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In the directed graph above, pick any vertex and follow the arrows in sequence blue-red-red three times. You will finish at the green vertex. Similarly, follow the sequence ...

If f_1(x), ..., f_s(x) are irreducible polynomials with integer coefficients such that no integer n>1 divides f_1(x), ..., f_s(x) for all integers x, then there should exist ...

The silver ratio is the quantity defined by the continued fraction delta_S = [2,2,2,...] (1) = 2+1/(2+1/(2+1/(2+...))) (2) (Wall 1948, p. 24). It follows that ...

A simple point process (or SPP) is an almost surely increasing sequence of strictly positive, possibly infinite random variables which are strictly increasing as long as they ...

The maximal number of regions into which space can be divided by n planes is f(n)=1/6(n^3+5n+6) (Yaglom and Yaglom 1987, pp. 102-106). For n=1, 2, ..., these give the values ...

Stern's diatomic series is the sequence 1, 1,2, 1,3,2,3, 1,4,3,5,2,5,3,4, (1) ... (OEIS A002487) which arises in the Calkin-Wilf tree. It is sometimes also known as the fusc ...

Given a Lucas sequence with parameters P and Q, discriminant D!=0, and roots a and b, the Sylvester cyclotomic numbers are Q_n=product_(r)(a-zeta^rb), (1) where ...

A Thâbit ibn Kurrah number, sometimes called a 321-number, is a number of the form K_n=3·2^n-1. The first few for n=0, 1, ... are 2, 5, 11, 23, 47, 95, 191, 383, 767, ... ...

Given rods of length 1, 2, ..., n, how many distinct triangles T(n) can be made? Lengths for which l_i>=l_j+l_k (1) obviously do not give triangles, but all other ...

A two-regular graph is a regular graph for which all local degrees are 2. A two-regular graph consists of one or more (disconnected) cycles. The numbers a_n of two-regular ...