A nonagonal hexagonal number is simultaneously a nonagonal number 
and hexagonal number 
and therefore satisfies the Diophantine
equation
 |
(1)
|
Here, 
and 
are positive integers. Completing
the square and rearranging gives
 |
(2)
|
Defining 
and 
gives the Pell-like equation
 |
(3)
|
This has positive integer solutions 
, (9, 3), and (19, 7), giving the family of solutions
(5, 1), (9, 3), (19, 7), (61, 23), (135, 51), (299, 113), .... These give solutions
which are integers in 
and 
of 
, (10, 13), (39025, 51625), ... (OEIS A048916
and A048917), giving the nonagonal hexagonal
numbers 1, 325, 5330229625, 1353857339341, 22184715227362706161, ... (OEIS A048918).
