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 sides is an irrational number, no equilateral triangle is Heronian. However, there is a unique sequence of Heronian triangles that are "almost equilateral" because the three sides are of the form "n" − 1, "n", "n" + 1. A method for generating all solutions to this problem based on continued fractions was described in 1864 by Edward Sang, and in 1880 Reinhold Hoppe gave a closed-form expression for the solutions. The first few examples of these almost-equilateral triangles are listed in the following table : Subsequent values of "n" can be found by multiplying the previous value by 4, then