Triangular Numbers

A friend reminded me that I hadn't explained triangular numbers here or on my web site, so taking care of both in a limited way today...

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 n^th 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 T_n = 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.

This means that they are also expressible as (n+1 choose 2)

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 T_n is equal to the sum of the cubes of the natural numbers from 1 to n,

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And one final note on the triangular numbers, the infinite sum of their reciprocals is an integer, ... how sweet, and unexpected, is that? I think you can figure out what it is..but if you get stuck and need a hint, or the answer... just drop a note