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

[Creighton Kenneth Dement]

> 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.

Unfortunately, I cannot address every statement right now. Let me try to
go over some of the more important points brought up.

vesseq(X), jes, les, tes, … vespos, vesneg, along with the identities
ves = jes + les + tes, etc. are all clearly (and easily, in my opinion)
defined here http://www.scribd.com/doc/14790151/Floretions-2009
(you can even see these visually by clicking on "ves", "tes", etc. in the
list selection box on the top left at
http://www.fumba.eu/sitelayout/floretion.html )

Mr. Munafo and others do have a valid point, which is that parts of the
documentation are confusing and/or incomplete. Before I continue, I would
like to thank him and others diving into the subject!

> "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.

Most of the confusing documentation comes from the very beginning
(2004/2005), when I was mezmerized by "all the different paths a floretion
could be taken down" and would often wake up each morning with a new idea.

In some cases I was trying to show
when you do “A”, you get a 4th order sequence
when you do “B”, you get a 2nd order sequence
when you do “C” this you get something very strange

and simply choosing one or more representative sequences from each class
(because I certainly didn't have enough time to submit them all).

My “old, dead website” has been discussed on the seqfan list (a few years
ago). What happened is that I had a private website with several pages
dedicated to explaining the terminology and giving examples, but where I
had to pay what became an overpriced monthly fee to maintain it. After a
year or two, I ran into financial trouble and could no longer pay the
monthly rates. Apparently, the site was then shut down very quickly and
the domain name sold to another "party". Years later, some data is still
retrievable but it seems several pages are missing and/or are
non-recoverable.

> 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.

When my site was taken offline, a member of the seqfan list wrote to Neil
and me that my site and been taken down and and that (yikes!) malicious
code was now being executed from it. Needless to say, I was quite upset.
At that point, Neil asked me what he should do with all the (more than
200, I believe) references to my homepage. I agreed that he should remove
all links as soon as possible and he did. This explains why some sequences
say “See sequences in context” or “See FAMP” in the comments but with no
proper link. Formerly, there would have been a link to my website with a
batch of related sequences (i.e. all generated by the same floretion under
the same conditions). There was also a link to download FAMP (the
Floretion Algebra Multiplication Program). I have since tried to recreate
an “online version of FAMP” here
http://www.fumba.eu/sitelayout/floretion.html (however, a lot of the
functionality is still missing for lack of time and in some cases security
reasons).

One of the most confusing statements is surely what is meant by a
"transform of the zero-sequence". First, I will agree that if the
consensus is this is totally nonsense, then by all means let's call it
something else!

Here's the idea: start with any floretion X and let (c(n)) be any sequence
of integers we would like to transform.

Define
Y = X + (c(0) – ves(X))ee
and Y(n+1) = Y^n + (c(n) – ves(Y^n))ee
where “ee” is the unit vector.

Then
ves(Y(n+1)) =  ves(Y^n + (c(n) – ves(Y^n))ee)
=  ves(Y^n) + ves((c(n) – ves(Y^n))ee)
= ves(Y^n) + ves(c(n)ee) – ves(Y^n)
= ves(c(n)ee)
= c(n)

The identity ves = jes + les + tes always holds, thus:
c(n-1) = ves(Y^n) = jes(Y^n) + les(Y^n) + tes(Y^n)

My terminology (which is certainly open to improvements) would say here
that the sequence of numbers jesfor(X) = (jes(Y), jes(Y^2), …,) is the
jesfor-transform of the sequence (c(n)) with respect to the floretion X.

One of the easiest choices for the sequence to be transformed is, of
course, the zero-sequence itself. This leads to the strange sounding (but
consistent, imho) formulation "... jesfor-transform of the zero-sequence".

Force transforms are interesting (there are at least two types- I can't
remember which one is which off hand) and I basically abandoned the entire
subject for lack of time. Substituting jesfor(X) (or one of the other
generated sequences, below it is 4dia[I]seq) for the new sequence to be
transformed led to this past seqfan post:

*************

Generalized Sequence Convergence?

Let J be some subset of the set of all integer sequences, T[ ]: J -> J be
some mapping "transform", and c an integer sequence.

Define T^n as T applied n times and T[c](m) as the m-th term of the
sequence T[c]. Assume that for each integer sequence c in J exists integer
sequence d so that for each m in naturals exists p in naturals so that for
all n > p : T^n[c](m) = d(m)

In that case, we can define T*[ ]: J -> integer sequences, T*[c] = d for
some new transform T*. Ex. T[c](m) = floor(c(m)/2), then T*[c] =
(0,0,0,0,...) for all c. Below, I give a T* such that, apparently (and
without proof), for all c: T*[c] = A000045 (Fib). However, it is the
procedure itself that is stressed here and not any particular result.

[snip]

O-th iteration *********

1vesseq: -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1,

4tesseq: -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1,

4lesseq: -5, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3,

1jesseq: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

4dia[I]seq: 1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1,


1st iteration ********* (LoopType: dia[I] )

1vesforseq: 1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1,

4tesforseq: 7, 7, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1,

4lesforseq: -5, -7, -7, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3,

1jesforseq: 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

4dia[I]forseq: 1, 3, 3, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1,


2nd iteration ********* (LoopType: dia[I] )

1vesforseq: 1, 3, 3, 1, -1, 1, -1, 1, -1, 1, -1,

4tesforseq: 7, 15, 21, 13, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1,
1, -1, 1, -1

4lesforseq: -5, -7, -17, -21, -11, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3,
-3, 3, -3, 3, -3

1jesforseq: 0, 1, 2, 3, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

4dia[I]forseq: 1, 3, 5, 9, 7, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1,
-1, 1, -1

... [snip] ...

8th iteration ********* (LoopType: dia[I] )

1vesforseq: 1, 3, 5, 11, 21, 43, 85, 171, 339, 649, 1103, 1505, 1471, 897,
255, 1, -1, 1, -1, 1, -1

4tesforseq: 7, 15, 29, 59, 117, 235, 469, 939, 1869, 3613, 6359, 9329,
10399, 8001, 3711, 769, -1, 1, -1, 1, -1

4lesforseq: -5, -7, -17, -31, -65, -127, -257, -511, -1025, -2037, -3923,
-6813, -9731,

1jesforseq: 0, 1, 2, 4, 8, 16, 32, 64, 128, 255, 494, 876, 1304, 1488,
1184, 576, 128, 0, 0, 0, 0

4dia[I]forseq: 1, 3, 5, 11, 21, 43, 85, 171, 341, 681, 1327, 2401, 3711,
4481, 3839, 2049, 511, 1, -1, 1, -1

Jacobsthal Sequence!

***************


On to the next question:

> 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.

Let's see how to generate A100545:
Proposition 3.3 (Pure Quaternions and Second Order Recurrence Relations)
states:

Let Y = A'i + B'j + C'k and E = .25('i + i' + ii + jj + kk + jk + kj + ee)

Then all sequences generated by the floretion E*Y satisfy the linear
recurrence relation:

a(n) = -A*a(n-1) – B*C*a(n-2)

and, in general, Proposition 3.4 states

Let Y =  A'i  + B'j + C'k + Di' + Ej' + Fk' + G'ii' + H'jj' + I'kk' +
J'ij' + K'ik' + L'ji' + M'jk' + N'ki' + O'kj'

then all sequences satisfy

a(n) = (-A-D+G+H+I+M+O)*a(n-1) + ((A+D-M-O)(G+H+I)
+(N+E-B-K)(J+C-F-L))*a(n-2)

Now let's really try and find the sequence. Go to
http://www.fumba.eu/sitelayout/floretion.html and, on the list at the
right, choose "Basic Fibonacci". You should see that the floretion "E"
defined in the above in Prop. 3.3 is automatically set in the top row. The
second row contains the numbers (-1,-1,1) which corresponds to the
quaternion Y = -'i -'j + 'k

Will prop 3.3 suffice to find the sequence A100545? Well, we see it
satisfies the recurrence a(n)=3*a(n-1)-a(n-2) so let's try the quaternion
Y = -3'i + 'j + 'k  by changing the numbers in the second row to (-3,1,1)
and hitting the button "Go Python!" then scrolling down the page to see a
list of sequences generated by E*Y.

In this case, this returns a batch of sequences which all satisfy the
recurrence relation a(n) = 3*a(n-1) – a(n-2), but not the sequence A100545
example we are looking for. That said, we also chose the simplest possible
example. The next simplest case (using Prop. 3.4) would be to choose Y =
-2'i + 'j + 'k - i'

Scan down the batch of sequences and you should see these lines:
2mixseq: [1, 3, 8, 21, 55, 144, 377, 987, 2584, 6765]
4mixposseq: [3, 8, 23, 61, 160, 419, 1097, 2872, 7519, 19685]
4mixnegseq: [-1, -2, -7, -19, -50, -131, -343, -898, -2351, -6155]

And there's our sequence... not only that, the (dynamic) identity
mix = mixpos + mixneg returns A055273 + A100545 = 2*A001906 (bisection of
the Fibonacci sequence)

With all that said, there may be a few sequences which have to be deleted
because the documentation on them has been completely lost. As stated
above, often I was choosing a representative sequence from a sea of
possibilities. A good example of this is the sequence Neil and I
considered for a themesong: A124856. I will have to check with past posts
to see where the documentation is, and if it can't be found, it should be
deleted, imo.

Sincerely,
Creighton