[seqfan] Re: Fractal sequence A087088

[M. F. Hasler]

On Tue, Jul 21, 2020 at 5:19 AM Frank Adams-watters wrote:

> I have tried to push that idea before, and gotten nowhere with it. For
> what it's worth, I'll support it now.
> Perhaps what we need is a word for the looser, non-rigorous definition.
> Might I suggest "fractaloid"? If that's too ugly, I'm open to other
> suggestions.
>

I favour much Jean-Paul's view on this subject :
let's not mix up "fractal" and "self-similar" ;
"reproduces itself modulo some shift / scaling / rotation"
is clearly (couldn't be more) an instance of "self-similarity"
(a convincing reason to use this terminology rather than any other!)
and not (a priori) related to "fractal", even if one might object that
a) some fractals are self-similar
 (but most of them only (very) approximately)
b) the adjective "fractal" (maybe) isn't well-defined yet and there could
be a way of including "our case" in a (new and/or better(?)) definition.
(That looks quite arbitrary / contrived to me, given the well-established
notion of self-similarity.
Sounds much like "I want to call it 'fractal' and not 'self-similar'
whatever reasons suggest the contrary.")

- Maximilian



> -----Original Message-----
> From: David Sycamore via SeqFan <seqfan at list.seqfan.eu>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Cc: David Sycamore <djsycamore at yahoo.co.uk>
> Sent: Sun, Jul 19, 2020 6:45 am
> Subject: [seqfan] Re: Fractal sequence A087088
>
> Looks like there is a variety of different perspectives on this. Would it
> perhaps be helpful to both contributors and editors if oeis were to have
> it’s own unambiguous statement of precisely what constitutes a fractal
> sequence?
> Perhaps the terms and definitions used by Kimberling et al could be taken
> as standard?
>
> Best regards
> David.
>
> > On 19 Jul 2020, at 07:28, Frank Adams-watters via SeqFan <
> seqfan at list.seqfan.eu> wrote:
> >
> > That definition is both too strong and too weak.
> >
> > It is too weak, because any infinitive sequence is fractal by that
> definition. (An infinitive sequence is a sequence of positive integers that
> contains every positive integer infinitely often.)
> >
> > Too strong, because, for example in the current instance A087088, the
> transformation getting the original sequence back is not just taking a
> subsequence; it also involves an arithmetic operation (subtracting one).
> A087088 is infinitive, and hence fractal by that definition; but this would
> be based on a different transformation than the one given in the definition
> of the sequence.
> >
> > Franklin T. Adams-Watters
> >
> >
> > -----Original Message-----
> > From: Éric Angelini <eric.angelini at skynet.be>
> > To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> > Cc: David Sycamore <djsycamore at yahoo.co.uk>
> > Sent: Thu, Jul 16, 2020 10:06 am
> > Subject: [seqfan] Re: Fractal sequence A087088
> >
> > I’ve always used the definition:
> > Fractal seq S = seq S containing
> > an infinite amount of copies of S.
> > There are a lot of ways to show/
> > fix/decide/etc. how to highlight
> > a single copy.
> >
> > à+
> > É.
> > Catapulté de mon aPhone
> >
> >
> >>> Le 16 juil. 2020 à 16:11, David Sycamore via SeqFan <
> seqfan at list.seqfan.eu> a écrit :
> >> Coming a bit late to this discussion, I have a question concerning the
> definition of what is meant by a “fractal” sequence? Could there be more
> than one different interpretation of this term?
> >> According to one definition, currently described in oeis, a sequence is
> fractal if, when all first occurrences are removed, what remains is the
> original sequence (which means that it contains a proper subsequence
> identical to itself).
> >> Removal of all first occurrences from A087088 gives:
> >> 1, 2, 1, 3, 1, 2,1, 4, 2, 1, 3, 1...
> >> which is not the same as the original.
> >> Can anybody explain why A087088 is considered to be fractal?
> >> Best regards,
> >> David.
> >>>> On 14 Jul 2020, at 07:12, Allan Wechsler <acwacw at gmail.com> wrote:
> >>> If I were asked to write a sequence title from scratch, I think I
> would
> >>> dispense with "simplest", and say something like "Positive ruler-type
> >>> fractal sequence with 1's in every third position." I think only this
> >>> sequence satisfies that description.
> >>> A "fractal" sequence can be constructed upon any "skeleton" of 1's, as
> long
> >>> as there are an infinite number of entries that are _not_ 1.
> >>>>>>> On Mon, Jul 13, 2020 at 12:42 PM Neil Sloane <njasloane at gmail.com>
> wrote:
> >>>> Allan, that is an excellent point.  So maybe the sequence should say
> >>>> something like "simplest two-step-insertion fractal" ?
> >>>> Best regards
> >>>> Neil
> >>>> Neil J. A. Sloane, President, OEIS Foundation.
> >>>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> >>>> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway,
> NJ.
> >>>> Phone: 732 828 6098; home page: http://NeilSloane.com
> >>>> Email: njasloane at gmail.com
> >>>>> On Mon, Jul 13, 2020 at 12:37 PM Allan Wechsler <acwacw at gmail.com>
> wrote:
> >>>>> To return to the claim of "simplest" sequence with this property; we
> are
> >>>> in
> >>>>> the difficult position of trying to read the mind of the person who
> was
> >>>>> making that claim. I think they had some notion of "simplicity" in
> mind
> >>>> for
> >>>>> which the statement was arguably true, but as it stands it is hard
> to see
> >>>>> what that notion was. The point about the ruler functions is a
> strong one
> >>>>> -- A001511 can be given a homologous four-step definition exactly
> >>>> analogous
> >>>>> to the one given for A087088, using gaps of one undefined place
> instead
> >>>> of
> >>>>> two. One is simpler than two, isn't it?
> >>>>> But even A000027, the positive integers, displays the required
> property.
> >>>>> Remove the only 1; decrement all other entries; behold. In what
> sense is
> >>>>> A087088 simpler than A000027? I think the author(s) had some
> additional
> >>>>> constraints in mind. But if I were shown the title only, and asked to
> >>>>> reconstruct the sequence, I would probably produce A000027.
> >>>>> On Mon, Jul 13, 2020 at 12:05 PM Frank Adams-watters via SeqFan <
> >>>>> seqfan at list.seqfan.eu> wrote:
> >>>>>> This sequence and A163491 are ordinal transforms of each other.
> >>>>>> Franklin T. Adams-Watters
> >>>>>> -----Original Message-----
> >>>>>> From: Peter Munn <techsubs at pearceneptune.co.uk>
> >>>>>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> >>>>>> Sent: Mon, Jul 13, 2020 9:01 am
> >>>>>> Subject: [seqfan] Fractal sequence A087088
> >>>>>> Hello seqfans,
> >>>>>> A087088 claims to be "the simplest nontrivial sequence" such that
> >>>>> removing
> >>>>>> every "1" gives the same result as adding 1 to every term. Ruler
> >>>>>> sequences, such as A001511, share this property, so does anyone
> have a
> >>>>>> clear idea how "simplest nontrivial" might be defined?
> >>>>>> And can anyone shed light on the reason its offset is 3? [1]
> >>>>>> Best Regards,
> >>>>>> Peter
> >>>>>> [1] Apart from the b-file, the rest of the sequence is written as
> >>>> though
> >>>>>> the offset is 1 (so formulas are strictly incorrect). The
> relationship
> >>>> to
> >>>>>> A244040 contributed by Edgar and Van Alstine is neatest with offset
> 1
> >>>> or
> >>>>>> offset 0. A relationship I discovered recently (comment in
> >>>>>> https://oeis.org/A024629) is clearly neatest if the offset is 1,
> >>>> whilst
> >>>>> my
> >>>>>> work on symmetry (https://oeis.org/history/view?seq=A087088&v=25)
> and
> >>>>> with
> >>>>>> A335933 suggests an OEIS-incompatible offset of 1.5 .
> >>>>>> As we are only now starting to refer from other sequences to terms
> of
> >>>>>> A087088, it seems a good time to settle on a good offset. Unless
> anyone
> >>>>>> knows a good reason for keeping it as 3, offset 1 seems better.
> >>>>>> --
>