[seqfan] Online Floretion Multiplier / Re: 2-d partitions, rotationally symmetric pieces

[Creighton Kenneth Dement]

Dear Seqfans,

Actually, I wanted to wait at least one more week to announce the "Online
Floretion Multiplier" as the databank connection, various buttons along
with several other basic features have yet to be completed. Also, it has
currently only been tested on Mozilla/FireFox.

However, the statement from today:

t(n) = [ 0 1 1 1 1 3 3 13 13 59 59 269 269 1227 1227 5597 5597 25531 ... ]
     = A052984((n - 3) / 2)
.. which I'm delighted to discover are the "Kekule numbers for certain
benzenoids", satisfy the recurrence:
  a(n) = 5a(n-1)-2a(n-2), a(0) = 1, a(1) = 3
.. and have several interesting comments for formulae.

gives me perhaps a nice opportunity to talk about what it will be able to do.

To begin, any interesting second order recurrence can be looked up /
cross-checked with the help of floretions. Note the emphasis on
"interesting"
in the last sentence: we are not completely free in setting the initial
values, but can hope that if the sequence is "important",
there will be a floretion which generates it!  Each result found will
return a list of related sequences.
For higher order sequences and sequences in other categories, there will
be a databank.

Here are the current 5 floretion integer sequence category types:

Sequence Type: 4th Order Linear Recurrence or less
Floretion Algorithm Method: Static (this is the easiest to reproduce, yet
determining general formula can still be a challenge- for an example, see
theorem, below)
Are there established connections to Triangular/Fib/Pell numbers? : yes

Sequence Type: (mostly) 16th order or less
Floretion Algorithm Method: Dynamic
Are there established connections to Triangular/Fib/Pell numbers? : yes

Sequence Type: Necklace/Prime Numbers
Floretion Algorithm Method: Necklace
Are there established connections to Triangular/Fib/Pell numbers? : yes

Sequence Type: Geometric/Elliptical/Musical
Floretion Algorithm Method: Add in fractional parts at defined stages.
(Ex. A108618)
Are there established connections to Triangular/Fib/Pell numbers: no (at
least not yet)

Sequence Type: Transformations
Floretion Algorithm Method: Subtract "ves" part, add sequence to be
transformed in "tes" part. (Ex. A108300)
Are there established connections to Triangular/Fib/Pell numbers: yes

Now returning to the example, let us first assume 1 3 13 59 269 is an
interesting sequence. :)

*********Begin Theorem:

If E = 0.25('i + i' + 'ii' + 'jj' + 'kk' + 'jk' + 'kj' + 'ee')   [Note:
'ee' is unit]

and

X = 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'

(coefficient of ee is 0, i.e. tes(X) = 0)

then all floretion-generated static sequences (i.e. ves, jes, les, ...)
have the form

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)

********* End Theorem

Pick any numbers which lead to the recurrence a(n) = 5*a(n-1) - 2*a(n-2):


-A-D+G+H+I+M+O = 5
(A+D-M-O)(G+H+I) + (N+E-B-K)(J+C-F-L) = -2


//He we set the -5a(n-1) condition of the recurrence
A = -1
D = -4
G+H+I+M+O = 0

//Setting the -2a(n-2) condition of the recurrence
G = H = I = 0
M+O = 0
B = K = 1
J = 1
C = F = L = 0

We can now use a prototype of the Online Floretion Multiplier to see this:
1. Go to http://www.fumba.eu/sitelayout/Floretion.html

2. Click on the button "Unlock Row A" to make the row editable. Note you
can skip this step along with steps 3 and 4 by selecting "Load Fibonacci"
from the select box- the values will be inserted automatically.

3. We will enter E = 0.25('i + i' + 'ii' + 'jj' + 'kk' + 'jk' + 'kj' + ee)
for Row A. To do this, enter "0.25" in the columns 'i (i with an arrow to
the left), i', 'ii', etc.

4. Enter X = 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'
with appropriate values for A, B, ..., N, O, (see above) for Row B.

Note: The Python script which calculates the result
can also handle variables, but this is currently deactivated to prevent
the program from taking up too much server time.

5. Hit the button "Multiply" and check the result in the scrollable text
area, below.

Scrolling down the text area, we find
1diaItesseq: [0, -1, -5, -23, -105, -479, -2185, -9967, -45465]	 ->
A107839 (Kekule numbers for certain benzenoids.)
2diaJtesseq: [-1, -1, -3, -13, -59, -269, -1227, -5597, -25531]	 ->
A052984 (Kekule numbers for certain benzenoids.)
2diaKtesseq: [5, 21, 95, 433, 1975, 9009, 41095, 187457, 855095] ->
unlisted sequence

It is now easy to see how one might conclude, based on symmetry of the
situation alone, that the unlisted sequence [5, 21, 95, 433, 1975...]
has something to do with "Kekule numbers for certain benzenoids." (Note we
also have several identities relating these sequence such
as diaI + diaJ + diaK = jes + fam, etc.). I will submit this sequence.

Sincerely,
Creighton