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.

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.)
    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 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 3x + 1 problem

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

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

Except for the ones marked with an asterisk (*), the published solutions and problem proposals below were collaborations with Marty Getz.

  • * Proposed problem 2164, 96 (1) February 2023, p. 89.
  • * 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.
  • 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

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Updated 7 March 2023