Editing Centered square number (section)

== 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 a consequence of (2''n''&nbsp;&minus;&nbsp;1)<sup>2</sup>&nbsp;+&nbsp;(2''n''<sup>2</sup>&nbsp;&minus;&nbsp;2''n'')<sup>2</sup>&nbsp;=&nbsp;(2''n''<sup>2</sup>&nbsp;&minus;&nbsp;2''n''&nbsp;+&nbsp;1)<sup>2</sup>.