[seqfan] Re: Fractal sequence A087088

David Sycamore djsycamore at yahoo.co.uk
Wed Jul 15 21:54:58 CEST 2020

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,

> 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.
>>>> --
>>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>> --
>>>> Seqfan Mailing list - http://list.seqfan.eu/
>>> --
>>> Seqfan Mailing list - http://list.seqfan.eu/
>> --
>> Seqfan Mailing list - http://list.seqfan.eu/
> --
> Seqfan Mailing list - http://list.seqfan.eu/

More information about the SeqFan mailing list