Triangular number

A triangular number is a number that can be arranged in the shape of an equilateral triangle (by convention, the first triangular number is 1):

+ x

x x + + x x

x x x x x x + + + x x x

10:

   x               x
  x x             x x
 x x x           x x x
+ + + +         x x x x

15:

x x x x x x x x x x x x x x x x x x x x + + + + + x x x x x

21:

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x + + + + + + x x x x x x

Since each row is one unit longer than the previous row it can be seen that a triangular number is the sum of consecutive integers.

The formula for the nth triangular number is ½n(n+1) or (1+2+3+...+ n-2 + n-1 + n).

It is the binomial coefficient

It can also be shown that for any n-dimensional simplex with sides of length x, the formula

will accurately show the number of that simplex. For example, a tetrahedron with sides of length 2 has a number of , or 4. (Note: A tetrahedron can be created by taking a number, getting the triangle of that number, and then adding to it all the triangles of the numbers before it, so a tetrahedron of 2 would have 2 triangled=3 plus 1 triangled=1 =4.)

One of the most famous triangular numbers is 666, also known as the Number of the Beast. Every perfect number is triangular.

The sum of two consecutive triangular numbers is a square number. This can be shown mathematically thus: the sum of the nth and (n-1)th triangular numbers is {½n(n+1)} + {½(n-1)n}. This simplifies to (½n²+½n) + (½n²-½n), and thus to n². Alternatively, it can be demonstrated diagrammatically, thus:

x + + +
x x + +
x x x +
x x x x

x + + + +
x x + + +
x x x + +
x x x x +
x x x x x

In each of the above examples, a square is formed from two interlocking triangles.