# All Science Fair Projects

## Science Fair Project Encyclopedia for Schools!

 Search    Browse    Forum  Coach    Links    Editor    Help    Tell-a-Friend    Encyclopedia    Dictionary

# Science Fair Project Encyclopedia

For information on any area of science that interests you,
enter a keyword (eg. scientific method, molecule, cloud, carbohydrate etc.).
Or else, you can start by choosing any of the categories below.

# Centered square number

A centered square number is a centered figurate number that represents a square with a dot in the center and all other dots surrounding the center up to a certain city block distance. Like this:

*   *     *       *
***   ***     ***
*   *****   *****
***   *******
*     *****
***
*


The centered square number for n is given by the formula

n2 + (n - 1)2.

In other words, a centered square number is the sum of two consecutive square numbers. The following pattern demonstrates this formula:

X     X       X         X
X0X     X0X       X0X
X     X0X0X     X0X0X
X0X     X0X0X0X
X       X0X0X
X0X
X


The formula can also be expressed as

${{(n^2 + 1)} \over 2}$

but n has to be odd in this case.

*   ***  *****  *******
**   *****  *******
***    *******
****


The first few centered square numbers are

1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, 313, 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

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

All centered square numbers and their divisors have a remainder of one when divided by four. Hence all centered square numbers and their divisors end with digits 1 or 5 in base 6, 8 or 12.