Pascal's triangle Construct a "d"-dimensional triangle (a 3-dimensional triangle is a tetrahedron) by placing additional dots below an initial dot, corresponding to P(1) = 1. Place these dots in a manner analogous to the placement of numbers in Pascal's triangle. To find P("x"), have a total of "x" dots composing the target shape. P("x") then equals the total number of dots in the shape. A 0-dimensional triangle is a point and a 1-dimensional triangle is simply a line, and therefore P("x") = "1" and P("x") = "x", which is the sequence of natural numbers. The number of dots in each layer corresponds
Heronian triangle with sides in arithmetic progression, and isosceles Heronian triangles. The list of primitive integer Heronian triangles, sorted by area and, if this is the same, by perimeter, starts as in the following table. "Primitive" means that the greatest common divisor of the three side lengths equals 1. Lists of primitive Heronian triangles whose sides do not exceed 6,000,000 can be found at A shape is called equable if its area equals its perimeter. There are exactly five equable Heronian triangles: the ones with side lengths (5,12,13), (6,8,10), (6,25,29), (7,15,20), and (9,10,17). Since the area of an equilateral triangle with rational