Centered square number

That is, each centered square number equals the number of dots within a given city block distance of the center dot on a regular square lattice.

While centered square numbers, like figurate numbers in general, have few if any direct practical applications, they are sometimes studied in recreational mathematics for their elegant geometric and arithmetic properties.

This can be easily seen by removing the center dot and dividing the rest of the figure into four triangles, as below: The difference between two consecutive octahedral numbers is a centered square number (Conway and Guy, p.50).

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 6, 8, and 12.