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A rootfinding algorithm for cubics
(Proceedings of the American Mathematical Society, 2013)Newton's method applied to a quadratic polynomial converges rapidly to a root for almost all starting points and almost all coefficients. This can be understood in terms of an associative binary operation arising from 2 x ... 
Integrating across Pascal's triangle
(Mathematical Analysis and Applications, 2011)Sums across the rows of Pascal's triangle yield powers of 2 while certain diagonal sums yield the Fibonacci numbers which are asymptotic to powers of the golden ratio. Sums across other diagonals yield quantities asymptotic ... 
A Lyness equation for graphs
(Journal of Difference Equations and Applications, 2012)The Lyness equation, x(n+1)=(x(n)+k)/x(n1), can be though of as an equation defined on the 2regular tree T2: we can think of every vertex of T2 as a variable so that if x and z are the vertices adjacent to y, then x,y,z ... 
A New Parameterization of Ford Circles
(Pi Mu Epsilon Journal, 2014)Lester Ford introduced Ford Circles in 1938 in order to geometrically understand the approximation of an irrational number by rational numbers. We shall construct Ford circles by a recursive geometric procedure and by a ... 
Complex Descartes Circle Theorem
(American Mathematical Monthly, 2014)We present a short proof of Descartes Circle Theorem on the curvaturecenters of four mutually tangent circles. Key to the proof is associating an octahedral configuration of spheres to four mutually tangent circles. We ... 
A short proof and generalization of Lagrange's theorem on continued fractions
(American Mathematical Monthly, 2011)We present a short new proof that the continued fraction of a quadratic irrational eventually repeats. The proof easily generalizes; we construct a large class of functions which, when iterated, must eventually repeat when ... 
Square Roots of 2x2 Matrices
(2010)This paper is designed to pique the interest of undergraduate students who are familiar with the concepts of linear algebra. We investigate five methods of computing square roots of twobytwo matrices. Each method gives ... 
On Stern's Diatomic Sequence 0,1,1,2,1,3,2,3,1,4,...
(American Mathematical Monthly, 2010)We investigate several of the many interesting properties of the title sequence. In particular, we focus on the combinatorics of the sequence (e.g., what the numbers count), some parallels with the Fibonacci sequence, some ... 
Sums across Pascal's triangle modulo 2
(Congressus Numerantium, 2010)We consider sums of the binomial coefficients C(i + j, i) modulo 2 over lines ai + bj = n. Many interesting sequences (old and new) arise this way.