A centered polygonal number is a figurate number in which layers of polygons are drawn centered about a point instead of with the point at a polygon vertex.
For a polygon with 
vertices, the centered
polygonal number after 
layers is
 |
where 
is a nonnegative integer. The central point
contributes 1, and layer 
contributes 
points.
Putting 
, the condition that 
be a square number 
is equivalent to
 |
where 
, 
,
and 
is a positive integer. For nonsquare 
, this is a generalized Pell
equation. When 
,
however, 
for every 
.
