9/24/2002 - REACH Meeting (2hrs).

9/25/2002 - Did the hw problem 2Xn tilings. 2hrs.

9/26/2002 - REACH meeting. Later worked out the 1 X n case of tilings.

9/28/2002 - Read a good portion of generatingfunctionology. Later, worked on
the 3Xn tiling case. (6.5 hours)

This Week's Hours: 10.5 | Running total: 10.5

10/1/2002 - Tried playing more with the 3xn recursion to make the five equations
out. This was not pretty. Proceeded to attend the REACH meeting. (3 hrs.)

10/3/2002 - Read Propp's Article.REACH meeting. (3hrs).

This Week's Hours: 6 | Running total: 16.5

10/8/2002 - Worked on finding a combinatorial interpretation of the negative
Fibonnacis.Rather unsuccessful. (1hr). Attended the REACH meeting. (2hrs).

10/10/2002 - Read the Propp article on Aztec diamond tiling.Took notes at REACH
meeting. Typed them up. (1 + 2 + 1 = 4 hrs.)

This Week's Hours: 7 | Running total: 23.5

10/15/2002 - Before REACH meeting looking over the Markov number paper written by that Musiker(?).(1.5 hrs) REACH meeting. (2hrs).

10/17/2002 - 2 hrs. at REACH meeting

10/18/2002 Read Rui's Proof, made some notes on stuff I thought needed to be
corrected (2 hrs).

10/19/2002 Got Rui's response, worked on the problem some more (1 hr.).

This Week's Hours: 8.5 | Running total: 32

10/22/02 Looked into some generalizations of the Markov Sequence. Found that for that case that S(n)*S(n-3) = 2S(n-1)^2 + S(n-2)^2 seems to yield in some cases at least integral values. In particular one branch counts 3 by 2(n-1) tilings. Attended REACH meeting. (3hrs total).

10/24/02 Attended REACH meeting. Started exploring the Markov like recurrence in which there are four terms. Trying to find a combinatorial model. (2 hrs)

This Week's Hours: 5 | Total Hours: 37

10/29/02 Attended REACH meeting. Worked more on the four term Markov-like recurrence. (2 hrs).

10/31/02 Found a graph that has 131 matchings, what I was looking for during the last meeting to see how to extend the 4-term Markov-like recurrence to another form of matchings. Unfortunately, it is not obvious to me how to create the next term in the series having 17291 = (131^2 + 11^2 + 3^2) matchings. Attended the REACH meeting. Tried to see how it worked with a 5-term Markov-like recurrence, but I am not having much luck with that one either. (3.5 hours)

This Week's Hours: 5.5 | Total Hours: 42.5

11/4/02 Tried simplifying the Markov-like recurrence in multiple variables by just iterating two numbers, and I appear to be able to get graphs that have the same number of matchings by creating a staircase of "units." However, adjoining more elements does not seem to work out.(3.5 hrs)

11/5/02 Attended REACH meeting, tried to get the 4-variable Markov generalization to work. (2 hrs)

11/7/02 Attended REACH meeting. More work in trying to generalize the Markov numbers. (2 hrs)

This Week's Hours: 7.5 | Total Hours: 50

11/12/02 - Looked at the linear recurrence version of the Markov numbers proof. Attended REACH meeting. (4 hrs)

11/14/02 - Attended REACH meeting. Looked at making cuts in order to prove a linear recurrence, which would be another proof of the combinatorial intepretation of the Markov numbers.(2 hrs) 11/16/02 - Worked out a large example of the Markov recurrence with 169 matchings to see how to prove the linear recurrence with cuts. The key is that if we have M(ABB) + M(A)(monomial) = (x^2+y^2+z^2)M(AB)M(B), the M(A)(monomial) term represents a matching of A with the only matching of BB that does not have a common cut between the B-parts. Put up on the web page a general idea of why the linear recurrence with cuts should work out nicely. Pretty sure I'll figure out a proof soon. (5 hrs).

This Week's Hours: 11 | Total Hours: 61

11/19/02 - Attended REACH meeting. Worked on coming up with a clean way of proving the linear recurrence with cuts. (2 hrs)

11/21/02 - Attended REACH meeting. Saw a proof of the Markov recurrence based on condensation. Managed to figure out how to prove the linear recurrence with cuts as well. (4 hrs)

This Week's Hours: 6 | Total Hours: 67

11/25/02 - Put up a fairly thorough proof of the Markov relation based on the linear recurrence and cuts. It will need to be redone if it is to be included in a paper.(2 hours)

11/26/02 - Attended REACH meeting. Put up on the web more details of the proof of the linear recurrence with cuts. (2 hrs)

This Week's Hours: 4 | Total Hours: 71

12/3/02 - REACH Meeting (2hrs). Worked on writing up the paper.

12/5/02 - REACH Meeting(2hrs). Worked on writing up the paper.

12/7/02 - Wrote up a draft of the background material dealing with the paper. (3 hrs).

12/8/02 - More paper writing work (2hrs).

This Week's Hours: 9 | Total Hours: 80

12/10/02 - REACH Meeting and paper writing.

12/12/02 - REACH Meeting and paper writing.

This Week's Hours: 4 | Total Hours: 84

12/17/02 - REACH Meeting and paper writing.

This Week's Hours: 2 | Total Hours: 86

1/2/03 - Paper writing.

This Week's Hours: 3 | Total Hours: 89