[seqfan] detective work related to Creighton Dement's "Floretions"

[Robert Munafo]

I read this and got curious:

On Mon, 9 Nov 2009, Michael Porter wrote:

> I did an OEIS search on "floretion" and the first two pages consisted of:
> 1. sequences related to floretions with few references or links, and 2. big
> sequences like the prime numbers or the squares which have some note about
> floretions buried in them.? It wasn't very useful for someone looking to
> learn more about floretions.
>

I used "Sloandora" (mrob.com/pub/math/sloandora) to find all the floretion
sequences and related sequences in OEIS (using an August 2007 copy of the
eisBTfry00000.txt files).

As reported by Michael, there are lots of OEIS entries that have little or
no explanatory references, and a few (like A000330, A001541, A001792,
A001834) that have a reference buried in the middle of their dozens or
hundreds of other references.

There are many sequences in OEIS that contain %o fields with text like "FAMP
Code: 1tesforzapseq[(.5i' + .5j' + .5'ki' + .5'kj')*(.5'i + .5'j + .5'ik' +
.5'jk')]". I have not found any information on how to take such text and
actually convert it into an integer sequence.

With Google searches I was able to find an old, dead website that is still
available through web.archive.org:

   http://www.crowdog.de/20801/home.html

 From that archive I was able to download a JAVA archive of "FAMP version
Moonlice", along with some cryptic "documentation" in the form of the web
page www.crowdog.de/Explanations.html

"1tesforzapseq" is part of an obscure naming system, combining a prefix with
one or more three-letter "syllables" representing different types of
calculations that can be performed in each step of an iterative process.

A few OEIS entries give tantalizing clues as to what the syllables are. For
example A100213 states:

*... The elements 'i, 'j, 'k, i', j', k', 'ii', 'jj', 'kk', 'ij', 'ik',
'ji', 'jk', 'ki', 'kj', e ("floretions") are members of the quaternion
product factor space Q x Q /{(1,1), (-1,-1)}. "pos" sums over positive
coefficients of the above basis vectors.*

Similarly A100215 states:

*... "ves" sums over all floretion basis vector coefficients for each n.*

and A100216 states:

*... "les" sums over coefficients belonging to basis vectors which squared
give the unit e (excluding e itself).*

An actual coherent explanation of an algorithm can be found in *A108618*,
which details the procedure by which the sequence can be calculated. Its %n
says the algorithm described is "version: 'tes'" which is another
"syllable". *A109620* has another example. Those who understand the jargon
of groups or related algebras might prefer the description in *A096252*.

I also found A119953, which refers to the PDF file at
www.research.att.com/~njas/sequences/a119953.pdf, which (on pages 2,3)
appears to define syllable "jes".

Sequence A108930 seems to provide more clues, including how the "spiral"
like graphical plots have been produced.


We have inaccurate (or at best, confusing) use of standard terminology in
some comments. For example, A103135 calls itself *"A floretion-generated
sequence which emerges as a transformation of A000004"*, with a %E line from
NJAS stating *"Definition not clear to me. A000004 is the zero sequence!
njas"*. Its %o line notes that the "vesforcycseq", "4lesforseq" and "vesseq"
iteration algorithms, each using the same initial vector of 16 elements,
give three different sequences one of which is A000004. This is probably
what the "transformation of A000004" comment means. If that were a valid use
of the word "transformation", then we could say that A000079 is a
transformation of A000004 because both can be produced by the iteration
A(n)=2*A(n-1) with different initial terms. A102871 and A105580 make similar
confusing claims. A101386, written earlier, makes the same error but also
explains its nonstandard word usage.

Similarly, A105770 makes the bizarre statement "*This sequence is
'tesrokseq' at the link 'Sequences in Context'.*" and also *"Link to
Sequences in Context contains futher details on the 'roktype' used"*. The
writer clearly did not understand what the "Sequence in context:" links are.
A105660 is similar.

A few, like A119560, give only part of a FAMP formula without even referring
to FAMP.


As for what floretions are useful for -- it appears to be an iterative
process similar to that of multiplying quaternions, but with many different
types of "multiplication". Each quantity has 16 components, rather than 4
for normal quaternions. There appears to be a general purpose computation
capability similar in both versatility and obscurity to John Conway's
FRACTRAN (look that up in Google). It is difficult to see what a given
formula will produce, and it is also difficult to write a formula that will
produce a given desired output.

This notion of universal computation seems evident from the fact that so
many different types of sequences can be produced. This is also believable
because Creighton Dement defines a large set of primitive functions
("syllables") and each step of the iteration has 16 (or perhaps 32)
quantities that serve as the "current state" of the calculation.

There is evidence (from comments added to sequences like A001834, A005251,
A005319, A007483, A046717, A049611, A088137 and A094297) that Creighton
Dement generated huge numbers of floretion sequences using his software,
with different combinations of "syllables" and initial quaternion vectors,
checking each result against existing sequences in OEIS to find matches.

I spent a few hours in Sloandora collecting these results. It found a lot of
sequences that were authored by Creighton Dement but have nothing to do with
his floretion work. After a while I started down-rating sequences like
A100545 and A113166 that say they were floretion-generated but provide no
further clues about the floretion definition or algorithm. Here is a list of
the sequences to which I gave a positive rating. In general, the more
interesting ones are the ones near the beginning of this list, which I have
ordered with the longest entries (by total byte size) first:
  A108618 A109620 A119953 A101386 A108930 A104769 A105058 A100216 A097947
A113249 A103145 A102702 A103196 A090390 A100683 A109437 A101463 A096252
A108051 A105660 A104563 A104005 A106157 A119954 A126626 A104161 A108946
A104934 A106603 A105968 A108619 A100887 A113224 A085903 A102301 A113067
A110064 A097924 A098212 A108898 A120743 A105580 A103135 A102871 A110210
A117154 A111569 A104771 A102296 A104770 A109363 A110048 A104004 A111955
A111926 A100888 A107363 A108766 A108791 A116483 A110212 A110613 A100215
A100886 A110311 A102841 A105082 A111573 A038519 A116484 A113066 A110283
A113225 A110282 A110063 A110281 A110280 A111663 A102714 A100213 A110279
A107307 A110062 A105966 A103646 A110213 A100828 A104237 A111645 A108986
A111642 A111352 A110683 A102129 A107663 A111662 A111644 A110211 A105951
A110684 A111640 A111641 A111643 A131039 A105343 A110307 A094297 A110687
A110686 A111639 A099163 A110310 A110685 A105579 A111954 A110689 A111572
A111574 A115032 A111734 A110309 A110688 A114160 A100212 A099868 A109731
A110294 A104681 A110308 A111571 A110679 A105578 A103645 A109794 A109765
A110225 A099867 A108931 A108391 A114161 A111914 A109781 A108057 A108390
A105225 A105576 A023554 A110526 A109782 A131041 A108620 A105077 A131040
A108306 A099256 A105671 A111915 A106664 A108985 A102214 A105577 A106691
A110047 A110614 A059165 A105770 A049332 A107854 A114619 A105963 A110050
A110274 A109430 A105163 A104522 A117153 A110527 A108765 A110052 A110051
A109340 A129905 A102285 A110046 A102239 A109609 A114696 A103177 A110293
A109610 A109165 A111570 A109362 A110528 A111566 A119560 A111567 A104187
A102865 A109785 A106666 A109793 A109438 A114689 A111587 A102785 A109599
A109784 A109803 A103644 A109164 A114697 A101348 A111665 A105964 A109787
A111666 A109786 A110906 A107853 A107850 A107852 A114688 A107851 A114647
A109359 A109360 A109249 A110151 A107849 A112533 A110158 A098111 A116698

P.S. Because it was submitted by Mr. Dement, I also found A124856,
"Candidate sequence for theme song for OEIS", which lacks a defining formula
or floretion but really does suit its description. Perhaps he composed it.
At any rate, you should listen to the Mpeg file!


-- 
 Robert Munafo  --  mrob.com