[seqfan] Re: Fractal sequence A087088

[Frank Adams-watters]

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.
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