[seqfan] Re: Fractal sequence A087088

[Éric Angelini]

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.
>>>>> --
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