A number which is simultaneously a nonagonal number 
and a triangular number 
and therefore satisfies the Diophantine
equation.
 |
(1)
|
Completing the square and rearranging gives
 |
(2)
|
Defining 
and 
gives the Pell-like equation
 |
(3)
|
This has unit solutions 
, (9, 3), and (19, 7), which lead to the family of
solutions (5, 1), (9, 3), (19, 7), (61, 23), (135, 51), (299, 113), (971, 367), ....
The corresponding integer solutions in 
and 
are 
, (10, 25), (154, 406), (2449, 6478), ... (OEIS A048907 and A048908),
giving the nonagonal triangular numbers 1, 325, 82621, 20985481, 5330229625, 1353857339341,
... (OEIS A048909).
