The triangular numbers are known back to before Pythagoras (500ish BCE). They are simply the sums of the natural or counting numbers 1 + 2 + ... + n is the nth triangular number, so the sequence of triangular numbers is 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, etc. An explicit formula for the triangular numbers is Tn = n(n+1)/2. For example, T10= (10)(11)/2 = 55.
The triangular numbers appear as the third number in each row of Pascal's triangle.
One of the interesting properties of the triangular numbers is that any two consecutive ones form a square number.... in the image below, T4 (in red) is added to T5 in blue, to form S5, the fifth square number or 25.
It is also known that any even perfect number, and all the ones we know of are even, is a triangular number. The first few are 6, 28, 496.
It is also known that the digit root in base ten of any triangular number is either 1 or 0. That means that all triangular numbers are either divisible by three, or have a remainder of one when divided by three. An easy test of any number to see if it is triangular is to multiply by eight and add one... if the result is a perfect square, then the number is triangular.
It is also interesting that the square of a triangular number Tn is equal to the sum of the cubes of the natural numbers from 1 to n,
And one final note on the triangular numbers, the infinite sum of their reciprocals is an integer, 2... how sweet, and unexpected, is that?
The Pyramidal (or more appropriatly, tetrahedral) Numbers can be thought of as the sum of a series of triangular numbers, so Pn = 1+3+6 +... +Tn. If you imagine stacks of oranges or cannonballs, with each layer arranged as a triangle, this is how many there would be in n layers.
The sum is given by Pn=n(n+1)(n+2)/6.