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.

A triangular number may be expressed according to its position, n, as (n+1)n/2 (ie. sum of consecutive numbers)
8 times the triangular number plus 1 is therefore 4n(n + 1)+1 = 4n^2 + 4n + 1 = (2n + 1)^2
which is a square of the number's place doubled plus 1.

Posted by retiarius on 2004-01-15 09:04:28