Question

What are square and triangular numbers?

Answer 1

A triangular number or triangle number counts the objects that can form an equilateral triangle.
The nth triangle number is the number of dots composing a triangle with n dots on a side, and is equal to the sum of the n natural numbers from 1 to n. The sequence of triangular numbers starting at the 0th triangular number, is:
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406 …


A square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself.
For example, 9 is a square number, since it can be written as $3\times 3$.
The usual notation for the formula for the square of a number n is not the product $n\times n$, but the equivalent exponentiation $n^2$, usually pronounced as "n squared".
Square numbers are non-negative. For example, $\sqrt{9}=\pm 3$, so 9 is a square number.
The number m is a square number if and only if one can compose a square of m equal (lesser) squares.
The numbers whose dot patterns can be arranged as triangles are called triangular numbers. The sum of two consecutive triangular numbers is a square number. This happens because they form a square on combining.