User:Edward Krogius

I am a Math teacher.

HERE IS MY CONTRIBUTION.

I just did some calculations and conjured that the

Squared triangle numbers OEIS : A001110 can be derived from the Pell numbers A000129 as evenly indexed Pell numbers divided by 2 squared ..

I suggest this comment.

A001110 : a(n) = (A000129(2k)/2)^2, k in {0,1,2,..}

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

Now replacing

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.

1. 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()