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Generalizes the secant method of root finding by using quadratic 3-point interpolation q=(x_n-x_(n-1))/(x_(n-1)-x_(n-2)). (1) Then define A = ...

Let A^~, B^~, ... be operators. Then the commutator of A^~ and B^~ is defined as [A^~,B^~]=A^~B^~-B^~A^~. (1) Let a, b, ... be constants, then identities include [f(x),x] = 0 ...

With n cuts of a torus of genus 1, the maximum number of pieces which can be obtained is N(n)=1/6(n^3+3n^2+8n). The first few terms are 2, 6, 13, 24, 40, 62, 91, 128, 174, ...

If r is a root of the polynomial equation x^n+a_(n-1)x^(n-1)+...+a_1x+a_0=0, where the a_is are integers and r satisfies no similar equation of degree <n, then r is called an ...

The number of bases in which 1/p is a repeating decimal (actually, repeating b-ary) of length l is the same as the number of fractions 0/(p-1), 1/(p-1), ..., (p-2)/(p-1) ...

The net graph is the graph on 6 vertices illustrated above. It is implemented in the Wolfram Language as GraphData["NetGraph"]. The bipartite double graph of the net graph is ...

(dy)/(dx)+p(x)y=q(x)y^n. (1) Let v=y^(1-n) for n!=1. Then (dv)/(dx)=(1-n)y^(-n)(dy)/(dx). (2) Rewriting (1) gives y^(-n)(dy)/(dx) = q(x)-p(x)y^(1-n) (3) = q(x)-vp(x). (4) ...

This is sometimes knows as the "bars and stars" method. Suppose a recipe called for 5 pinches of spice, out of 9 spices. Each possibility is an arrangement of 5 spices ...

The Lucas polynomials are the w-polynomials obtained by setting p(x)=x and q(x)=1 in the Lucas polynomial sequence. It is given explicitly by ...

By way of analogy with the eban numbers, uban numbers are defined as numbers whose English names do not contain the letter "u" (i.e., "u" is banned). Note that this ...