User talk:Charles R Greathouse IV/Chase sequences

It'll take a while to untangle, but I hope the real example becomes a historical example (e.g., "used to be defined as...")

Here's one that's not as bad and thus hopefully easier: A213717, just came up with Michel making a minor spelling correction. Alonso del Arte 17:52, 7 January 2013 (UTC)

Indeed. I haven't yet been able to understand any of the sequences in the web, though.

As for A213717, it's much easier. It references two sequences, each of which are defined directly. One is a subsequence of the other, and A213717 asks for their difference (or relative complement, which is the same in this case).

Charles R Greathouse IV 19:02, 7 January 2013 (UTC)

The terms of A213717 are those nonleaves (i.e. branching vertices) of the binary beanstalk that are located in its "tendrils", i.e. the finite side-trees. This will be all clearer the day I finally draw a proper illustration of the whole thing. By "binary beanstalk" I mean the whole set of nonnegative integers arranged as a planar binary tree, where the relation $a=b-A000120(b)$ defines the edges between parent and child nodes. (Note that for any nonnegative integer $a$, there is either two solutions to that or no solutions at all, thus this creates a binary tree. Those $a$ with no solutions are the leaves of the tree). In somewhat looser language, "beanstalk" may refer only to the subset A179016 which consists only of the nodes in the infinite trunk of the said tree. — Antti Karttunen 22:49, 13 July 2013 (UTC)

Circular chase sequences?

Let's hope there are no circular chase sequences (dogs chasing their tail sequences?)... — Daniel Forgues 02:59, 16 February 2013 (UTC)

I've come across examples in the past. Of course just because a sequence is impredicative doesn't mean it's ill-defined.... Charles R Greathouse IV 18:00, 17 February 2013 (UTC)

Should we have the keyword "impredicative"? I will create the categories Category:Impredicative sets and Category:Impredicative sequences. I've learned a new term!

Impredicativity—Wikipedia.org.

— Daniel Forgues 00:27, 18 February 2013 (UTC)

Response

Come on guys, what's the big deal? Yes, in some case there's a lot's of "infrastructure"-sequences required, before we even get into anything interesting. See e.g. [1]. And yes, many of the derived sequences could have more meaningful names than just "partial sums", "bisection", "intersection of ... and ...". But these names can always be improved later, I'm not allergic to that. Of course I could leave all the intermediate sequences out, but if I computed the "leaf-sequence" with a single monolithic imperative program, with many variables updated in the loop, it would be much harder to grasp and see that it's correctly computed. BTW, I never create "circular chase sequences" (How that would be even possible?!). However, because of the insistence of T. D. Noe that all cross-refs like a(n) = Axxxxxx(n)-Ayyyyyy(n) should be moved to Formula-section, that kind of mistaken impression is possible at first glance. When in doubt, it's always the Scheme-code I include in the program-section that computes and defines the sequence. — Antti Karttunen 17:37, 2 July 2013 (UTC)

Of course, one way to detangle the mess like A179833 a little, would be to employ neologisms, e.g. I could name sequences like A179830 and A179831 and as "zipper-sequences" of the "flame-pyramid" sequence A122242, etc. (Cf. "EKG-sequence" for a precedent, for giving names based on the appearance of the sequence.) — Antti Karttunen 17:50, 2 July 2013 (UTC)

Hi Antti! I'm very interested in finding ways to, as you say, detangle clumps of sequences. Because you created them, I assume that they are interesting -- that's one of the reasons I chose your sequences as an example. (Sorry!) But in their present state it's hard to figure out why. (Sadly, this is not uncommon: it's playing out right now at A182221, where Ribas has a sequence which is surely interesting but there is no explanation given. Fortunately he's writing a paper and should add it to the sequence when it's complete, but until then everyone is in the dark as to its meaning.)

Usually I try to 'detangle' starting at the ends. If there is an interesting and reasonably well-understood sequence Axxxxxx, it should be possible to study the direct derivatives of that sequence and add good descriptions. Sometimes you can add a comment which gives an informal description of the sequence: "This functions measures the tendency of divisors of a number to cluster".

I'm not automatically opposed to arbitrary names of the type you mention, but I don't prefer them in general. A person can only keep so many in their head, while definitions are quite a bit more self-parsing.

You may have other good ideas for detangling and otherwise explaining sequences, which I would be very interested to hear.

Charles R Greathouse IV 20:33, 3 July 2013 (UTC)

Sorry for the late answer. Yes, sequences like A179833 might be left into a complete mystery, unless I start documenting them better. Essentially, what is happening that when we are iterating function A122237, which is essentially constructed from a bijection A082358 (which in turn is related to converting between different ways of encoding binary trees, see comments at A085184) and A057548 which in this context adds an additional vertex at the root of the binary tree (so it's not genuine bijection anymore, but instead the tree grows with each step, which you can see for example in A122242, where that tree is drawn line by line as in its Dyck word manifestation (just as a balanced binary string, with black and white pixels), it grows by two bits/pixels per every iteration. Now, what conjecture at A179831 essentially says, is that is that "the central zipper" you see in the middle, stays there, at least after certain point. Now, what it really is about (at least I surmise), is some kind of an attractor subtree, to which more branches are now and then glued to a slowly growing stable sub-configuration. See also similar figure drawn for the same function, but starting iterating from a slightly different initial value, and iterated for longer time (use zoom). In retrospect, when seeing that figure, where the central "zipper" (or "skyscraper") just keeps on widening, my conjecture at A179831 and A179776 might be a bit lame, and more relevant question would be at what specific points (or intervals, if taking first differences) the said "skyscraper" widens, i.e. attaches more stable sub-branches into it. I hope I will myself understand the process better after I have coded some better visualization routines for this. Inferring the tree structure from the Dyck word drawn as a row of pixels is not the best way to go... Hope this helped, — Antti Karttunen 23:20, 13 July 2013 (UTC)

This is fascinating, I'll have to read it more carefully when it's not too late.

Unrelated to the A179833 cluster, I just came across a bunch of sequences implicitly (or, in some cases, explicitly) related to A078310. It's yet another case of "I don't know why this is interesting, though I trust the author".

Charles R Greathouse IV 06:00, 16 July 2013 (UTC)