SSL Minutes for Tuesday, Nov. 11, 2003
Notetaker for today: Martin
Notetaker for Thursday: Stephen
Snackbringer for today: Sam
Snackbringer for Thursday: Martin
There was no specific agenda for this meeting. Instead, Jim met with
members of SSL from 2:30 to about 3:35 during which members brainstormed,
then snacks were had (consisting of Baked Doritos, Zebra Cakes, and dnL
carbonated beverage product), then additional brainstorming occurred while
Jim ran to catch his plane to Seattle.
Martin conceived of the following math joke: A student is at a blackboard
drawing various figures and then runs and gets his professor.
"Professor!" he exclaims. "I have noticed that for every quadrilateral I
draw, the angles add up to 360 degrees!" The professor replies, "Yes,
that's a 4-gon conclusion."
Paul and Martin looked at octagon-square snakes. Paul looked at snakes of
the following form:
___ ___
/ \ / \
o--o \ o--o \ .
/ \ \ / \ \ .
o o--o o--o...
| | | | |
o o--o o--o...
\ / / \ / / .
o--o / o--o / .
\___/ \___/
which he thought looked like bugs. The sequences did not show up in
Sloane, aside from the number of matchings of the snake up to each
progressive square, which turned out to be Sloane A084326,
a(n)=(1/2)*sum(k=0,n,binomial(n,k)*F(3*k)) where F(k) denotes
the k-th Fibonacci number.
Paul was somewhat interested in proving a bijection.
Martin, in his first transcription of the notes, accidentally connected
the "extra edges" to the closer of the two vertices in each horizontal
octagon segment instead of the farther one. This gave him a different
sequence for the number of matchings up to each square, which actually
appears as several highly related sequences on Sloane, but also an
additoinal intriguing one, A061509. Its description is: "Write n in
decimal, omit 0's, replace each digit k by k-th prime, raise to k-th power
and multiply." E.g. decimal 123 appears as 2^1 + 3^2 + 5^3. Every term
of the snake sequence seems to appear in this sequence, though I have no
proof of this; this sequence certainly contains additional terms. It's
probably because every term of the snake sequence is "highly composite"
which actually shouldn't be too hard to prove.
Martin looked at things like the following:
o--o o--o o--o
/ \ / \ / \
o o o o
| | | |
o o o o
\ / \ / \ /
o--o o--o o--o
/ \ / \ / \
o o o o
| | | |
o o o o
\ / \ / \ /
o--o o--o o--o
/ \ / \ / \
o o o o
| | | |
o o o o
\ / \ / \ /
o--o o--o o--o
He also looked at filled staircases which included and did not include the
center diamonds, and looked at both the matchings and number of spanning
trees of these graphs. He did not find anything in Sloane. He also
noticed that the sequences found in the Musiker-Propp paper are not in
Sloane.
In class, Abby and Paul, for fun, looked at snakes like the following:
o----o
/| |\
o | | o
|\| |/|
| o----o |
|/| |\|
o | | o
\| |/
o----o
.
.
.
whose matchings (2, 10, 30, 107, 344...) do not appear in Sloane.
Sam was interested in Laurent things and what they count. Specifically,
he wanted to know what work the REACH group had done with Somos sequences.
Emilie was interested in "trilinear coordinates." Stephen showed how one
goes about finding rational solutions. We examine the unit circle, and
lines which originate at (-1,0) and pass through the upper right quadrant
of the circle. The line passes through the y axis at (0,a) and touches
the circle at b. y = ax + a is the line's equation. If we make a
rational, then b will be rational.