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

[Brad Klee]

Hi Kevin,

See also:

https://oeis.org/draft/A307672

Taking Mod[-A307672,3] gives this sequence
Gosper calls "magicd" (see G4G link therein),
and it has the same digit construction you
mention.

I agree, it is possible to prove this function
from the replacement rules.

At first I did not understand the sign convention,
but now it makes sense b/c incrementing +/- goes
between even / odd, and then the rules are either
cyclic or anti-cyclic depending on parity of the
pre-image.

If we take out the parity switch, then we can get
back to A053838 pretty quickly, but otherwise I
think we are stuck with needing six symbols.
Does it factor well enough to produce the parity
separate from the offset? I don't know.

Perhaps more interesting and worthwhile, is to try
and answer the big-picture question: under what
conditions does a substitution system produce the
same output as a function on the base-n digits?

Gosper's example gives a non-standard example
that pushes into unknown territory, so my opinion
is now that yes, there seems to be something
unusually interesting here.

Cheers,

--Brad