## Math

I’ve been exploring new backwaters in elementary mathematics, and visiting some old ones, in my spare time since 1972.

#### The periambic constellation

Some of this topic’s history and background may be found here.

- “Modest triangles and periambic points.”
*International Journal of Geometry*,**10,**no. 1 (2021). - “What’s new with altitudes and perpendicular bisectors?”
*Mathematics Magazine*,**93,**no. 5 (2020), pp. 347–351. https://doi.org/10.1080/0025570X.2020.1822703. - “The periambic constellation: altitudes, perpendicular bisectors, and other radical axes in a triangle.”
*Forum Geometricorum*,**17**(2017), pp. 383–399. MR3733035.

#### Continued powers and continued function compositions

- “A chronology of continued square roots and other continued compositions, through the year 2016.” Version 1 posted at arXiv.org on 19 July 2017. Version 2 posted at arXiv.org on 15 June 2018.

(More about this chronology can be found here.) - “Continued reciprocal roots,”
*Ramanujan Journal*, online 4 August 2014;**38**(2) November 2015, pp. 435–454. MR3414500. Here’s a preprint. - “Continued powers and a sufficient condition for their convergence,”
*Mathematics Magazine*,**68**(5) December 1995, pp. 387–392. MR1365650.

(An error in Example III, noted in a Letter to the Editor by J. Nichols-Barrer,**69**(3) June 1996, p. 238, is corrected in my Letter to the Editor,**69**(4) October 1996, p. 316.)On page 388, in the third displayed equation, the plus-or-minus sign should be a minus. (An error introduced**Errata:***after*the galleys were corrected…) - “Continued powers and roots,”
*Fibonacci Quarterly*,**29**(1) February 1991, pp. 37–46. MR1089518.

(A preprint of this paper was published in 1989 as Technical Report Number 89-02 by the Department of Mathematical Sciences, University of Alaska Fairbanks.)**Errata:**

p. 37, end of first paragraph, change to “When*p =*0, the expression is identically*x_*0 (provided that the terms are not all 0).”

p. 45, end of first partial paragraph, change to “We therefore infer the convergence of C_0^\infty, which completes the proof.”

p. 45, 4th displayed equation should be C_0^\infty (2, 1/4 + 2^{*-i*})

#### Real-valued cycles in the 3*x* + 1 problem

- “Families of maximal perigees,”
*Far East Journal of Dynamical Systems*,**12**(2) March 2010, pp. 77–94, MR2666414. (Here’s a preprint.) - “Parameter-independent structure in periodic orbits of an iterated function system on the real line,”
*Publicationes Mathematicae Debrecen*,**76**(1-2) 2010, pp. 51–65, MR2598171. (Here’s a preprint.) - I gave a talk in April 2006 about my work on the 3
*x*+ 1 problem. - Tables of 3
*x*+ 1 perigees,*n =*2, . . . , 20 are available upon request (in Excel or text format).

#### Generalizations of the butterfly problem

- “Quadrangles, butterflies, Pascal’s hexagon, and projective fixed points,”
*American Mathematical Monthly*,**87**(3) March 1980, pp. 197–200, MR0562923.

(In the*Monthly***90**(3) August 1983, p. 473, Leon Gerber points out that Theorem 1 of this paper appears in “Poncelet’s theorems, ‘Traite des proprietes projectives des figures'” (Paris, 1822, article 513) and in Cremona’s*Elements of Projective Geometry*(1913).) - “A double butterfly theorem,”
*Mathematics Magazine*,**49**(2) March 1976, pp. 86–87, MR0397526.

#### Integer sequences

My good friend and mathematical conspirator Marty Getz and I jointly submitted these sequences to the Online Encyclopedia of Integer Sequences:

- A072255 Number of ways to partition {1,…,
*n*} into arithmetic progressions, where in each partition all the progressions have the same common difference and have lengths greater than or equal to 2. (8 July 2002). - A053732 Number of ways to partition {1,…,
*n*} into arithmetic progressions of length >= 1. (13 Feb 2000)

Related to these, we also submitted a comment on the sequence

- A000325 2^
*n*–*n*. (21 May 2005)

Back in 1999 I took an undergraduate course in combinatorics taught by Jill Faudree (see below as well). Thanks to a project in that class, I became interested in linear combinations of binomial coefficients. Returning to this topic in a feeble way, we have the following trapezoids of dot products of row *j* (signs alternating) with sequential (*j* + 1)-tuples read by rows in Pascal’s triangle for *j* = 3, . . . , 9:

In 2020 I had occasion to need series solutions to certain trinomial equations. For a few of these series, I found the coefficients in the OEIS, presented under guises unrelated to trinomials, and I offered comments thereon:

For other trinomial series solutions, I submitted the following new sequences of coefficients:

#### Journal problems and solutions

Marty Getz and I collaborated on a number of solutions to journal problems. We lucked out a few times with the *Mathematics Magazine*:

- Solution without words to problem 2016, “A regular octagon with one-third the area of another,”
**91**(2) April 2018, p. 153. - Solution to problem 2013, “Counting lattice points in a tetrahedron,”
**91**(1) February 2018, p. 74. - Solution to problem 1785, “Summing floor powers,”
**81**(5) December 2008, p. 380. - Solution to problem 1653, “A trapezoid in a triangle,”
**76**(4) October 2003, pp. 319–320. - Solution to problem 1650, “Cutting a polygon into rhombi”,
**76**(3) June 2003, pp. 235–236. - Solution to problem 1627, “A generalization of the arbelos,”
**75**(3), June 2002, p. 231. - Solution to problem 1598, “Periodic transformations of
*n*-tuples,”**74**(2), April 2001, p. 159. - Solution to problem 1585, “A timely sum, but not a square,”
**73**(5) December 2000, p. 405. - Solution to problem 1574, “Similar triangles in a convex quadrilateral,”
**73**(3) June 2000, p. 241. - Solution to problem 1545, “A divisibility condition on consecutive terms of a sequence,”
**72**( 2) April 1999, p. 151.

In the *College Mathematics Journal*:

- Solution to problem 1001, “The limit of the sum of two integrals,”
**45**(3) May 2014, p. 224.

We even sneaked a couple into the *Monthly*:

- Proposed problem 11005,
**110**(3) April 2003, page 340.- The solution is printed as “Partitioning by arithmetic progressions,”
**112**(1) January 2005, p. 89-90. The published solution is incomplete, though; see A072255 and A000325 in the Online Encyclopedia of Integer Sequences for an explanation. This problem originated as a spin-off of a project I did in Jill Faudree’s nifty undergraduate combinatorics class in the fall of 1999. Marty generalized my original question and found the solution, and I came up with a proof of his result.

- The solution is printed as “Partitioning by arithmetic progressions,”
- Editor’s comment on problem 10823, “Periodicity under shift and complement,”
**109**(8) October 2002, p. 762.

And although I was not involved, it’s my pleasure to also mention

- Maya Genaux and Marty Getz, solution to problem 840, “Area of a circle segment and length of its chord,”
*College Mathematics Journal*,**38**(5) November 2007, pp. 395–396.

#### Misc

- In memory of Donald R. DeWitt, 1937 – 2017.
- I received a Knuth check (what’s that?) for a suggestion about the function “
*x*mumble*y*,” found on page 83 of*Concrete Mathematics*. - A thumbnail, or perhaps hangnail, biography of Fermat de Pierre.

*Updated 5 Sept 2021*