20 citations · 22 across the 10 of their papers we have counts for
10 papers
Periodic Weighted Sums of Binomial Coefficients
Greg Dresden, Yike Li
Using elementary methods, we establish old and new relations between binomial coefficients, Fibonacci numbers, Lucas numbers, and more.
On the Brousseau sums
Gregory Dresden
We start with new convolution formulas for involving only the binomial coefficients. Then, we use those to find direct formulas for the sums …
Chebyshev Polynomials, Sliding Columns, and the -step Fibonacci Numbers
Greg Dresden
We give a direct and intuitive proof (via sliding some columns up and down) of the following interesting fact: if we write out the Chebyshev polynomials in a chart and take the sum…
On the Middle Coefficient of a Cyclotomic Polynomial
Gregory Dresden
In this article, we provide a short and elementary proof of the following result: For the middle coefficient of is either zero (when is a power of ) or a…
Two Irrational Numbers That Give the Last Non-Zero Digits of and
Gregory Dresden
If we form a decimal where the nth digit is the last non-zero digit of (likewise, the last non-zero digit of ), we obtain an irrational number
Finite subgroups of the extended modular group
Gregory Dresden, Prakriti Panthi, Anukriti Shrestha +1
We show that in the extended modular group PGL(2,Z) there are exactly seven finite subgroups up to conjugacy; three subgroups of size 2, one subgroup each of size 3, 4, and 6, and…
Cubic Polynomials, Linear Shifts, and Ramanujan Cubics
Gregory Dresden, Prakriti Panthi, Anukriti Shrestha +1
We show that every monic polynomial of degree three with complex coefficients and no repeated roots is either a (vertical and horizontal) translation of or can be composed…
When is the sum of two squares?
Greg Dresden, Kylie Hess, Saimon Islam +5
Using Fermat's two squares theorem and properties of cyclotomic polynomials, we prove assertions about when numbers of the form can be expressed as the sum of two integer…
A simplified Binet formula for k-generalized Fibonacci numbers
Gregory P. Dresden
We present a particularly nice Binet-style formula that can be used to produce the k-generalized Fibonacci numbers (that is, the Tribonaccis, Tetranaccis, etc). Furthermore, we sho…
A New Approach to Rational Values of Trigonometric Functions
Gregory P. Dresden
It is well known that sin(aπ/b), cos(aπ/b), etc., are only rational numbers for a few select integers a and b. We show that this is equivalent to the fact that only for d = 1,2,3,4…