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.