Math &c

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

The periambic constellation

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

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 J., online August 4, 2014; 38 (2) November 2015, pp. 435–454. MR3414500. Here’s a preprint.
  • Continued powers and a sufficient condition for their convergence,” Math. Mag., 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) Oct. 1996, p. 316.)
    Errata: On page 388, in the third displayed equation, the plus-or-minus sign should be a minus. (An error introduced after the galleys were corrected…)
  • Continued powers and roots,” Fibonacci Quart., 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 3x + 1 problem

Generalizations of the butterfly problem

  • Quadrangles, butterflies, Pascal’s hexagon, and projective fixed points,” Amer. Math. Monthly, 87 (3) March 1980, pp. 197–200, MR0562923.
    (In the Monthly 90 (3) Aug. 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,” Math. Mag., 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

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:

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) Dec. 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.
  • 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 Math. Journal, 38 (5) November 2007, pp. 395–396.

Misc


Updated 21 Sept 2019.