Editing Centered square number (section)

{{Short description|Centered figurate number that gives the number of dots in a square with a dot in the center}}
{{Use American English|date=March 2021}}
{{Use mdy dates|date=March 2021}}
{{no footnotes|date=January 2016}}

In [[elementary number theory]], a '''centered square number''' is a [[Centered polygonal number|centered]] [[figurate number]] that gives the number of dots in a [[Square (geometry)|square]] with a dot in the center and all other dots surrounding the center dot in successive square layers. That is, each centered square number equals the number of dots within a given [[Taxicab geometry|city block distance]] of the center dot on a regular [[square lattice]]. While centered square numbers, like [[figurate number]]s in general, have few if any direct practical applications, they are sometimes studied in [[recreational mathematics]] for their elegant geometric and arithmetic properties.

The figures for the first four centered square numbers are shown below:

:{|
|- align="center" valign="middle" style="line-height: 0;"
|[[Image:GrayDot.svg|16px]]
| &nbsp; &nbsp;
| &nbsp; &nbsp;
|[[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]]
| &nbsp; &nbsp;
| &nbsp; &nbsp;
|[[Image:GrayDot.svg|16px]]<br>[[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]]<br>[[Image:GrayDot.svg|16px]]
| &nbsp; &nbsp;
| &nbsp; &nbsp;
|[[Image:GrayDot.svg|16px]]<br>[[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: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]]<br>[[Image:GrayDot.svg|16px]]
|- align="center" valign="top"
| <math>C_{4,1} = 1</math>
| &nbsp; &nbsp;
| &nbsp; &nbsp;
| <math>C_{4,2} = 5</math>
| &nbsp; &nbsp;
| &nbsp; &nbsp;
| <math>C_{4,3} = 13</math>
| &nbsp; &nbsp;
| &nbsp; &nbsp;
| <math>C_{4,4} = 25</math>
|}

Each centered square number is the sum of successive squares. Example: as shown in the following figure of [[Floyd's triangle]], 25 is a centered square number, and is the sum of the square 16 (yellow rhombus formed by shearing a square) and of the next smaller square, 9 (sum of two blue triangles):

[[File:centered_square_numbers_vs_triangular_numbers.svg|thumb|Centered square numbers (in red) are in the center of odd rows of Floyd's triangle.]]