[seqfan] Re: Problem reproducing A019989, A019990, and A019991

[Brad Klee]

On Mon, Apr 15, 2019 at 10:30 AM Marc LeBrun <mlb at well.com> wrote:
> I was hoping it was trying to be some kind of ternary Thue-Morse

Yes, it does look something similar to a ternary Thue-Morse, but also
a bit worse, as really the function abcABC1 involves six sequences,
with function "c" A019991 growing out-of-step. If we are really going
to make any sense out of this, it will probably help to spell out a few
more details, and consider another "threequence" at an intermediary
level of difficulty.

Thue-Morse A010060 splits by the Odious and Evil numbers (haha),
A000069 and A019989 respectively. A similar "Gosper-style" recursion
appears to produce both sets in unison:

TMiterate[sets_] := With[{ind = {{1, 2}, {2, 1}}},
  Union @@ (2 sets[[#]] + {0, 1}) & /@ ind]
TMdata = Union @@ # & /@ Transpose[
    NestList[TMiterate, {{0}, {1}}, 10][[All]]];
TableForm[TMdata[[All, 1 ;; 30]]]

0, 3, 5, 6, 9, 10, 12, 15, 17, 18 ...
1, 2, 4, 7, 8, 11, 13, 14, 16, 19 ...

If you want a "threequence" that generalizes upon Thue-Morse,
the following seems to do just well enough:

TM3iterate[sets_] :=
 With[{ind = {{1, 2, 3}, {3, 1, 2}, {2, 3, 1}}},
  Union @@ (3 sets[[#]] + {0, 1, 2}) & /@ ind]
TM3data = Union @@ # & /@ Transpose[
    NestList[TM3iterate, {{0}, {1}, {2}}, 10][[All]]];
TableForm at TM3data[[All, 1 ;; 30]]

Starting with:
0, 4, 8, 11, 12, 16, 19, 23, 24, 28 ...
1, 5, 6, 9, 13, 17, 20, 21, 25, 29 ...
2, 3, 7, 10, 14, 15, 18, 22, 26, 27 ...
( none of these appear in OEIS )

Notice that this analogy preserves that each column should
contain a permutation of three consecutive numbers. This
property does not hold for A019989, A019990, A019991.

Beyond this, it seems more apt to call Gosper's original
"threequence", a "parity-preserving sexaquence", if one is
not afraid to invent and use absolutely frivolous language.

Cheers, Brad