SSL Meeting
02/26/04

Notetaker today: Emilie
Snacks today: Hal
Notetaker next time: We volunteered Brendan; however Carl volunteered 
    if Brendan can't make the 24 hour turnaround
Snacks next time: Paul

When the meeting began everybody pretty much just started working on what 
they had been doing on Tuesday.

Sam announced this new formula for the Pn(x) and Qn(x) when a=b=-c=1

Pn(x) = sum(binomial(2^n, k)*((the (2^n-k)th Fibonacci number)-1)*x^(2^n-k)
Qn(x) = sum(binomial(2^n, k)*(the (2^n-k-1)th Fibonacci number)*x^(2^n-k)

When we moved into the computer lab they verified that these equations agreed
with what they previously had for Pn(x) and Qn(x) using Maple.

They decided at the end of the meeting that the decisive thing they needed to
prove was:
sum(binomial(2^n,k1)*binomial(2^n,k2)*(F(k1)*F(N-k1)-2*F(N-k1)))=
binomial(2^(n-1),N)*F(N) 
where F(m) =  the mth fibonacci number.

They also simplified the Pn(x) and Qn(x) a little more for a=b=-c=1
Pn(x) = sum(binomial(2^n,2^n-k')*F(k'-1)*x^k'
Qn(x) = sum(binomial(2^n,2^n-k')*(F(k'-1)-1)*x^k'

Paul and Emilie discussed the recurrences they were trying to prove 
integrality on.  Neither of them had found a linear recurrence that 
would satisfy the quadratic recurrence of 
a(n)*a(n-7)=a(n-1)*a(n-6)+a(n-3)+a(n-4)

Paul had been looking for other similar palindromic recurrences of the form:
a(n)*a(n-k)=a(n-l)*a(n-k+l)+a(n-m)+a(n-k+m)
It turns out that for each (l,k-l) pair there is a unique (m,k-m) pair.

The particular sequence of the pair of pairs (4,k-4)->(2,k-2) is interesting.
Paul's conjecture is that for k odd and k->infinity all the sequences converge 
to a single sequence. 

When Jim joined the meeting around snacktime we talked about a couple 
administrative issues.

1. Should we all get keys to B107?
People's thoughts were: 
-not completely necessary as most people have access to maple whenever they 
 want through the undergrad.math.wisc.edu
-only if we want to work together
-Jim, John, and Steven will be available if we want them to come down and 
 unlock the door for us. email them when you want to go in and then knock 
 on their doors.

2. Can someone volunteer to write up the cube snake stuff??
-the square or twice a square conjecture for integers and generating function.
-Jim wants this to be written up even though it's not completely solved so that
 we can gime it to the "mathematical community" as an open problem
-Paul volunteers to do the write up

3. On Tuesday we will discuss what is going on with all our projects, why they
are important, and how they are going. This is "sort of" a practice for 
Wednesday's VIGRE meeting. 
Also, about the Vigre meeting, I (Emilie) need someone to take notes at the 
Monday pre-meeting since I can't go to that. Sam volunteers to do this.

All the logistics are out of the way, we continue to work on what I described 
above in the "mathy" section of the notes.

See you Tuesday!