More fractals


This is mainly for David Sycamore and Maximilian Hasler (and also for Arthur Janta-Polczynski, now in heaven, who invited me in his garage avenue Montana, Brussels, exactly half a century ago, to listen to Ummagumma).

> SeqFan Mailing List, Dec 25th, Eric A.
Hello SeqFans,
If we call R the result of [a(n)-a(n+1) when a(n) is prime], then the successive such Rs will rebuild S itself:
S = 1, 5, 4, 11, 6, 13, 9, 19, 8, 29, 23, 10, 31, 22, 37, 18,...
S should be the lexicographically earliest seq of distinct positive terms with a(1)=1 and the obligation for nonprimes to be followed by at least one prime. [This is why a(2) = 5 here, and not a(2) = 4].
Best,
É.
-----
Dec 25th, Maximilian Hasler
Hello Eric,
my very simple program below confirms your terms and gives more:
(1, 5, 4, 11, 6, 13, 9, 19, 8, 29, 23, 10, 31, 22, 37, 18, 41, 33, 43, 14, 47, 24, 59, 49, 61, 30, 67, 45, 53, 16, 73, 55, 79, 38, 83, 50, 71, 28, 89, 75, 101, 54, 109, 85, 103, 44, 97, 48, 107, 46, 149, 119, 127, 60, 113, 68, 131, 78, 137, 121, 139, 66, 151, 96, 163, 84, 167, 129, 157, 74, 173, 123, 179, 108, 181, 153, 191, 102, 193, 118, 199, 98, 197, 143, 223, 114, 211, 126, 227, 124, 229, 185, 233, 136, 251, 203, 239, 132, 241, 195, ...)
I will submit it (...)
-----
Dec 25th, David Sycamore
Nice work Éric and Maximilian
Two questions :
If k is a term, can it arise more than once ?
If so, when the first occurrence of every k is erased does the original sequence reappear ? (as I understand it that is the normal behaviour of a fractal sequence, but I am no expert in this) ?
Best
David. 
-----
Dec 25th, Maximilian Hasler
It's now proposed as  https://oeis.org/draft/A330614.
The graph should look interesting (not yet visible on OEIS, I'll upload the b-file after validation). I added a comment on its (apparently) fractal nature.
DS> If k is a term, can it arise more than once ?
MH: no, as Eric says, "sequence of * distinct* positive integers ...", also in the definition of  https://oeis.org/A330614
DS> (as I understand it that is the normal behaviour of a fractal sequence, but I am no expert in this) ?
MH: well, a "fractal" in the usual sense is such that a "zoom" gives back the original; for integer sequences it is rather a certain subsequence, or as here, a sequence constructed somehow else from the original one.
(e.g., using first differences or so; here, the sequence (a(n)-a(n+1))_{n such that a(n) is prime}. I don't know whether there is a universally accepted definition of "fractal sequence", I think it is not a rigid but rather "flexible" concept.
-----
Dec 25th, David Sycamore
Right ok, thanks Maximilian. 
I know what you mean about the zoom thing (like Mandelbrot set ), but definitions of fractal that I have seen on Wikipedia for example mention the erasure requirement (which first means multiple appearances of any term k). Also, another property often mentioned is that a fractal sequence contains itself as a proper subsequence. Now there are examples of sequences which obey the latter condition but not the former, and I have assumed (maybe wrongly?) that they cannot be called fractal. 
I did not much follow your argument about the slopes of various graphs of subsequences, but I only read it through quickly a couple of times. It would be good to see the graph of this sequence, so as to see the prominence of these  various behaviours,  but I guess that has to wait, like the b-file, for approval. 
-----
Dec 25th, Maximilian Hasler
Hi David,
yes, maybe the simplest is to avoid claiming that the sequence is "fractal". It has the described "self-reproducing" property, that's all. (The "erasure" requirement could be seen in the fact that one only considers indices of prime terms to reproduce the sequence, so in some sense the composite terms are "erased". 
But not the terms of the sequence itself, but of its first differences are used.  So the erased terms are somehow still present in the result.) The  graph has a "fractal" property in yet another sense: 
It consists of "rays" of subsequences (roughly, but not exactly, every other term of the initial sequence, then every other term of the remaining sequence, and so on),
and they lie on a slope which is in-between the previous two slopes. Again, I think you can see some kind of "fractality" here, but it's not as simple as that.
PS: Below is a plot of the first 10'000 points; the largest y-value is 48'661. So you see the "main ray" (the primes) with a slope of around 5, and below the ray of composite terms of roughly half the slope,
and roughly in-between these the third ray, and so on.

-----
Dec 26, Eric Angelini
Hello David and Maximilian,
many thanks to both of you.
I would like to illustrate below my understanding of the word "fractal". Here is where I discovered that notion, 15 years ago: "Kimberling's paraphrases: if n = (2k-1)*2^m then a(n) = k".
And indeed, the idea of "erasure" and "fractal" began to be linked from there in my head (like in David's, I guess). Then came a doubt: "Why would one erase the first occurrences of 1, 2, 3, 4, 5, ... in a sequence S to recover S? Wouldn't a true "fractal sequence" contain an infinite amount of copies of _itself_ instead of an infinite amount of copies of _the succession of the first occurrences of the integers >0_?!"

I then worked on that idea: to produce sequences F, apparently chotic, but fractal in reality – where fractal meant "to contain an infinite amount of copies of F". I saw rapidly that 3 elements where involved: 
A. the starting sequence F; 
B. the terms that will be erased in F; 
C. the terms that will not be erased in F. 
You can play with the elements A, B and C in multiple ways – the graal for me being to find a sequence where A, B and C would be identical (and apparently chaotic) with a not too complicated "erasure law" (which I changed into a "underline law" for clarity). One example I'm proud of is the "Toblerone sequence" illustrated here on my blog (in French) and there (OEIS, English). A bunch of others can be find here (see the Xrefs).

Here is an example I designed especially yesterday for David where A and B are the same (not C):

Erasing law:
"Underline in S every term that comes after an even number: the underlined term rebuild S itself."
(I always want my laws to be simple enough for a 10-year old kid to understand and have fun with.)
S = 1, 2, 1, 3, 4, 2, 1, 5, 6, 3, 7, 8, 4, 2, 1, 9, 10, 5, 11, 12, 6, 3, 13, 14, 7, 15, 16, 8, 4, 2, 1, 17, 18, 9, ...
Check:
Even terms in yellow:
S = 1, 2, 1, 3, 4, 2, 1, 5, 6, 3, 7, 8, 4, 2, 1, 9, 10, 5, 11, 12, 6, 3, 13, 14, 7, 15, 16, 8, 4, 2, 1, 17, 18, 9, ...
Underline the terms immediately after a yellow one:
S = 1, 2, 1, 3, 42, 1, 5, 6, 3, 7, 842, 1, 9, 10, 5, 11, 126, 3, 13, 14, 7, 15, 16842, 1, 17, 18, 9, ...
Forget the yellow color and put in bold the underlined terms:
S = 1, 21, 3, 4, 21, 5, 63, 7, 8, 4, 21, 9, 105, 11, 12, 6, 3, 13, 147, 15, 16, 8, 4, 21, 17, 189, ...

We see that the bold terms rebuild S. And between them the pattern (decided by me in tribute to Clark Kimberling, my god) is obvious (and could be something else – the primes, for instance).
I guess a 10000-term graph would show the fractality I'm fond of.
Best,
É.
P.-S. another [A and B, not C] example here. An example of [A and C, not B] here


____________________
December 27th 2019 update

From: David Sycamore Date: 27 December 2019 at 15:37:27  GMT To: seqfan@list.seqfan.eu Subject: Fractal sequence?


What is a  “fractal” sequence?
When writing A329221 some weeks ago I found that it has a proper subsequence identical to itself and for this reason thought that it must be a fractal sequence. Then I found the Wikipedia definition (see link in sequence entry) and discounted this sequence as fractal because it is not true in this case that erasure of the first occurrence of every term recovers the original sequence. At the time  I wasn’t entirely happy about that definition, which seemed to be too exclusive, but felt that I had to acknowledge it.

Now I’m wondering again about this definition, especially following submission of A330614 by Éric and Maximilian. This is described as fractal because it has the familiar “zoom” quality, associated with containing a proper subsequence copy of itself. One big difference between these two sequences is that whereas in A329221 every integer recurs infinitely many times, in A330614 every occurrence of every number is unique, so if the first occurrence of every term is erased, what we end up with is { }. Maximilian argues that the “zoom” behaviour is such a distinctive characteristic that with a somewhat “flexible” attitude to the definition it can be said that A330614 has fractal qualities. I agree with him, but not being an expert and wanting to get to the bottom of this, I decided to put the question out here, in case other members of the list may be able to throw some light on it.

Regarding the Wikipedia definition I don’t really see why erasure of the first occurrence is so necessary. First, it precludes any sequence like A330614, which does not have this specific property and secondly what is so special about the first occurrence, which is just one way of erasing a subsequence? If we say instead that an infinite sequence is fractal if it has an infinite proper subsequence, removal of which recovers the original, then we have preserved the (erasure) concept of the Wikipedia version whilst leaving the choice of subsequence open.

What I’m  suggesting here is that the main fractal characteristic (the “zoom”) is perhaps more important than how you actually get to it. In some cases it might be by removal of every first occurrence, in others it might be a different subsequence which gets rubbed out. The point is that by finding the right thing to erase, you’re back to square one.
I wonder if oeis should have its own standard definition of a fractal sequence rather than relying on other sources? Also I’m asking if there should be a keyword linked to these sequences. Would a library of fractal sequences be worth having in the data base?
Best,
David.
____________________
Maximilian Hasler:
[...]
> I just want to insist that
– first, I don't want to "claim" anything, e.g., this or that *is* fractal or means "fractal"...Life is complicated enough without adding complications artificially...And even in mathematics, the same word  (and notation) may have different more or less widely accepted  meanings / definitions. 
[Examples: "natural numbers" or "counting numbers"  denoted ℕ, with or without 0. (Some even say that "whole numbers" must be >= 0 although "integer" exactly means "whole number" and I really can't see how negative integers might not be "whole" which to me means that they don't have a fractal part. But I simply accept that some authors use such a contradictory name / meaning). Or, "round(x)" : in some programming languages (maybe even the very popular Python),  (n+0.5) is rounded towards the nearest EVEN (or ODD) integer!! Weird, isn't it? Or "ring" means for some "ring with unit", for some it means "pseudo-ring" (not necessarily with unit). There are zillions of such (and probably better) examples.]
The most important is that, when the precise definition is really important, then it has to be defined where it is used. second, I'd not even claim that the graph of this sequence has the "zoom" property, what specialists call "self-similarity" (and even that may well have different definitions depending on the author).The property it has is that of subsequences of "(roughly) every other of the (remaining) terms forms a sequence which is between the two previous subsequences and larger [smaller, every other time] than the previous such subsequence".
Also (somewhat a variant of 1), I think "fractal" does and should be allowed to have several distinct meanings. I'm not sure we need a very strict rule concerning this. (There are other things where a rule would be useful (and "natural") on OEIS but NJAS says that we already have too many of these.)
Anyway, let's not get a headache about this...
Best wishes for a pleasureful end of 2019 and transition into 2020 !

Maximilian


Commentaires

Posts les plus consultés de ce blog

A square for three (chess)

Le tripalin se présente

Some strings au cinéma Galeries