Definition:Square Number/Definition 4

Definition

Square numbers are defined as the sequence:

$\forall n \in \N: S_n = \map P {4, n} = \begin{cases} 0 & : n = 0 \\ \map P {4, n - 1} + 2 \paren {n - 1} + 1 & : n > 0 \end{cases}$

where $\map P {k, n}$ denotes the $k$-gonal numbers.

Examples of Square Numbers

The first few square numbers are as follows:

Sequence of Square Numbers

The sequence of square numbers, for $n \in \Z_{\ge 0}$, begins:

$0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, \ldots$

Also see

• Equivalence of Definitions of Square Number

• Results about square numbers can be found here.