Editing Centered square number (section)

== Relationships with other figurate numbers ==

Let ''C''<sub>''k'',''n''</sub> generally represent the ''n''th [[Centered polygonal number|centered ''k''-gonal number]]. The ''n''th centered square number is given by the formula:

:<math>C_{4,n} = n^2 + (n - 1)^2.</math>

That is, the ''n''th centered square number is the sum of the ''n''th and the (''n'' – 1)th [[square number]]s. The following pattern demonstrates this formula:

:{|
|- align="center" valign="middle" style="line-height: 0;"
|[[Image:GrayDot.svg|16px]]
| &nbsp; &nbsp;
| &nbsp; &nbsp;
|[[Image:RedDot.svg|16px]]<br>[[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]]<br>[[Image:RedDot.svg|16px]]
| &nbsp; &nbsp;
| &nbsp; &nbsp;
|[[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]]
| &nbsp; &nbsp;
| &nbsp; &nbsp;
|[[Image:RedDot.svg|16px]]<br>[[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]]<br>[[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]]<br>[[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]]<br>[[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]]<br>[[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]]<br>[[Image:RedDot.svg|16px]]
|- align="center" valign="top"
| <math>C_{4,1} = 0 + 1</math>
| &nbsp; &nbsp;
| &nbsp; &nbsp;
| <math>C_{4,2} = 1 + 4</math>
| &nbsp; &nbsp;
| &nbsp; &nbsp;
| <math>C_{4,3} = 4 + 9</math>
| &nbsp; &nbsp;
| &nbsp; &nbsp;
| <math>C_{4,4} = 9 + 16</math>
|}

The formula can also be expressed as:

:<math>C_{4,n} = \frac{(2n-1)^2 + 1}{2}.</math>

That is, the ''n''th centered square number is half of the ''n''th odd square number plus 1, as illustrated below:

:{|
|- align="center" valign="bottom" style="line-height: 0;"
|[[Image:GrayDot.svg|16px]]
| &nbsp; &nbsp;
| &nbsp; &nbsp;
|[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:MissingDot.svg|16px]]<br>[[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]]
| &nbsp; &nbsp;
| &nbsp; &nbsp;
|[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]]<br>[[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]]<br>[[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]]
| &nbsp; &nbsp;
| &nbsp; &nbsp;
|[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]]<br>[[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]]<br>[[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]]<br>[[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]]
|- align="center" valign="top"
| <math>C_{4,1} = \frac{1 + 1}{2}</math>
| &nbsp; &nbsp;
| &nbsp; &nbsp;
| <math>C_{4,2} = \frac{9 + 1}{2}</math>
| &nbsp; &nbsp;
| &nbsp; &nbsp;
| <math>C_{4,3} = \frac{25 + 1}{2}</math>
| &nbsp; &nbsp;
| &nbsp; &nbsp;
| <math>C_{4,4} = \frac{49 + 1}{2}</math>
|}

Like all [[centered polygonal number]]s, centered square numbers can also be expressed in terms of [[triangular number]]s:

:<math>C_{4,n} = 1 + 4\ T_{n-1} = 1 + 2{n(n-1)},</math>

where

:<math>T_n = \frac{n(n+1)}{2} = \binom{n+1}{2}</math>

is the ''n''th triangular number. This can be easily seen by removing the center dot and dividing the rest of the figure into four triangles, as below:

:{|
|- align="center" valign="middle" style="line-height: 0;"
|[[Image:BlackDot.svg|16px]]
| &nbsp; &nbsp;
| &nbsp; &nbsp;
|[[Image:RedDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:BlackDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:RedDot.svg|16px]]
| &nbsp; &nbsp;
| &nbsp; &nbsp;
|[[Image:RedDot.svg|16px]]<br>[[Image:RedDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:BlackDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:RedDot.svg|16px]]<br>[[Image:RedDot.svg|16px]]
| &nbsp; &nbsp;
| &nbsp; &nbsp;
|[[Image:RedDot.svg|16px]]<br>[[Image:RedDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:RedDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:BlackDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:RedDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:RedDot.svg|16px]]<br>[[Image:RedDot.svg|16px]]
|- align="center" valign="top"
| <math>C_{4,1} = 1</math>
| &nbsp; &nbsp;
| &nbsp; &nbsp;
| <math>C_{4,2} = 1 + 4 \times 1</math>
| &nbsp; &nbsp;
| &nbsp; &nbsp;
| <math>C_{4,3} = 1 + 4 \times 3</math>
| &nbsp; &nbsp;
| &nbsp; &nbsp;
| <math>C_{4,4} = 1 + 4 \times 6</math>
|}

The difference between two consecutive [[octahedral number]]s is a centered square number (Conway and Guy, p.50).

Another way the centered square numbers can be expressed is:

:<math>C_{4,n} = 1 + 4 \dim (SO(n)),</math>

where 

:<math>\dim (SO(n)) = \frac{n(n-1)}{2}.</math>

Yet another way the centered square numbers can be expressed is in terms of the [[centered triangular number]]s:

:<math>C_{4,n} = \frac{4C_{3,n}-1}{3},</math>

where 

:<math>C_{3,n} = 1 + 3\frac{n(n-1)}{2}.</math>