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Let f(1)=1, and let f(n) be the number of occurrences of n in a nondecreasing sequence of integers. then the first few values of f(n) are 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, ... ...

The squeeze theorem, also known as the squeezing theorem, pinching theorem, or sandwich theorem, may be stated as follows. Let there be two functions f_-(x) and f_+(x) such ...

Let N be an odd integer, and assume there exists a Lucas sequence {U_n} with associated Sylvester cyclotomic numbers {Q_n} such that there is an n>sqrt(N) (with n and N ...

A polygon whose vertices are points of a point lattice. Regular lattice n-gons exists only for n=3, 4, and 6 (Schoenberg 1937, Klamkin and Chrestenson 1963, Maehara 1993). A ...

O_h is the point group of symmetries of the octahedron having order 48 that includes inversion. It is also the symmetry group of the cube, cuboctahedron, and truncated ...

The Thomson problem is to determine the stable equilibrium positions of n classical electrons constrained to move on the surface of a sphere and repelling each other by an ...

A bicentric quadrilateral, also called a cyclic-inscriptable quadrilateral, is a four-sided bicentric polygon. The inradius r, circumradius R, and offset x are connected by ...

The complete elliptic integral of the second kind, illustrated above as a function of k, is defined by E(k) = E(1/2pi,k) (1) = ...

A composite number n is a positive integer n>1 which is not prime (i.e., which has factors other than 1 and itself). The first few composite numbers (sometimes called ...

Consider an arbitrary one-dimensional map x_(n+1)=F(x_n) (1) (with implicit parameter r) at the onset of chaos. After a suitable rescaling, the Feigenbaum function ...