[seqfan] Re: Fractal sequence A087088

Frank Adams-watters franktaw at netscape.net
Thu Jul 16 17:37:08 CEST 2020

```"Fractal" seems to be used in the loose sense that "you perform some elementary operations on it and get back the original sequence". (Of course, this doesn't include trivial cases like "add one to every term and then subtract one from every term".) The process that fractal sequences, narrowly defined, are invariant under is called "lower trimming"; the process under which A087088 is invariant is called "upper trimming". I like to call sequences invariant under upper trim "co-fractal", though I don't know if anyone else uses this term. These are closely related: the ordinal transform of a fractal sequence is co-fractal, and the ordinal transform of a co-fractal sequence is fractal.

-----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: Wed, Jul 15, 2020 2:54 pm
Subject: [seqfan] Re: Fractal sequence A087088

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.
>>>> --
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