1:
+ x
3:
x x + + x x
6:
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
The formula for the nth triangular number is ½n(n+1) or (1+2+3+...+ n2 + n1 + n).
It is the binomial coefficient
It can also be shown that for any ndimensional simplex with sides of length x, the formula
<math> \frac {(x)(x+1)...(x+(n1))} {n!} </math>
will accurately show the number of that simplex. For example, a tetrahedron with sides of length 2 has a number of <math> \frac {(2)(2+1)(2+2)} {6} </math>, 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.
See also: square number, polygonal number.
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