The numbers, which can be both triangular and square, are called triangular square numbers.

**E.g.** 36 is both triangular and square. 6, on the other hand, is triangular, but
not square; 25, is a square, but not triangular.

Triangular numbers are given by the formula n(n+1)/2 where n >=1. The following programme, generates first 20 triangular numbers

Triangular Square Numbers can be derived in the following way.

(n)(n+1)/2 = m^{2} where m and n are integers. The left-hand side denotes a triangular number
and the right-hand side denotes a square number.

(n^{2} + n)/2 = m^{2}

n^{2} + n = 2m^{2}

Using the completing the square method,

(n + 1/2)^{2} - 1/4 = 2m^{2}

(2n + 1)^{2} - 1 = 8m^{2}

Let y =2m and x = 2n +1

Then, x^{2} -1 = 2y^{2} where x represents an odd number and y, an even number.

**x ^{2} - 2y^{2} = 1**

This is **Pell Equation**

Find pairs of (x,y) which satisfy the Pell Equation and the half of y value in each pair is the square root of a
**Triangular Square** numbers.

Now, in order to generate first few **triangular square numbers**, please press the button.

This is a vast collection of tutorials, covering the syllabuses of GCSE, iGCSE, A-level and even at undergraduate level.
They are organized according to these specific levels.

The major categories are for core mathematics, statistics, mechanics and trigonometry. Under each category, the tutorials are grouped according to the academic level.

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