I am a Math teacher.
HERE IS MY CONTRIBUTION.
I just did some calculations and conjured that the
I suggest this comment.
This is because
Trianle = Square T_m = S_n m*(m+1)/2 = n*n m*m + m = 2*n*n (m+1/2)*(m+1/2) = 2*n*n + 1/4 (2m+1)(2m+1) = 8n*n + 1 (2m+1)(2m+1) = 2*(2n)*(2n) + 1
x=2m+1 and y=2n
we get the Pells equation
x*x - 2y*y = 1
Now this is solved by even Pells numbers for y. n = y/2 and the numbers looked for are n*n = (y/2)*(y/2)
I tested it with a python code also.
- Trianglenumbers as squares in Python
endNumber = 40
def pell2k(n): # Even Pell numbers P(2k), k in N
if n < 2: return 2*n return 6*pell2k(n-1) - pell2k(n-2)
def output(): # get the squares
print p2k tsk =  for x in p2k: tsk.append( (x/2)**2 ) print tsk return
p2k = map(pell2k, range(0, endNumber, 1)) output()