Pronic number

A rectangular figure, square, or number pronische number is a number that is the product of two successive natural numbers. For example, a rectangle number. The first square numbers are

For some authors, the zero is not a square number, so the number sequence only begins with the two.

The name square number is derived from a geometric property. If one blocks to form a rectangle, one side by 1 is longer than the second, then the number of the stones of a square number. Because of this relationship with a geometric figure are the rectangular numbers to the figurate numbers, which include the triangular numbers and square numbers belong.

Calculation

The -th square number is calculated according to the formula

Th the rectangle number is the sum of the first straight natural numbers.

( This is similar to the law of formation of the square numbers, which are the sums of the first odd natural numbers. )

Relations to other figurate numbers

The -th square number is twice the th triangular number.

Properties

• All square numbers are even numbers.
• Only the rectangle number that is a prime number, is the 2

Series of the reciprocals

The sum of the reciprocals of all square numbers is 1

Generating function

The function

Contain in their expansion (the right side of the equation ), respectively the number of rectangular -th coefficient of the. Why it is called the generating function of the square numbers.