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
Triangular number Triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle, as in the diagram on the right. The th triangular number is the number of dots in the triangular arrangement with dots on a side, and is equal to the sum of the natural numbers from 1 to . The sequence of triangular numbers , starting at the 0th triangular number, is The triangle numbers are given by the following explicit formulas: where formula_2 is a binomial coefficient. It represents the number of distinct pairs that can be selected from objects, and it is read