Preprint version of chapter 7 .
Papers relevant to the subject can be found at Stern Sequence Papers. Email me if you forget the password. The batch from the third week is now up. No new papers for the fourth week. Homework grading!
F 3/15 -- JV talks more about continued fractions; a few other points on the ?-Function and its derivative. Spring Break comes at a good time.
W 3/13 -- The weird antediluvian proof of Lagrange's Theorem that every quadratic irrational has an eventually periodic continued fraction; more on the ?-Function and ways to avoid computation.
M 3/11 -- Infinite continued fractions and the Brocot array and the Minkowski ?-Function.
F 3/9 -- HW 2 due. Most of class period used to go over the problems.
W 3/7 -- Zombie instructor returns with (finite and infinite) continued fractions and their application to Sternstuff.
M 3/5 -- Class cancelled, instructor sick.
F 3/2 -- JV gave an hour-long presentation on continued fractions, suggested by his dissertation work. Homework 2 duedate delayed by a week.
W 2/29 -- More on continued fractions, instructor's flu got the better of him halfway through class.
M 2/27 -- The beginning of finite continued fractions.
F 2/24 -- Applications of linear recurrences to sums of powers of polynomials; that is, section 3.8 in the notes. Nothing to do with the Stern sequence, but there are number-theoretic connections.
W 2/22 -- The asymptotics of the Stern sequence, in various ways. Handout above. More on the introduction to simple finite continued fractions
M 2/20 -- Class had a guest, RS. Discussion of the new papers in the vault. More on the behavior of the summatory functions and their variations. Handout (see top). and a start on simple finite continued fractions.
F 2/17 -- Homework 1 returned, with comments. The final version of chapter two was distributed (see top), along with another set of pictures (see top)
W 2/15 -- The class was devoted to a proof that if f(N) denotes the number of n < N so that s(n) is a multiple of three, then |f(N) - N/4| is bounded by a constant times N^{1/2}
M 2/13 -- More on recurrences. I think I followed the notes I haven't put up yet.
F 2/10 -- Oliver gave some interesting data about the behavior of the iterated Stern sequence: see his beamer slides. I started on constant coefficient recurrences. I promised that a draft of chapter two of the notes, suitable for your corrections, would be up by Saturday afternoon, and it was!
W 2/8 -- More on the topic of 2/6 and also a bit on the inverse of T(x), which also had some less-pretty pictures, but still not bad.
M 2/6 -- A seminar-style day, with material presented and proofs, handwaving, and pretty pictures. The pretty pictures are at asymptotic evidence. Oh that Stern generating function, what peculiar behavior you have near the unit circle!
F 2/3 -- Homework collected and solutions distributed (link above); there also will be supplemental comments distributed when the homework is returned.) We pretty much finished the generating function material in the parts of chapter 2 that have already been posted above. Yikes, I can talk faster than I can write.
W 2/1 -- Typos in the first exercise set: Problem 9 has a missing star in the last sum. Problem 8 probably needs you to consider r=1 separately. Your answers can be as detailed as you like. I'd like to see you do some problems; you don't have to do all. If you like a problem and want to generalize it in some way, that will make me happy. I've now gotten through close to the end of the chaprter two notes: we now know the generating function for s(n).
M 1/30 -- Chapter one notes distributed (see above), with a few comments about additions which hadn't been discusssed in class. New papers in the Vault were described. Chapter 2 has begun, with about the first four pages covered. First homework is actually due on Friday, though a few people thought it was today.
F 1/27 -- Completion of the discussion of chapter 1, a version of which will be distributed in class on Monday. Topics not in the notes included the construction of Williams and Browne (the original of which is in the vault) and De Rham's construction, which will be put in next week.
W 1/25 -- Serious apologies. I thought I'd linked to the new draft of the notes, but I copied the old link. The new draft is now up. Corrections by Friday would be welcomed for the hand-out version. Today's class featured a remix of sections 1.5 and 1.8, followed by 1.6. There is some material from later chapters which will appear in the final version of chapter 1.
M 1/23 -- There are now seven articles on the site. Only handout: Greg's better version of the Stern graph. We have now covered through roughly p.11 of the first chapter of the notes. I will have a final version of chapter one notes distributed on M 1/30 (to all) and a preliminary version of at least parts of chapter two up on the web by then.
F 1/20 -- Second day of class. The paper site is up and password-protected, and the first two papers there are Stern's 1858 original and Lehmer's 1928 survey, along with some historical information. Let me know if you want/need a hard copy. I have now covered through the middle of p.8 of Chapter One. I separately distributed the exercises for Chapter One, which are pp.17-18 above. They are tentatively due F 2/3. There was one other handout. Second pictures of the Stern sequence . This doesn't include the picture I drew partially by hand of a Stern graph. A much better version will be distributed in class on Monday, courtest of Greg Puleo.
W 1/18 -- First day of class. Three handouts: Course Organization, plus the Course Blurb, together on one sheet, and First pictures of the Stern sequence . I realized too late that I hadn't distributed the paper by Lehmer, which will be on the history section of the site. I covered roughly the first four pages of the classnotes. Will try to be smoother on Friday.