Centered square number

A square number centered is a Nombre illustrated centered which can be represented by a Carré having a point in its center and other points placed around the center with some distance (partner to standard 1.) as in the following example:

*** *** *** * ***** ***** *** ******* * ***** *** * -->

For entire strictly positive, the centered square number of row N is equal to:

$n^2 + (N - 1) ^2 \,$ .

In other words, a centered square number is the sum of two square numbers consecutive. The following figure illustrates this fact well:

X X X X X0X X0X X0X X X0X0X X0X0X X0X X0X0X0X X X0X0X X0X X -->

$1 + 4 \ cdot \ frac {N \ cdot (n+1)}{2}$ .

The centered square numbers can be also written in the following form (where N is an odd entirety):

$\,$ .

The first four values of N (1,3,5,7) are illustrated below. The figure is formed by considering a square of N not by points, and by selecting half of the points, starting from the left higher corner, until the central point included.

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

The few first 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 the centered square numbers are odd, and in bases 10 we can notice that their figures of the units follow the model 1-5-3-5-1.

All the centered square numbers and their dividers have remainder equal to one when they are divided by four. Thus all the centered square numbers and their dividers end in figure 1 or 5 bases of it 6, 8 and 12.

Also see

square Number

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