A number which is simultaneously a nonagonal number 
and octagonal number 
and therefore satisfies the Diophantine
equation
 |
(1)
|
Completing the square and rearranging gives
 |
(2)
|
Defining 
and 
gives the Pell-like equation
 |
(3)
|
The first integral solutions in 
and 
are 
, (425, 459), (286209, 309141), (192904201, 208360351),
... (OEIS A048922 and A048923),
giving the nonagonal octagonal numbers 1, 631125, 286703855361, 130242107189808901,
... (OEIS A048924).
