Various people outside the class have expressed an interest in
following along with the course as it progresses. You are welcome to
do so and to download copies of linked documents. It would be helpful
to me if you'd send comments or otherwise let me know that you are
doing this.
Classmember Jason Benda has written a nifty and useful Stern calculator, which
is available to readers of this webpage:
Stern Calculator.
W 3/1 -- On M and W I distributed the two pieces of Fourth set, supplemental , and a mathematica's chart, not yet scanned. The paper du jour was by C.G. Paradine from the Mathematical Gazette of 1956.
M 2/20 -> M 2/27 -- I've fallen behind, so I will just write one big paragraph here. The handouts include Comments on the First Problem Set, Second Problem Set and Fourth set of notes (this was distributed in two pieces)>. The texts distributed include two of my papers "Continued fractions and an annelidic pde", from 1983 and "Some extremal problems for continued fractions" from 1985, notes on the Stern-Brocot array from the excellent book Concrete mathematics by Graham, Knuth and Patashnik and a short note on the Minkowski ?-Function by Hancock. I have also had Tori Corkery scan: 2/13 handout and 2/15 handout . Let me know if I've forgotten anything.
F 2/17 -- See W 2/15 for the tex'd handout; otherwise we had a selection from E. Lucas' Theorie des Nombres (1891), a historical note on Lucas MacTutor Biography of Lucas. (I couldn't find anything on the genealogy page about him.) Beyond the material covered in the notes, we started to get into continued fractions, and will be working in that direction for the next few classes.
W 2/15 -- The big handout for today and Friday is Third Supplemental Set, Part 2. The first 5 pages were distributed on W and the last 6 on F. The content is variable, and reflects the discussion of the week; namely, the moments of the Stern sequence and the behavior of the Stern sequence mod d. There is also some discussion of roots of unity and various analytic inequalities satisfied by the means. I also distributed some computer data (to be scanned), on multiples of d among the first 15 rows of the diatomic array, for d up to 15. The other handout is a xerox of a 1990 paper called "Some binary partition functions", written by the instructor's favorite author.
M 2/13 -- Four handouts today! A note for Tim Lesaulnier about how I could write the TeX character "chi" without it being too low to easily show subscripts, and a brilliant Stern sequence calculator which is linked above the diary. Also, a three-page Mathematica printout on s(n) mod 3 (to be scanned and linked). Finally, a short note from G. T. Williams and D. H. Browne from a 1947 issue of the American Mathematical Monthly. Readers from UIUC websites could download it, starting from the JSTOR. homepage.
F 2/10 -- I became extremely ambitious and wrote Third Supplemental Set. The handouts today included a barely legible, and not-very-good series of 1979 articles from the Fibonacci Quarterly on the original Stern paper, and the Math Reviews review (by Lehmer!) pointing out the weakness.
W 2/8 -- New stuff on the Stern sequence mod 3, not yet written up. Today's handout was the original Stern paper from 1858, which I'll have scanned at some point. I can now provide a link to some Mathematica material on Graphs of u(6n+j). On to more matrix-related recurrences.
M 2/6 -- Most of the time was spent reviewing the homework. The solutions were distributed: First homework solutions. There are at least two corrections known, one serious. The induction in problem 7 requires that the theorem be restated for "sufficiently large r". I advertised my Number Theory seminar talk on Th 2/10, 1pm, 241 AH with a discussion of a finite product that will show up on the revisions to the third set of notes. Finally, I distributed a copy of D. H. Lehmer's 1929 paper on the Stern sequence (unlinked) and historical information about him, taken from MacTutor Biography of D. H. Lehmer and Mathematical Genealogy of D. H. Lehmer.
F 2/3 -- Two comments from after class. In #7, to be precise, I should have said that the polynomial has degree at most d. And the exact degree turns out to be the number of non-zero coefficients minus 1. Also, the highest order coefficient has to be positive, because we haven't defined s(n) for negative n. In #8, if you're stuck on how to prove the pattern, you might want to look mod 6. Otherwise, I went through much of the third set of notes, with some additional material that will show up on Monday, together with the hw solutions and a start of handouts of papers from the literature.
W 2/1 -- Two handouts for the day: Second set of supplemental corrections, and Third set of notes. These notes cover linear recurrences and some applications to Stern sequences. Also, Tori Corkery has turned the three Mathematica handouts into .pdf files. Specifically, here are Table of s(n) up to n=412 from the first day, the handout Basic pictures of the Stern sequence from the second day and the Pictures of s(n)/s(n+1) that filled out the last page of an earlier handout.
M 1/30 -- It's always a dangerous thing when the professor starts to present material that he's just been working on. The probability of errors is quite high. For this reason, don't read Second set of supplemental notes and corrections without looking at the entry for W 2/1. The main topic of the day was the behavior of the generating function on rays to the circle of convergence.
F 1/27 -- A more amusing application of generating functions, and the closed form for the Stern sequence. Some new material at the end relating to the coefficients of the reciprocal of the generating function. These will be in notes early next week.
W 1/25 -- A day for generating functions and not particularly the Stern sequence. The new handout is Second set of notes. Keep those errors coming in!
M 1/23 -- Since there are graduate exams on Sat. 2/4/06, the due date for the first problem set has been moved to M 2/6. Handout is 4 pages of First set of supplemental notes and corrections, with a few tables filling out the bottom part of the 4th page, and not linked in this handout.
F 1/20 -- Second day. Handout of 6 pages of Mathematica printouts related to the Stern sequence. I'm not sure how to link the original file. Additional material not in the notes included a computation of s(2006). A couple of minor errors fixed in the webpage.
W 1/18 -- First day of class. Handouts included a table of the Stern sequence up to 412 (unlinked), the Class Organization, First set of notes and First problem set. A continuously updated set of corrections to the handouts can be found here: Corrections to the notes.