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.