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

Sat Jan 25 15:08:49 CET 2014

On Sat, Jan 25, 2014 at 2:05 AM, Neil Sloane <njasloane at gmail.com> wrote:
> Just to respond to a couple of Antti's questions:
>
>
>> 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>
>
> ???
>
> Absolutely! Good idea - because otherwise one has no idea whether
> a graph is interesting.
>
> You could even say:
> <a href="http://oeis.org/wiki/Index_to_OEIS:_Section_Gra#graphs_plots">Index
> entries for sequences with interesting graphs or plots</a> [mentions this
> sequence]
>

Those subheadings are of course of very subjective nature, and the
categories might overlap. A229037 is already in two.
As what comes to "fractal froth" for A135141, maybe it would allow
other descriptions as well? I remember seeing that kind of texture
also in some real world photos. Microbial growth, maybe? Or fabricated
textile of some sort?

Of course how the Scatter Plot image will look like depends a lot on
for how many terms it has been drawn. It seems that the image one sees
at:
http://www.youtube.com/results?search_query=The+OEIS+Movie
is slightly different from what Plot will nowadays draw:
http://oeis.org/A135141/graph

BTW, anybody invents a good visual category for this one:
http://oeis.org/A233270/graph ?
I would not place it under "bounces", like the elegant
http://oeis.org/A162499/graph

>
>
>> (BTW, I'm not anymore entirely happy with that word "entangling",
> maybe "interweaving" would have been better? )
>
> if you mean an ababababa... alternation, "interleaving" is the usual term

Well, that is what occurs in Murthy's permutation:

A073846: a(1) = 1 and then every even term is prime and every odd term
is composite.

However, when we do similar thing recursively, using the emerging
permutation itself to specify the order of both interleaved
bisections, we get:

A227413: a(1)=1, a(2n)=nthprime(a(n)), a(2n+1)=nthcomposite(a(n)).

which is the inverse permutation of Matylla's A135141.

Now, the other pair of complementary sequences does not need to be
"even numbers/odd numbers" (although that is certainly one of the most
fundamental divisions there are, and might serve as our "central
relaying station" / "interface" when composing such permutations with
each other), so we might as well have:

a(p_n)=nthluckynumber(a(n)), a(c_n)=nth_unluckynumber(a(n)), where p_n
= n-th prime, c_n = n-th composite number.
(And maybe requiring some fiddling with index-offsets to result a
bijection), and vice versa for the inverse permutation

so it's not just simple "interleaving" anymore, but still some
metaphoras from craftworks bring forth in my mind. But maybe it's
still not yet so hopelessly messed up that it would be "entangled",
like some Gordion's knot? Because those permutations are still very
well defined, and can be "unraveled" with their inverse permutations.
But English is not my mother tongue, so these words probably have
slightly different connotations for a native speaker.

>
>> BTW, I'm not submitting these by myself, because my mental backburner
> is still full with dozens of _other_ interesting sequences...
>
> They sound like they should be in the OEIS, so if
> you can send them in, that would be worth doing. You may have
> done most of the real work already!

Yes, I will be slowly submitting these, unless somebody else is
faster. It helps if one already has certain "tool sequences" at hand,
like:

A065855 "Number of composites <= n." = n-A000720(n)-1
and their "GF(2)[X]-analogues": A091245 & A091226.

But I guess such are still missing for Lucky/Unlucky & Ludic/Nonludic pairs.

Yours,

Antti

>
> Best regards
>
> Neil
>
>
> On Fri, Jan 24, 2014 at 6:08 PM, Antti Karttunen <antti.karttunen at gmail.com>
> wrote:
>>
>> > 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>
>> > Content-Type: text/plain; charset=ISO-8859-1
>> >
>> > 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
>
>
>
>
> --
> Dear Friends, I have now retired from AT&T. New coordinates:
>
> 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
>