[seqfan] Re: Quantum Rotation Correlation Sequences

[Olivier Gerard]

Dear Brad,

Unfortunately, the policy at seqfan is not to forward attachments.
This is both a security measure and a courtesy to the hundreds
of members of which only a few might be interested by a given subject.

People interested can ask you (or me as I saved these files before
the server stripped them automatically) to send them directly off-list.

Nevertheless you say


> 3. Other series that do not occur in the OEIS appear to have periodic first
> differences, which is not all that surprising. As the notebook shows, the
> periods seem to be:
>
> 12 for octahedral A2
> 10 for icosahedral T1 & T2
> 15 for icosahedral G
>
> Does this mean there is automatically a generating polynomial? Who knows...
>
>
Periodic first differences implies that there is a rational generating
polynomial
function.

In the case of the new sequence   Octahedral A2 starting

0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1,
1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2,
2, 2, 2, 3, 2, 2, 3, 3, 2, 3, 3, 3,
3, 3, 3, 4, 3, 3, 4, 4, 3, 4, 4, 4,
4, 4, 4, 5, 4, 4, 5, 5, 4, 5, 5, 5, 5, 5, 5, 6, ...
the rational generating function is

x^3/(1 - x^3 - x^4 + x^7)

As you are a Mathematica user, you can check that by generating the first
500 terms

CoefficientList[Series[x^3/(1 - x^3 - x^4 + x^7), {x, 0, 500 - 1}], x]
There are several ways to derive such a function.

In Mathematica version 7 and above you can try FindGeneratingFunction[ seq,
var]

but depending on your version, this will not always give you the simplest
form
and you will have to find the good compromise between number of terms
and computing time.  You can also do it by hand or semi-automated.


> If there is any interest to get rotational correlations into OEIS, this
> code could be extended to other finite symmetry groups with relative ease.
>

Please submit the 3 sequences currently missing in the OEIS
with their respective G.f.

Take model on the already existing sequences and other entries
for  Molien series and add relevant cross-refs from your sequences
to the older ones.
You can also add a link to a webpage of yours where you publish
your research files.

Use a little modification of your code to generate a b-file for each.

With my best regards,

Olivier Gérard