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.

It is given that:

T(1) = 1st triangular number = 1

T(2) = 2nd triangular number = 1+2

.........................

........................

T(n) = nth triangular number = 1+2+....+n

Now, we observe that T(n) is the sum of an arithmetic sequence of n terms with each of the first term and the common difference being equal to 1.

Therefore, T(n) = (n/2)*(1 + n*1)) = (n/2)*(n+1) = n(n+1)/2

Accordingly,

8*T(n) + 1

= 8*(n(n+1)/2) + 1

= 4*(n^2) + 4n + 1

= (2n+1)^2

Thus, 8 times the nth triangular number increased by 1 equals (2n+1)^2.

Hence, the proof.