TRIANGLE Numbers that are Perfect Squares

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Other Links : PELL NUMBERS FORMULAE ,  SUMS OF POWERS,  FIBONACCI FORMULAE, RAJESH's MATHS PAGE, TRIANGLE NUMBERS THAT ARE PERFECT SQUARES

Also see Sums of Cubes , Sums of Powers - Ramanujan and his Number 1729

The solutions to the above equation is given by Sets T and Y where

T = {1, 6, 35, 204, 1189, 6930, .........}

Y = {1, 8, 49, 288, 1681, 9800, ........}

So, we can express the solution using the following relations:

The following are some of my findings that give formulae to find the kth, 2kth, etc.

	T1
	T2
	T3
	T4
	T5
	T6
	T7
	T8
	T9
	T10
	T11
	T12
	T13
	T14
	T15

Note : T is a actually contained in Pell Numbers P = {1, 2, 5, 12, 29, 70, 169, 408, ......} in more than one way.  If you take the even terms out of P and write it as {2, 12, 70, 408, .....} = 2 {1, 6, 35, 204, .....} = 2T.   This means that you can substitute (1/2)P(2k) in all the formulae for T(k) to get PELL Numbers Formulae. Also consider Pell numbers P = {1, 2, 5, 12, 29, 70, .....} and its associated numbers Q = {1, 3, 7, 17, 41, .....} The Pell numbers P(n) and their associated numbers Q(n) satisfy P(n+2)=2P(n+1)+P(n), P(0)=0, P(1)=1; Q(n+2)=2Q(n+1)+Q(n), Q(0)=1, Q(1)=1.  Note that P(n).Q(n) = T(n).  So this means that we can arrive at formulae for Q(n) since we have formulae for T(n) and P(n).