[s_] := DeleteDuplicates[Flatten[Map[{If[StringEndsQ[#, "1"], # <> "0", Nothing], # <> #, StringReplaceList[#, {"111" -> "0","00" -> ""}]}&, s]]];
With[{rowmax = 9}, Map[Length, NestList[MIUStepDW3, {"1"}, rowmax]]]

Python

from itertools import islice
def occurrence_swaps(w, s, t):
    out, oi = [], w.find(s)
    while oi != -1:
        out.append(w[:oi] + t + w[oi+len(s):])
        oi = w.find(s, oi+1)
    return out
def moves(w): # moves for word w in MIU system, encoded as 310
    nxt = []
    if w[-1] == '1': nxt.append(w + '0')        # Rule 1
    if w[0] == '3': nxt.append(w + w[1:])       # Rule 2
    nxt.extend(occurrence_swaps(w, '111', '0')) # Rule 3
    nxt.extend(occurrence_swaps(w, '00', ''))   # Rule 4
    return nxt
def agen(): # generator of terms
    frontier = {'31'}
    while len(frontier) > 0:
        yield len(frontier)
        reach1 = set(m for p in frontier for m in moves(p))
        frontier, reach1 = reach1, set()
print(list(islice(agen(), 10))) # Michael S. Branicky, Jan 14 2024

Formula

a(n) <= A368947(n).

Extensions

a(10)-a(11) from Michael S. Branicky, Jan 14 2024