On this page, you will find links to various Stern-related papers.
: M. A. Stern, Ueber eine zahlentheoretische Funktion,
J. Reine Angew. Math., 55 (1858), 193--220. (German).
D. H. Lehmer, On Stern's diatomic series,
Amer. Math. Monthly, 36 (1929), 59--67.
Dilcher and Stolarsky, 1
: K. Dilcher and K. B. Stolarsky, A polynomial analogue to the
Stern sequence. Int. J. Number Theory 3 (2007), no. 1, 85–103.
- Reznick 1:
B. Reznick, Regularity properties of
the Stern enumeration of the rationals. J. Integer Seq. 11 (2008),
no. 4, Article 08.4.1, 17 pp., arXiv -- math.NT/0610601.
- Anders, Dennison, Lansing
K. Anders, M. Dennison, J. W. Lansing and B. Reznick, Congruence
properties of binary partition functions, to appear in Ann. Combin.
: B. Hayes, On the teeth of wheels, The American Scientist,
July-August 2000, Volume 88, Number 4, p. 296.
: D. Rowe, "Jewish Mathematics" at Gottingen in the Era of Felix Klein,
Isis, Vol. 77, No. 3 (Sep., 1986), pp. 422-449.
Classic papers (But see week 6)
- Our scanner decided to make separate .pdf's for each page of this
article by G. Halphen from Bull. Soc. Math Paris, vol. V, 1877: Page one, Page two, Page three, Page four, Page five, Page six: (French).
- Commentary by Eduard Lucas
on Halphen's paper, in the same journal, same year, same language.
: H. Minkowski, Zur Geometrie der Zahlen,
talk at 1904 International Congress of Mathematicians (German).
- According to
this 2007 blog post,
Minkowski may have given the first talk ever using slides. The
article reproduces several of the slides.
: H. Hancock, Development of the Minkowski Geometry of Numbers,
Macmillan, 1939. In the 1930s, Hancock wrote two volumes summarizing
and describing Minkowski's papers on the subject; this is what he
said about the ? function.
Dilcher and Stolarsky
: K. Dilcher and K. B. Stolarsky,. Stern polynomials and
double-limit continued fractions. Acta Arith. 140 (2009),
119–134. (Linked are galley-proofs).
Stern, but not by name
G. T. Williams and D. H. Browne, A family of integers and a theorem
on circles, Amer. Math. Monthly, 54 (1947), 534--536.
- In two letters, one and
two, the great
computer scientis E. W. Dijkstra rediscovers the Stern sequence and
several properties discussed in the first chapter. He only mentions
other appearances at the end. This is taken from: E. W. Dijkstra,
Selected Writings on Computing: A Personal Perspective,
New papers from the arXiv
and Mendes France: J.-P. Allouche and M. Mendes France,
Stern-Brocot polynomials and power series, preprint.
- Coons and
Shallit: M. Coons and J. Shallit, A pattern sequence approach to
Stern's sequence, preprint.
From the Monthly
- Calkin and
and Wilf: N. Calkin and H. Wilf, Recounting the
Rationals, American Mathematical Monthly, 107 (2000), 360-- 363.
and Escribano: M. Benito and J. Escribano, An easy proof of
Hurwitz' Theorem, American Mathematical Monthly, 109 (2002),
916--918. [Uses the Brocot array]
A monthly problem by D. Knuth with a different definition of t(n).
Rationals, American Mathematical Monthly, 110 (2003), 642-643.
- Sloane and
Sellers: On non-squashing partitions, Discrete Mathematics,
294 (2005), 259--274.
Stern in 1906
: The entry on Stern in the 1906 reference work: "The Jewish
B. Reznick, Some extremal problems for continued fractions, Illinois
J. Math, 29 (1985), 261--279. [My first reference to Sternstuff
(pp.270-271); reference  has become the classnotes.]
Classic papers (though not directly Stern)
- Guy1: R. Guy,
The strong law of small numbers, American Mathematical Monthly, 95
(1988), pp. 697-712.
R. Guy, The second strong law of small numbers, Mathematics Magazine, 63
(1990), pp. 3-20. (To both: "Nice looking sequence you've got
starting there. Think it will continue? In the immortal words of Clint
Eastwood, you've got to ask yourself one question: 'Do I feel
lucky?' Well, do ya, grad student?")
Recent papers, largely from "Integers" (on-line, so no page numbers)
S. H. Chan, Analogs of the Stern sequence, Integers, 11 (2011)
#A26. (Song is a 2005 student of Bruce Berndt's.)
R. Bacher, Twisting the Stern sequence, arXiv:1005.5627. (Probably
will not be published.)
M. Coons, On some conjectures concerning Stern's sequence and its twist,
Integers, 11 (2011) #A35.
M. Coons, A correlation identity for Stern's sequence, 12 (2012)
More on twisting the Stern sequence
J.-P. Allouche, On the Stern sequence and its twisted version, arXiv 1202.417v1.
Better .pdfs of the 19th century French papers from Week 2
(Tip o' the chapeau to Jean-Paul Allouche )
Local authors (that is, references for section 3.8 and 4.1)
B. Reznick, On the length of binary forms, to appear in Quadratic and
Higher Degree Forms , (K. Alladi, M. Bhargava, D. Savitt, P. Tiep,
eds.), Developments in Math. Springer, New York,
Reznick4: B. Reznick,
Sums of even powers of real
linear forms, Mem. Amer. Math. Soc., Volume 96, Number 463, March,
1992 (MR 93h.11043). Webpage contains links to each chapter,
separately scanned and posted.
B. Reznick, Continued fractions and an annelidic pde,
Math. Intelligencer, 5 (1983), 61-63. (MR 86e.11007). Written while
waiting for tenure to be approved. Never cited anywhere else. Only
paper in MathSciNet to use the word "annelidic"