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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 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 ...