[seqfan] Re: Fractal sequence A087088

David Sycamore djsycamore at yahoo.co.uk
Sun Jul 19 13:45:52 CEST 2020

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

> 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.
>>>>>> --
>>>>>> 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/
>> --
>> 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