View source for Centered square number

{{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.]]

== 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>

== List of centered square numbers ==

The first centered square numbers (''C''<sub>4,''n''</sub> < 4500) are:

:[[1 (number)|1]], [[5 (number)|5]], [[13 (number)|13]], [[25 (number)|25]], [[41 (number)|41]], [[61 (number)|61]], [[85 (number)|85]], [[113 (number)|113]], [[145 (number)|145]], [[181 (number)|181]], [[221 (number)|221]], 265, [[313 (number)|313]], [[365 (number)|365]], 421, 481, 545, 613, 685, 761, 841, 925, 1013, 1105, 1201, 1301, 1405, 1513, 1625, 1741, 1861, 1985, 2113, 2245, 2381, 2521, 2665, 2813, 2965, 3121, 3281, 3445, 3613, 3785, 3961, 4141, 4325, … {{OEIS|id=A001844}}.

== Properties ==

All centered square numbers are odd, and in base 10 one can notice the one's digit follows the pattern 1-5-3-5-1.

All centered square numbers and their divisors have a remainder of 1 when divided by 4. Hence all centered square numbers and their divisors end with digit 1 or 5 in base [[Senary|6]], [[Octal|8]], and [[Duodecimal|12]].

Every centered square number except 1 is the [[hypotenuse]] of a [[Pythagorean triple]] (3-4-'''5''', 5-12-'''13''', 7-24-'''25''', ...). This is exactly the sequence of Pythagorean triples where the two longest sides differ by 1. (Example: 5<sup>2</sup> + 12<sup>2</sup> = '''13'''<sup>2</sup>.)

This is not to be confused with the relationship (''n'' – 1)<sup>2</sup> + ''n''<sup>2</sup> = ''C''<sub>4,''n''</sub>. (Example: 2<sup>2</sup> + 3<sup>2</sup> = '''13'''.)

== Generating function ==

The generating function that gives the centered square numbers is:
:<math>\frac{(x+1)^2}{(1-x)^3}= 1+5x+13x^2+25x^3+41x^4+~...~. </math>

== References ==
*{{citation
 | last = Alfred | first = U.
 | mr = 1571197
 | issue = 3
 | journal = Mathematics Magazine
 | pages = 155–164
 | title = {{math|''n''}} and {{math|''n'' + 1}} consecutive integers with equal sums of squares
 | jstor = 2688938
 | volume = 35
 | year = 1962| doi = 10.1080/0025570X.1962.11975326
 }}.
*{{Apostol IANT}}.
*{{citation
 | last = Beiler | first = A. H.
 | location = New York
 | page = 125
 | publisher = Dover
 | title = Recreations in the Theory of Numbers
 | year = 1964}}.
*{{citation
 | last1 = Conway
 | first1 = John H.
 | author1-link = John Horton Conway
 | last2 = Guy
 | first2 = Richard K.
 | author2-link = Richard K. Guy
 | mr = 1411676
 | isbn = 0-387-97993-X
 | location = New York
 | pages = [https://archive.org/details/bookofnumbers0000conw/page/41 41–42]
 | publisher = Copernicus
 | title = The Book of Numbers
 | year = 1996
 | url = https://archive.org/details/bookofnumbers0000conw/page/41
 }}.

{{Figurate numbers}}
{{Classes of natural numbers}}

[[Category:Figurate numbers]]
[[Category:Quadrilaterals]]