# [seqfan] Re: help needed with former sequence A138036

hv at crypt.org hv at crypt.org
Sun Sep 4 23:21:01 CEST 2011

```"N. J. A. Sloane" <njas at research.att.com> wrote:
:Dear Seqfans, I admit I don't speak Mathematica.
:Here is the definition of the former sequence A138036
:(now deleted from the OEIS, but you can read the History):
:
:Clear[a]; a = With[{n = 8, k = 3}, NestList[DeleteCases[Flatten[Map[Table[Append[ #, i - 1], {i, k}] &, # ], 1], {___, u__, v__} /; Sort[{u}] == Sort[{v}]] &, {{}}, n]]; Flatten[a]
:
:which apparently produces
:
:{{}},
:{{0}, {1}, {2}},
:{{0, 1}, {0, 2}, {1, 0}, {1, 2}, {2, 0}, {2, 1}},
:{{0, 1, 0}, {0, 1, 2}, {0, 2, 0}, {0, 2, 1}, {1, 0, 1}, {1, 0, 2}, {1, 2, 0}, {1, 2, 1}, {2, 0, 1}, {2, 0, 2}, {2, 1, 0}, {2, 1, 2}},
:{{0, 1,0, 2}, {0, 1, 2, 0}, {0, 1, 2, 1}, {0, 2, 0, 1}, {0, 2, 1, 0}, {0, 2, 1, 2}, {1, 0, 1, 2}, {1, 0, 2, 0}, {1, 0, 2, 1}, {1, 2, 0, 1}, {1, 2, 0, 2}, {1, 2, 1, 0}, {2, 0, 1, 0}, {2, 0, 1, 2}, {2, 0, 2, 1}, {2, 1, 0, 1}, {2, 1, 0, 2}, {2, 1, 2, 0}},
:{{0, 1, 0, 2, 0}, {0, 1, 0, 2, 1}, {0, 1, 2, 0, 1}, {0, 1, 2, 0, 2}, {0, 1, 2, 1, 0}, {0, 2, 0,1, 0}, {0, 2, 0, 1, 2}, {0, 2, 1, 0, 1}, {0, 2, 1, 0,2}, {0, 2, 1, 2, 0}, {1, 0,1, 2, 0}, {1, 0, 1, 2, 1}, {1, 0, 2, 0, 1}, {1, 0, 2, 1, 0}, {1, 0, 2, 1, 2}, {1, 2, 0, 1, 0}, {1, 2, 0, 1, 2}, {1, 2, 0, 2, 1}, {1, 2, 1, 0, 1}, {1, 2, 1, 0, 2}, {2, 0,1, 0, 2}, {2, 0, 1, 2, 0}, {2, 0, 1, 2, 1}, {2, 0, 2,1, 0}, {2, 0, 2, 1, 2}, {2,1, 0, 1, 2}, {2, 1, 0, 2, 0}, {2, 1, 0, 2, 1}, {2, 1, 2, 0, 1}, {2, 1, 2, 0, 2}},
:...
:
:which then became the sequence
:
:0, 1, 2, 0, 1, 0, 2, 1, 0, 1, 2, 2, 0, 2, 1, 0, 1, 0, 0, 1, 2, 0, 2, 0, 0, 2, 1, 1, 0, 1, 1, 0, 2, 1, 2, 0, 1, 2, 1, 2, 0, 1, 2, 0, 2, 2, 1, 0, 2, 1, 2, 0, 1, 0, 2, 0, 1, 2, 0, 0, 1, 2, 1, 0, 2, 0, 1, 0, 2, 1, 0, 0, 2, 1, 2, 1, 0, 1, 2, 1, 0, 2, 0, 1, 0, 2, 1, 1, 2, 0, 1, 1, 2, 0, 2, 1, 2, 1, 0, 2, 0, 1, 0, 2, 0, ...
:
:which was A138036
:
:which was recently deleted by the editors.
:
:But the sequence looks interesting. Can anyone supply a definition
:in English for it? Or is it indeed not worthy of being included?

I don't speak Mathematica; by eye, the first list looks like all
square-free sequences in a 3-symbol alphabet ordered first by length,
then lexically.

Looking back at the history, it seems to be a demonstration of this:

[T]his computation shows that [...] squares are unavoidable over
3 letters, since every word of length 8 turns out to contain them.

.. unfortunately swaddled in so much semi-mystical verbiage as to be
effectively encrypted.

Replacing the mysticism with a clear explanation would probably be
sufficient to make this worth keeping.

Hugo

```