Triangle numbers are calculated taking each integer plus all the ones before it. The first triangle number is 1, the second is 1+2 or 3, and the third is 1+2+3 or 6.

If you take 8 times a triangle number plus 1, the result will be a perfect square. This number also will be the square of the triangle number's place doubled, plus one.

For example, 6 is third in the triangle number sequence. (1, 3, 6...) This means 8 times 6 plus 1 = 49 equals 3 times 2 plus 1, squared, or 7 squared.

Prove why this works.

The n-th triangular number can be represented as a triangle such as:

X

XX

XXX

XXXX



and so on. Two such triangles can be joined to build a nx(n+1) rectangle. Four such rectangles can be joined to build a (2n+1)x(2n+1) square, with just the center cell missing. Thus, (2n+1) squared equals 8 times the n-th triangular number, plus 1.

Posted by e.g. on 2004-01-15 10:13:08