Number of rectangles

Twelve balls in three rows and four columns form a rectangle.

A square number , square number, or pronic number is a number that is the product of two consecutive natural numbers . For example, is a rectangle number. The first square numbers are ${\ displaystyle 12 = 3 \ cdot 4}$

0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, ... (sequence A002378 in OEIS )

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

The name rectangle number is derived from a geometric property. If you place stones in a rectangle , one side of which is 1 longer than the second, the number of stones corresponds to a number of the rectangle. Because of this relationship with a geometric figure , the rectangular numbers are among the figured numbers , which also include the triangular and square numbers .

calculation

The -th number of rectangles is calculated according to the formula ${\ displaystyle n}$ ${\ displaystyle P_ {n}}$

{\ displaystyle P_ {n} = n \ cdot (n + 1) = n ^ {2} + n}

The -th square number is the sum of the first even natural numbers . ${\ displaystyle n}$ ${\ displaystyle n}$

{\ displaystyle {\ begin {aligned} & 2 & = 2 & = 1 \ times 2 \\ & 2 + 4 & = 6 & = 2 \ times 3 \\ & 2 + 4 + 6 & = 12 & = 3 \ times 4 \\ & 2 + 4 + 6 + 8 & = 20 & = 4 \ cdot 5 \\ && \ vdots & \ end {aligned}}}

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

Relationships to other figured numbers

The -th square number is twice the -th triangular number . ${\ displaystyle n}$ ${\ displaystyle n}$ ${\ displaystyle \ Delta _ {n}}$

{\ displaystyle P_ {n} = 2 \ cdot \ Delta _ {n}}

properties

All square numbers are even numbers .

The only square number that is prime is 2.

{\ displaystyle \ lfloor {\ sqrt {P_ {n}}} \ rfloor \ cdot \ lceil {\ sqrt {P_ {n}}} \ rceil = P_ {n}}

Series of reciprocals

The sum of the reciprocal values of all rectangular numbers is 1.

{\ displaystyle \ sum _ {n = 1} ^ {\ infty} {\ frac {1} {n ^ {2} + n}} = 1}

Generating function

The function

{\ displaystyle {\ frac {2x} {(1-x) ^ {3}}} = 2x + 6x ^ {2} + 12x ^ {3} + 20x ^ {4} + \ ldots}

contains in its series expansion (right side of the equation) the -th square number as a coefficient of . It is therefore called the generating function of the rectangular numbers. ${\ displaystyle n}$ ${\ displaystyle x ^ {n}}$

Web links

Eric W. Weisstein : Rectangular number . In: MathWorld (English).