[seqfan] Sequences with nice graphs / "Entanglement permutations"

[Antti Karttunen]

> Message: 3
> Date: Tue, 21 Jan 2014 16:57:30 -0500
> From: Neil Sloane <njasloane at gmail.com>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>, Paul
>         Tek <paul.tek at mail.be>
> Subject: [seqfan] Sequences with nice graphs
> Message-ID:
>         <CAAOnSgRkpp-QkRP10_-az=qXHnaUrGou=4O32x0mqQ4DAeoJjg at mail.gmail.com>
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>
> Paul Tek's User Page on the OEIS wiki has a short
> list of sequences that have interesting graphs.
> For example:
> lacework: A003987
>
> Stimulated by this, I've created a subsection of
> the Index to the OEIS to list such sequences.
> This is in Section Gra of the Index, under
> graphs (or plots), sequences with interesting
>
> I used Paul's list and added a handful of others.
>
> Anyone who has watched Tony Noe's OEIS Movie will
> know that there are many sequences with lovely graphs.
> The best way to look at the frames of the movie is here:
> http://www.sspectra.com/math/MovieFrames/
>
> The purpose of this posting is to invite people
> to expand these entries in the Index.
>
> Of course, whether you think a graph
> looks like a waterfall or a hurricane is a matter of taste - but
> that's OK!
>
> This is all part of the plan to emphasize the visual aspects of
> the OEIS.
>
> Neil
>


I added two from my own list (which I admit has some bias to the seqs
submitted by myself, as I recall those ones the best):
http://oeis.org/wiki/User:Antti_Karttunen#My_favourite_graphs

with these two new "visually descriptive subheadings":

fractal froth: A135141
 and
waves: A218789

to
http://oeis.org/wiki/Index_to_OEIS:_Section_Gra#graphs_plots

I guess it's OK from now on to include in the links-section of the
sequences that appear in that list a link like:

<a href="http://oeis.org/wiki/Index_to_OEIS:_Section_Gra#graphs_plots">Index
entries for sequences with interesting graphs or plots</a>

???

-------

As what comes to A135141 by Katarzyna Matylla, (see
http://oeis.org/A135141/graph )
I found its beautiful graph from the still-frame of the OEIS-movie.
(I mean, the one which Youtube will show for example in its "side-video" lists).

Inspired by it, I submitted its inverse permutation A227413 last July,
as well as later, several generalized cases, i.e. "entanglement
permutations", see:

http://oeis.org/search?q=A135141&sort=&language=&go=Search

where one divides a set of natural numbers to two complementary
subsets, in two different ways, and then "entangles" them, in some
cases requiring extra +1 or -1 fiddling with indices, as to make the
resulting sequences permutations of nonnegative integers/natural
numbers.
(BTW, I'm not anymore entirely happy with that word "entangling",
maybe "interweaving" would have been better? )

First, I was hesitating as whether to submit any of those at all, as I
considered them just as further noise into the system, but then I
realized that we obtain also some "old friends", for example, when
entangling the even/odd pair with evil/odious pair (A001969/A000069)
we get Gray code and its inverse: A003188/A006068.

Also, when entangling even/odd pair with the complementary pair
A048724/A065621 (which are actually just evil and odious numbers
further permuted) I bumped into "Blue code": http://oeis.org/A193231
which itself is a very nice sequence.


So, for Katarzyna's A135141, those two pairs of complementary sets are
primes/composites and even/odd numbers, and the latter occur as one of
the pairs in all examples I have submitted so far. But of course one
does not need to delimit oneself to that. E.g. what comes if one
"entangles" pair primes/composites with pair composites/primes, in a
similar way?
Is it already in OEIS? (Should result a self-inverse permutation).

Also, as what comes to possibly graphically interesting variants, I
suggest permutations pairs like A135141/A227413 but instead of
primes/composites, use complementary pairs like the following,
obtained with various sieving processes:

A000959/A050505 (Lucky/Unlucky numbers),
A003309/A192607 (Ludic/Nonludic numbers),
A014580/A091242 (Irreducible/Reducible GF(2)[X] polynomials, evaluated at X=2),
etc.

Also, one could entangle these, not just with even/odd numbers pair,
but also with each other. What will result? (I mean, will they look
radically different from http://oeis.org/A135141/graph or are they
also "fractal froth" ?)

BTW, I'm not submitting these by myself, because my mental backburner
is still full with dozens of _other_ interesting sequences...


Yours,

Antti Karttunen