The squares of triangular numbers 1 and 6 are triangular numbers 1 and 36.

T1^2 = 1 * 1 = 1 = T1
T3^2 = 6 * 6 = 36 = T8

Are there additional triangular numbers whose squares form a triangular number?

The sum of two consecutive squares, less 1, is the square of the product of two consecutive whole numbers:

N = n(n+1)/2 = s^2 = (m(m+1)/2)^2  
2n^2+2n = m^2 (m+1)^2  
n^2+(n+1)^2-1 = (m(m+1))^2 

Posted by broll on 2015-01-12 12:51:54