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

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.

See also: regular square number.