[seqfan] Organization of dispersion-arrays.
Antti Karttunen
antti.karttunen at gmail.com
Tue Dec 15 05:57:17 CET 2015
(Warning: The following message is a little self-absorbed in dispersions
and is about their (sometimes missing) links. I will comment in another
posting about what Eric actually wrote about.)
On Mon, Dec 14, 2015 at 2:08 PM, Eric Angelini <Eric.Angelini at kntv.be>
wrote:
> > Interesting musings and a point of view, Eric.
>
> ... thanks, Antti
>
> > However, I think myself that the best "artificial laws"
> > are those that feel most "natural" to me.
>
> ... all points of view and tastes are welcomed! The benefit
> of an online digital database is that you can fulfill almost
> everyone -- collectors, searchers, creators. The key is of
> course the internal search engine, the way labels are given,
> the data structure, the indexes, etc. I guess we will have
> at some point in the future the possibility to make "voice
> queries", or "sound queries", or "picture queries", or
> "graph" comparisons, etc.
>
> >(Sieve of Eratosthenes and Ludic numbers sieve) co-create
> > something visually nice: https://oeis.org/A255422/graph
>
> ... indeed
>
> > "Spiro-Fibonacci numbers" [...] where would you place it
> > in your continuum of "natural" ... "artificial" laws?
>
> ... good point -- but the "natural/artificial" distinction
> was emphasized by me only in the hope that more seqs would
> be linked to each other based on their "building structure".
> This is, what are the bricks and what are the laws that rule
> the seq -- no matter if they are of interest "in the real
> world".
>
Hah, you are reading my thoughts here!
Just yesterday I was frustrated, when looking at Martin MÃ¸ller Skarbiniks
Pedersen's new A265650 where he links to another fractal sequence
http://oeis.org/A003603 "Fractal sequence obtained from Fibonacci numbers
(or Wythoff array)."
>From there I tried to immediately hop to the Wythoff array, that is,
http://oeis.org/A035513 (but whose A-number escaped me at the moment) but
alas, there is no link from A003603 to A035513 or vice versa!
So please people, whenever we have a "dispersion array" which is of a
fundamental nature (or which you think might later attain such status,
depending on the future interests of mathematicians), could we make sure
that from that array there will
be links to:
A) Transpose of that array, and please create it right away, if it's not
already there, because otherwise somebody else will create it sooner or
later, and then we have again two parallel timelines of formulae and
observations accruing from different people, about what is essentially the
same thing.
[Also, as these arrays are also permutations of natural numbers or almost
(sometimes beginning from 2), maybe also the inverse permutation, although
I think this is optional, an actual formula for which requires the two
functions mentioned next].
B) The column index and the row index. (That is, the sequence Acolind(n),
which tells in which column (resp. row for Arowind) n is located in the
array. The other one of these is always (or so I think) a fractal sequence,
like Pedersen's new http://oeis.org/A265650 (BTW, of what arrays it is
col/row index of ?)
C) First few rows and first few columns. Depending on the orientation of
the array (which of two transposes it is), the topmost row or the leftmost
column is the complementary sequence to the sequence whose dispersion this
array is, and correspondingly, the edgemost sequence on other axis gives
the iterates of the dispersion sequence itself.
D) The sequence whose dispersion this array is. Again, depending on the
orientation, it is also a function with which to proceed from each term
towards right or bottom, that is, to the next term on the same row or in
the same column.
Like e.g. I have done here:
http://oeis.org/A257505 "Square array A(row,col): A(row,1) = A256450(row),
and for col > 1, A(row,col) = A255411(A(row,col-1)); Dispersion of
factorial base shift A255411."
Now, I reckon that Clark Kimberling's original dispersion papers require
that the function to be dispersed should be monotonic (and also that the
complementary values should be laid in monotonic order at the left edge of
the array, those values working as the "seeds" from which the
dispersion-function starts generating each row).
However, I have found it useful to disperse also nonmonotonic functions,
like the prime-shift function A003961 in http://oeis.org/A246279
Sometimes the dispersion function can be created _a posteriori_. For
example, for an array like
http://oeis.org/A083140 "Sieve of Eratosthenes arranged as an array and
read by antidiagonals in the up direction; n-th row has property that
smallest prime factor is prime(n)."
for which it is:
http://oeis.org/A250469 "a(1) = 1; and for n > 1, a(n) = A078898(n)-th
number k for which A055396(k) = A055396(n)+1, where A055396(n) is the index
of smallest prime dividing n."
(although the function itself is not monotonic, each of the produced
columns and rows are, of course, because of the way A083140 was created).
(BTW, I actually concentrate more on the transposes of those arrays,
A246278 and A083221, there are more useful links there. Also on my todo
list, similar back-formation dispersion functions for the arrays
constructed from Lucky and Ludic sieves).
Just now in the editing state and my mental backburner I have a couple of
developments where the dispersing sequence and its complement are
back-formations from the permutations involving "bijective base-3 reverse",
A263273. Let's see how weird harvest I will get, when I start dispersing
from the seeds sown like this: https://oeis.org/A265341/graph
(The point here is that I am "somewhat artificially" viewing a base-3
related sequence as a _binary_ tree, just because it is possible to do so.
Actually, you can view _any_ permutation in such a way, but of course the
most of such views are ugly and do not lead anywhere, or lead only
half-way, cf. e.g. A253888).
Best regards,
Antti Karttunen
PS. And please keep me on To: or CC:-line, because otherwise it's hard for
me to reply, as then I will see your reply only later on the next
SeqFan-digest.
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